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i 




iy>c>^\d. ji,\ 



/ 






I 



' 






A TREATISE 



\ ON THE 



ANALYTICAL DYNAMICS 

OF PARTICLES AND RIGID BODIES 



/ 



CAMBRIDGE : 
AT THE UNIVERSITY PRESS 

1904 






A TREATISE 



ON THE 



ANALYTICAL DYNAMICS 



OF PARTICLES AND RIGID BODIES 



\ 



\ 



CAMBRIDGE : 
AT THE UNIVERSITY PRESS 

1904 



\ 



\ 

\ 



\ 



\ 



\ 



16 Kinematical Preliminaries [ch. i 

From these expressions we can at once deduce the values of ©i, ©j, ©„ 
in terms of the symmetrical parameters f , 17, ?, %, of § 9 ; for we have 



Similarly we have 






and we have cos ^ = - f* iy- ^« + ^ + X". 

Substituting these values in the Hquation w, = i^ + <^ cos #, we have 

The values of Wi and «*, can Bog^g ^^^ ^jjjjg obtained from this by the 
principle of symmetry ; and thus we -tcuiave the components of angular velocity 
given by the equations -ota 

17. Time-fiux of a vector wlioa^ components relative to moving axes are 
given. jT 

Suppose now that a vector quantity is speci6ed by its components f, r,, ? 
at any instant t with reference to the Instantaneous position of a nght-handed 
system of axes 0<n/^ which are themilves in motion : and let xt be required 
to find the vector which represents th| rate of change of the given vector. 

Let 0,,. 0,,, 0,., denote the comXents of the angular velocity of the 
system Oxyz, resolved along the instaAtaneous position of the axes Ox, Oy, Oz 
themselves. 

' The time-flux of the given vector is the (vector) sum of the time-fluxes 
of the components |, 97, ?, taken separktely. But if we consider the vector f , 
it is increased in length to f + ^dt inl the infinitesimal interval of time dt, 
and at the same time is turned by the motion of the axes, so that (owing to 
the angular velocity round Oy) it is displaced through an angle ©2 dt from its 
position in the original plane zOx, in the direction away from Oz, and also 
(owing to the angular velocity round Oz) it is displaced through an angle 
0), dt from its position in the original plane xOy, towards Oy. The coordinates 
of its extremity at the end of the interval of time dt, referred to the positions 



• "^^r- 



16. 17] 



KinemcUiccU PrdimitMries 



17 



of the axes at the commencement of the interval dt, are therefore (neglecting 
infinitesimals of order higher than the first) 

and .80 the components of the vector which represents the time-flux of f are 

Similarly the components of the vectors which represent the time-fluxes 
of the vectors tj and ^ are respectively 

- 6)317, 17, 6)117, 

and 6)2^, -6)if, t 

Adding these, we have finally the components of the time-fiux of the given 
vector in the form 

17 - f6)i -fi ^6),, 

This result can be immediately lapplied • to find the velocity and 
acceleration of a point whose coordinates (a?, y, z) at time t are given with 
reference to axes moving with an angular velocity whose components along 
the axes themselves at time^ are (ajj, toji^ 6),). 

For substituting in the above formijilae, we see that the components of 
the velocity are 

dP — y6)8 + ^6)3, y — <^«i-r^6)3, i — iC6)j + y6)i. 

Now applying the same formulae to the case in which the vector whose 
time-flux is sought is the velocity, we hjave the components of the accelera- 
tion of the point in the form 

j^ (^ - y«8 + -2:6),) - 6), (y - zi)i + xa^) + 6)2 (i - a?6)a + ywi), 
d 



\ 



dt 



(y — 2:0)1 + wco^) — 6)1 (i — axoi + ya>i) + 6)3 (^ — yo), -f za)^), 



-^ (i — X(Oi + y6)i) — 6)a (i? — y6)s + ^6)2) + 6)1 (y — zco^ + ^76)3). 



In th^ case in which the motion takes place in a plane, which we may 

take as the plane Oxy, there will be only two coordinates (x, y), and only one 

component of angular velocity, namely d, where is the angle made by the 

moving axes with their positions at some fixed epoch; the components of 

velocity are therefore (putting z, 6)1, 6)3, each equal to zero in the above 

expressions) 

x — yd and y + x6y 

and the components of acceleration are 

^ - 2yd -y'e - xd^ and y -f 2i?d+ xd- y6^. 

2 



W. D. 



18 



KinemcUical Preliminaries 



[CH. I 



Example. Prove that in the general case of motion of a Tigid body there is at each 
instant one definite point at a finite distance which regarded as invariably connected with 
the body has no acceleration at the instant, provided the axis of the body's screwing 
motion be not ins^ntaneously stationary in direction. (ColL EzauL) 

18. Special resolutions of the velocity and acceleration. 

The results obtained in the last article enable us to obtain formulae, 
which are frequently of use, relating to the components of the velocity and 
acceleration of a moving point in various special directions. 

(i) Velocity and acceleration in polar coordinates. 

Let the position of a point be defined by its polar coordinates r, 0, ^, 
connected with the coordinates (X, F, Z) of the point referred to fixed 
rectangular axes OXYZ by the equations - 

'X s= r g in 5 cos ^ 
F=r sin ^ sin ^ 
Z = rcQS0\ 

and let it be required to determine the components of velocity and 
acceleration of the point in the direction of the radius vector r, in the 
direction which is perpendicular to r and lies in the plane containing r and 
OZ (this plane is generally called t le meridian plane), and in the direction 
perpendicular to the meridian plane; these three directions are frequently 
described as the directions of r imreasing, 6 increasing, and ^ increasing, 
respectively. Take a line through tne origin 0, parallel to the direction of 
increasing, as a moving axis Ox; aijd take a line through 0, parallel to the 



direction of <f> increasing, as axis Oy, 
increasing as axis Oz. The three 



and a line parallel to the direction of r 
Eulerian angles which determine the 
position of the moving axes Oxyz with reference to the fixed axes OXYZ are 
{0,'4>y 0); so (§ 16) the components of angular velocity of the system Oxyz, 
resolved along the axes Ox, Oy, Oz, themselves, are 

fi>i == ■" ^ sin 0, (0^=6, ci>s = (^ cos 0, 

The coordinates of the moving point, referred to the moving axes, are 
(0, 0, r); and so by § 17 the components of velocity of the point resolved 
parallel to the moving axes are i 

r6, r<j> sin 0^ f , 

and the components of acceleration in the directions of increasing, 
(j) increasing, and r increasing, (again using the formulae of § 17) are 

-J- (r6) — r<f)^ sin 5 cos ^ + fd, or r6 + 2rd — rtp^ sin cos 0, 



id 



-r. (r^ sin 0) + r</) sin + r6^ cos ft or — , ~^ j^ (^ sin' 0<i>), 
az r sin cr az 



dt 
\and 



r'-r0*-'r<j>^sm^0. 



17, 18] Kinematical Preliminaries 19 

If the motion of the point in a plane, we can take the initial line in this 
plane as axis Oz, and the quantities denoted by r and in these formulae 
become ordinary polar coordinates in the plane; since ij> is now zero, the 
components of velocity and acceleration in the directions of r increasing and 
increasing are 

(r, rdl 

and (r - r^, r0 + 2rd), 

(ii) Velocity and acceleration in cylindrical coordinates. 

Consider now a point whose position is defined by its cylindrical 
coordinates z, p, <l>, connected with the coordinates (X, Y, Z) of the point 
referred to fixed rectangular axes OXYJZ by the equations 

Xapcos^, F=psin^, Z^2\ 

and let it be required to find the componjents of the velocity and acceleration 
of the point in the direction parallel to the axis of e, in the direction of the 
line drawn from the axis of z to the poin,t, perpendicular to the axis of z^ and 
in the direction perpendicular to these t^^o lines. These three directions are 
generally called the direction of z increfLsing, the direction of p increasing, 
and the direction of ^ increasing; and the coordinate <f) is called the ajsimuth 
of the point ' 

In this case we take moving axes Ook Oy, Oz, passing through the origin 
and parallel respectively to the directions of p increasing, ^ increasing, and z 
increasing. The components of angular velocity of the system Oxyz, resolved 
along the axes Oayz themselves, are clearly 

and the coordinates of the moving point, referred to the moving axes, are 
(p, 0, z). It follows by § 17 that the components of velocity of the point in 
these directions are ' 

and the components of acceleration are . 

ip-piP^ p4>-^2p<i>^ 1?). 

(iii) Velocity and acceleration in arc-coordinates. 

Another application of the formulae of § 17 is to the determination of the 
components of velocity and acceleration of a point which is moving in any 
way in space, resolved along the tangent, principal normal, and binormal, to 
its path. 

Consider first the case of a particle moving in a plane : and take lines 
through a fixed point 0, parallel respectively to the timgent and inward 
normal to the path, as moving axes Ox and Oy. These axes are rotating 

2—2 



r 



20 Kinematieal Prdtminaries [ch. i 

round with angular velocity <}>, where is the angle made by the tangent 
to the path with some fixed line in the plane. If v denotes the velocity of 
the point, a the arc of the path described at time t, and p the radius of 
curvature of the path at the point, we have 

ds da 

"'df I'-di' 

and theangular velocity of the ases can therefore be written in the form v/p. 

Since the components of the velocity parallel to the moving axes are 
(tJ, 0), it follows from § 17 that the comjKmeDts of the acceleration parallel to 



the same axes are ft), v- -) ■ Since f 



. __ d« dadv _ dv 
dt fit da ~ da' 

. it follows that the acceleration of t|ie moving point in the direction of the 

tangent to its path is v -j- , and the acceleration in the direction of the inward 

normal is — . I 

' . . I. . . ■ 

Now the velocity of a moving boint is determined by the knowledge of 
two consecutive positions of the moviug point, and the acceleration is therefore 
determined by the knowledge of three consecutive positions ; so even if the 
path of the point is not plane, it can for the purpose of determining its 
acceleration at any instant be reganded as moving in the osculating plane of 
its path, since this plane contains tbree consecutive positions of the point. 
Hence the componenta of acceleroHon of the point, in the dii'ectiona of the 
tangent, principal normal, and binormal to its path, are 



Vda' ^p- "y- 

(iv) Acceleration alcmg the radirta and tangent. 

The acceleratiou of a point which is in motion in a plane may be expressed 
in the following form* ; let r be the radius vector to the point from a fixed 
origin in the plane, p the perpendicular &om the origin on the tangent to the 
path, a the arc of the path described at time t, p the radius of curvature of 
the path at the point, and v or i the velocity of the point at time t ; and let 
' ' ote the product pv. Then the acceleration of the point can be reaolved 

omponenta — along the radius vector to the origin and -^j- along the 
•a to the path. 

* Dae to SiMwi, AtU dcUa R. Aee. di Torino, nv. p. 760. 



18] KinenuUieal Preliminaries 21 

For the acceleration can be resolved into componeDts vdv/ds along the 

tangent and t^/p along the normal ; now a vector F directed outwards along 

the radius vector can be resolved into vectors — Fp/r along the inward normal 

and F dr/ds along the tangent, so a vector ^/p along the inward normal can be 

7*v^ vt^ dv 

resolved into — inwards along the radius vector and — -r- along the tangent 

The acceleration is therefore equivalent to components 

dv ir^ dv 
i; T- H -T- along the tangent, 

and — inwards alonp: the radius vector. 

AV 
The latter component is -r , and the former can be written 

fp 

2 d« "*";> (fo ^^ 2p^ da ' ^^ p^ds' 
which establishes Siacci's result. 

Example 1. Determine the meridian^ normaf, and transverse components of the acoderor 
tion of a point moving on the surface of the dnchor-ring 

a;e(c + asin^)co6^, y«(c4o8ind)8in^, z^acoaB, 



centre of the anchor-ring and C the centre 

The polar coordinates of C relative to 

tive to (7 are (a, ^, ^} ; 80 the components 



Let P be the point ($^ <f>\ and let be the 
of the meridian cross-section on which P lies. 
are (c, ^}, and the polar coordinates of P re 
of acceleration of C relative to are 

cj>, transverse I 

and - ej>* outwards from the axis, i.e. - c<^' sin $ along the normal, 

anjd - c<^* cos 6 along the meridian. 

The components of acceleration of P relat(ive to C are 

'a$ — o^' sin $ cos along the meridian, 

. >» "t: (sin* 6 . 4>) transverse, 
sm $dt^ ^' 

- o^ - aij>^ sin' 3 normal. 

Thus finally the components of acceleration of P in space are 

oB-ic-^-a sin $) <^* cos $ along the meridian, 

• • • 

- o^ - o^* sin* $ - c0* sin $ normal, 
and oi + . -^-j- (sin* 6 . i) transverse. 



22 KinematiccU Preliminaries [cB. i 

Example 2. If the tangential and normal components of the acceleration of a point 
moving in a plane are conetant^ shew that the point describes a logarithmic spiral. 

In this case 

dv 
V ^ ssO) where a is a constant, 

so v^==as. 

Also — ssc, where c is a constant, 

P 

so «s Cpy where (7 is a constant, (t 

or «B C^, where ^ is the angle made by the tangent with a fixed line. 

Integrating this equation, we have 

where A and B are constants : and this is the intrinsic equation of the logarithmic spiral 

Example 3. To find the acoderation of a point which describes a logarithmic spiral with 
constant angvlar velocity about the pole. 

I AV 

By Siacci's theorem, the components bf acceleration are -=- along the radius vector 

and --« -r along the tangent; but if » is /the constant angular velocity, we have h^af^: 

so the components of acceleration are 

Q)V^ L 2a)V» dr 
p^p \ p!^ ds' 

Since - , - , and -j- are constant in the Ispiral, we see that each of these components of 
acceleration varies directly as the radius vector. 



MlSCELLANEpUS EXAMPLES. 

1. If the instantaYieous axis of rotation of a body moveable about a fixed point is fixed 
in the body, shew that it is also fixed in sptfoe, i.e. the motion is a rotation round a fixed 
axis. 

2. A point is referred to rectangular sizes Ox^ Oy<, revolving about the origin with 
angular velocity oo ; if there be an acceleration to ^so, yaO, of amount n'a>' x (distance), 
shew that the path relative to the axes can be constructed by taking (i) a point 
j7ssn\i/(n'~l), (ii) a uniform circular motion with angular velocity (n-I)» about this, 
and (iii) a uniform circular motion with angular velocity (^+l)a>, but in the opposite 
sense, about this last. (ColL Exam.) 

3. The velocity of a point moving in a plane is the resultant of a velocity v along the 
radius vector to a fixed point and a velocity v^ parallel to a fixed line. Prove that the 
corresponding accelerations are 

dv ^ wf . ^ dxl vxf 

-5-H cos^, and -5- H — , 

at r , dt r ^ 

6 being the angle that the radius vector makes with the fixed direction. (ColL Exam.) 



CH. i] Kinematical Preliminaries 23 

4 A point moves in a plane, and is referred to Cartesian axes making angles a, ^ with 
a fixed line in the plane, where a, ^ are given functions of the time. Shew that the com- 
ponent velocities of the point are 

i— ^cotO-a)-yj3cpsec(/3— a), y+y/3cotO-a)+ardcosecO— o), 

and obtain expressions for the component accelerations. (ColL Exam.) 

5. A point is moving in a plane : B is the logarithm of the ratio of its distances from 
two fixed points in the plane, and <f> is the angle between them : also 2k is the distance 
between the fixed points. Shew that the velocity of the point is 

^uf'^'^\ • (ColL Exam.) 

cosh - cos 9 ^ 

6. If in two different descriptions of a curve by a moving point, the product of the 
velocities at corresponding places in the two descriptions is constant, shew that the 
accelerations at corresponding places in the t^o descriptions are as the squares of the 
velocities, and that their directions make equal angles with the normal to the curve, in 
opposite senses. . (J. von Vieth.) 

7. A point is moving in a parabola of latus .^tum 4a, and when its distance from the 
focus is r, the velocity is v ; shew that its acceleration is compounded of accelerations R 
and y, along the radius vector and normal respectively, where 

R^v^, y=f-^±(vh), (Coll. Exam.) 

I 

8. Shew that if the axes of x and y rotate ;^ith angular velocities a»i, ci»2 respectively, 

and yft is the angle between them, the component accelerations of the point (jp, y) parallel 
to the axes are < 

X - jFo>i* — (^»i + 2i», ) cot ^i - (y«2 + Sywj) cosec ^, 

and y-yo)2'+(j?«i+2i»i)cosetj^+(yw2+2y«8)cot^. (Coll. Exam.) 

9. The velocity of a point is made up of components t«, i; in directions making angles 
By ff} with a fixed line. Prove that the componeints/, /' in these directions of the accelera- 
tion of the point will be given by 

/«ti - uB cot X"^^ cosec Xj 

/'= t? + ^ cosec V + v<^ cot ;^, 

X being the inclination of the two directions. ; 

r 

Being given that the lines joining a moving point to two fixed points are r, $ in length 
and By <t> in inclination to the line joining the two fixed points, determine the acceleration 
of the point in terms of a>, ^ , the rates of increase of By 0. (Coll. Exam.) 

10. If Ay By Che three fixed points, and the component velocities of a moving point P 

along the directions PAy PBy PC be «, v, and w ; shew that the accelerations in the same 

directions are 

/I co^APB\^ (\ COB APC\ 

''+"H?5 — PA-r''''[pc'—PA-)^ 

and two similar expressions. (ColL Exam.) 

11. The movement of a plane lamina is given by the angular velocity a> and the com- 
ponent velocities u, v of the origin resolved along axes Ox, Oy traced on the lamina. 



I 

24 Kinematical Prdiminaries [gh. i 

Find the component velocities of any point (4;, y) of the lamina. Shew that the equations 

at \v+xmj ' 

represent circular loci on the lamina ; one being the locus of those points which are pass- 
ing cusps on their curve loci in space and the other being the locus of the centres of curva- 
ture of the envelopes in space of all straight lines of the lamina. (ColL Exam.) 

12. Shew that when a point describes a space-curve, its acceleration can be resolved 
into two components, of which one acts along the radius vector from the projection of a 
fixed point on the osculating plane, and the other along the tangent ; and that these are 
respectively 



and l^^Tli^ 



where p is the radius of curvature, q the ^istance of the fixed point from its projection on 
the osculating plane, r and p are the distances of this projection from the moving point 
and the tangent, 7* is an arbitrary function (equal to the product oip and the velocity) and 
» is the arc. (Siacci.) 

13. A circle, a straight line, and a point lie in one plane, and the position of the point 
is determined by the lengths t of its tangent to the circle and p of its perpendicular to 
the line. Prove that, if the velocity of t]ie point is made up of components u, 1;, in the 
directions of these lengths and if their mutual inclination be B, the component accelera- 
tions will be 

ii - uv cos $/t, V -f uv/t, (Coll. Exam. ) 

14. A particle moves in a circular arck If r, / are the distances of the particle at P 
from the extremities ^1, ^ of a fixed choro, shew that the accelerations along AP, BP, are 
respectively 

5^ + ^ (r-»^cosa),i and ■^ + ^('^-»'C08«)f 

where r, if are the velocities in the directions of r, /, and a is the angle APB, 

A point describes a semicircle under 1 accelerations directed to the extremities of a 
diameter, which are at any point inversely is the radii vectores r, / to the extremities of 
the diameter. Shew that the accelerations are 

where a is the radius of the circle and V the velocity of the point parallel to the diameter. 

(Coll. Exam.) 

15. The motion of a rigid body in two dimensions is defined by the velocity (ti, v) of 
one of its points C and its angular velocity <a. Determine the coordinates relative to C of 
the point / of zero velocity, and shew that the direction of motion of any other point P is 
perpendicular to PL 

Find the coordinates of the point J of zero acceleration, and express the acceleration of 
P in terms of its coordinates relative to J. (Coll. Exam.) 



OH. i] Kinematical Preliminaries 25 

16. A point on a plane is moving with constant velocity V relative to it, the plane at 
the same time turning round a fixed axis perpendicular to it with angular velocity ». Shew 
that the path of the point is given by the equation 

^^ /:5 — r^^ -i« 
— = V^ — « +-cos^- ; 

r and 6 being referred to fixed axes, and a being the shortest distance of the point from the 
axis of rotation. (Coll. Exam.) 

17. The acceleration of a moving point Q is represented at any instant by am, where « 
is a fixed point and a describes uniformly a circle whose centre is a>. Prove that the 
velocity of Q at any instant is represented by Op^ where is a fixed point andj9 describes 
a circle uniformly ; and determine the path described by Q. 

(Camb. Math. Tripos, Part I, 1902.) 

18. A point moves along the curve of intersection of the ellipsoid -j + ra + ;5='-^> ^^'^ 

the hyperboloid of one sheet -5 — r + j^—^ "^ JT^ ~ ^» *°^ ^^ velocity at the point where 

the curve meets the hyperboloid of two sheets -S — + tt- — + -5 — — 1 is 

""^ aV-u b*-u c*-u 



t(a«-^)(6»-M)|:c«-/.)J ' 



where h is constant. Prove that the resolved part of the acceleration of the point along 
the normal to the ellipsoid is | 

h^abcjn-}.) 

19. A rigid body is rolling without sliding oi|i a plane, and at any instant its angular 
velocity has components oii, a>2, along the tangeijts to the lines of curvature at the point 
of contact, and cn^ along the normal : shew that tjhe point of the body which is at the point 
of contact has component accelerations / 



- R^l»i , - /^fli>2»3 > R^(0^^ + i22»i'. 



where R^^ R^^ are the principal radii of curvatuile of the surface of the body at the point 



of coiitact 






(Coll. Exam.) 



CHAPTER II. 

THE EQUATIONS OF MOTION. 

f 

19. The ideas of rest and moti&n. 

In the previous chapter we haye frequently used the terms " fixed " and 
" moving " as applied to systems, i So long as we are occupied with purely 
kinematical considerations, it is unnecessary to enter into the ultimate 
significance of these words; all that is meant is, that we consider the 
displacement of the " moving " systiems, so far as it affects their configuration 
with respect to the systems which kre called " fixed," leaving on one side the 
question of what is meant by absolute " fixity." 

When however we come to comider the motion of bodies as due to specific 
causes, this question can no longer be disregarded. 

In popular language the word ** fixed" is generally used of terrestrial 
objects to denote invariable position relative to the surface of the earth 
at the place considered. But ^he earth is rotating on its axis, and 
at the same time revolving rou^d the Sun, while the Sun in turn, * 
accompanied by all the planets, is moving with a large velocity along some 
not very accurately known directioik in space. It seems hopeless the^fore 
to attempt to find anything which can be really considered to be " at rfest." 

Accordingly in dynamics, although when we speak of the motion of bodies 
we always imply that there is some 86t of axes, or Jrame of reference as it may 
be called, with reference to which the motion is regarded as taking place, and 
to which we apply the conventional word "fixed," yet it must not be supposed 
that absolute fixity has thereby been discovered. When we are considering 
the motion of terrestrial bodies at some place on the earth's surface, we shall 
take the frame of reference to be fixed with reference to the earth, and it is 
then found that the laws which will presently be given are sufficient to 
explain the phenomena with a sufficient degree of accuracy ; in other words, 
the earth's motion does not exercise a sufficient disturbing influence to make 
it necessary to allow for its effects in the majority of cases of the motion of 
terrestrial bodies. 



19, 20] The Equations of Motion 27 

It is also necessary to consider the meaning to be attached to the word 
"time," which in the previous chapter stood merely for any parameter 
varying continuously with the configuration of the systems considered. We 
shall now assign a definite significance to this parameter by supposing that 
the angle through which the earth has rotated on its axis (measured with 
reference to the fixed stars, whose small motions we can for this purpose 
neglect), in the interval between two events, measures the time elapsed 
between the events in question. This angular measure can be converted 
into the ordinary measure in terms of mean solar hours, minutes, and seconds 
at the rate of 360 degrees to 24 x 3651/366^ hours. 

20. 7%« laws which determine motion. 

Considering now the motion of terrestrial objects, and taking the earth as 
the frame of reference, it is natural to begin by investigating the motion of a 
very small material body, or particle as we | shall call it, when moving in vacuo 
and entirely unconnected with surrounding objects. The paths described by 
such a particle under various circumstances of projection may be observed, 
and the methods of the preceding chapter enable us, from the knowledge 
thus acquired, to calculate the acceleration! of the particle at any point of any 
particular observed path. It is found tha^ for all the paths the acceleration 
is of constant amount, and is always directed vertically downwards. This 
acceleration is known as gravity^ and is geijierally denoted by the letter g ;. its 

amount is, in Great Britain, about 981 centimetres per second per second. 

I 

The knowledge of this experimental fact is theoretically sufficient to 
enable us to calculate the path of any free terrestrial particle in vacuo, when 
the circumstances of its projection are known : the actual calculation will 
not be given here, as it belongs more properly to a later chapter. 

The case of motion which is next in I simplicity is that of two particles 
which are connected together by an extr^ely light inextensible thread, and 
are free to move in vacuo at the .earth's! suiface. So long as the thread is 
slack, each particle moves with the acceleration gravity, just as if the other 
were not present. But when the thread is taut, the two particles influence 
each other's motion. We can now as before observe the path of one of the 
particles, and hence calculate the acceleration by which at any instant its 
motion is being modified. We thereby arrive at the experimental fact, that 
this acceleration can be represented at any instant by the resultant of two 
vectors, of which one represents the acceleration g and the other is directed 
along the instantaneous position of the thread. 

The influence of one particle on the motion of the other consists there- 
fore in superposing on the acceleration due to gravity another acceleration, 
which acts along the line joining the particles and which is compounded 
with gravity according to the vectorial law of composition of accelerations. 



28 The Equations of Motion [gh. n 

Denoting the particles by A and B, we can at any instant calculate, from the 
observed paths, the magnitudes of the accelerations /i and/, thus exerted by 
B on A and by il on £ respectively ; and this calculation immediately yields 
the result that the ratio of fi to f^ does not vary throughout tlie motion. On 
investigating the motions which result from various modes of projection, at 
various temperatures etc., we are led to the conclusion that this ratio is 
an invariable physical constant of the pair of bodies A and B*. 

We are led by a consideration of the motion of more complex systems 
to infer that the experimental laws just stated can be generalised so as 
to form a complete basis for all dynamics, whether terrestrial or cosmic. 
This generalised statement is as follows : If any set of mutually connected 
particles are in motion, the acceleration with which any one particle moves is 
the resultant of the acceleration with which it would move if perfectly free, and 
a/^celerations directed along the lines joining it to the other particles which 
constrain its motion. Moreover, to the several particles A, B, C, ,.., numbers 
''^At ^i}> ^c> ••• c^^ ^ assigned, Si^ch that the acceleration along AB due to the 
influence of B on A is to the acceleration along BA due to the influence of 
A on B in the ratio m^ : m,^. 2J%e ratios of these numbers mj^, m^, .., are 
invariable physical constants of thp particles. 

The evidence for the truth of this statement is to be found in the universal 
agreement of the calculations ba^ed on it, such as those given later in this 
book, with the results of observation. 

It will be noticed that only the ratios of the numbers m^, m^, m^, ... are 
determined by the law ; it is convenient to take some definite particle A as 
a standard, calling it the unit of n^ass, sCnd then to call the numbers m^/m^ , 
mc/m^, ... the masses of the other particles m^, ma .... 

The mass of the compound particle formed by uniting two or more particles 
is found to be equal to the sumj of the masses of the separate particles. 
Owing to this additive property o^ mass, we can speak of the mass of a finite 
body of any size or shape; and it will be convenient to take as our unit of 
mass the mass of the TT^th part of a certain piece of platinum known as the 
standard kilogramme; this unit will be called a gramme, and the number 
representing the ratio of the mass of any other body to this unit mass is 
called the mass of the body in grammes. 

21. Force. 

We have seen that in every case of the interaction of two particles A and 
B, the mutual influence consists of an acceleration y^ on A and an acceleration 
fs on B, these accelerations being vectors directed along AB and BA respec- 
tively, and being inversely proportional to the masses m^ and m^. It follows 

* The ratio is in fact equal to the ratio of the weight of B to the weight of A ; the ratio of 
the weights of two terrestrial bodies, as observed at the same place on the earth's surface, is a 
perfectly definite quantity, and does not vary with the place of observation. 



20-22] The Equatiom of Motion 29 

that the vector quantity tt^jifA is equal to the vector quantity m^f^^ but has 
the reverse direction. The vector tw^/^ is called the force exerted by the 
particle B on the particle A ; and similarly the vector m^/j, is called the force 
exerted by the particle A on the particle B, 

With this terminology, the law of the mutual action of a connected 
system of particles can be stated in the form : the forces exerted on each other 
by every pair of connected particles are equal and opposite. This is often 
called the Law of Action and Rea^ction, 

If the various forces which act on a particle ^1 as a result of its connexion 
with other particles are compounded according to the vectorial law, the 
resultant force gives the total influence exerted by them on the particle A ; 
this force divided by m^ is the acceleration induced in A by the other 
particles ; and the resultant of this acceleration and the acceleration which the 
particle A would have if entirely free (due to such causes as gravitation) 
is the actual acceleration with which the^ particle A moves. 

In general, if an acceleration represented by a vector / is induced in 
a particle of mass m by any agency, the vector mf is called the ybrce due to 
this cause acting on the particle ; and the resultant of all the forces due to 
various agencies is called the total force Voting on the particle. It follows 
that if {X, T, Z) are the components parallel to fixed rectangular axes of the 
total force acting on the particle at any ipstant, and {x, y, z) are the com- 
ponents of the acceleration with which its path is being des(7ribed at that 
instant, then we have the equations ' 

mx = X, my —Ymz^Z. 

Two other terms which are frequently used may conveniently be defined 
at this point. 

The product of the number which repi*esents the magnitude of the com- 
ponent of a given force perpendicular tp a given line L and the number 
which represents the perpendicular distanjce of the line of action of the force 
from the line L is called the moment of tne force about the line L. 

If the three components (X, F, Z) of the force acting on a single free 
particle are given functions of the coordinates {x, y, z) of the particle, the^ 
are said to define a field of force. 

22. Work, 

Consider now any system of particles, whose motion is either quite free or 
restricted by given connexions between the particles, or constraints due to 
other particles which are not regarded as forming part of the system. Let m 
be the mass of any one of the particles, whose coordinates referred to fixed 
rectangular axes in any selected configuration of the system are {x, y, z); and 
let (X, F, Z) be the components, parallel to the axes, of the total force 
acting on the particle in this configuration. 



-_ J^ 



J • 






30 The Eqitations of Motion [oh. n 



3v Let {x + Sa?, y + Sy, z + hz) be the coordinates of any point very near to 

the point (a;, y, z\ such that the displacement of the particle m from one 
point to the other does not violate any of the constraints (for instance, if m is 
constrained to move on a given surface, the two points must both be situated 
on the surface). Then the quantity 

is called the work done on the particle m by the forces acting on it in the 
infinitesimal displacement from the position {Xy y, z) to the position 

(a? + &r, y + Sy, ^ + hz\ 

This expression can evidently be interpreted physically as being the 
product of the distance through v/hich the particle is displaced and the com- 
ponent of the force (X, F, Z) along the direction of this displacement. 

Since forces obey the vectorial law of composition, the sum of the com- 
ponents in a given direction of Ojny number of forces acting together on a 
particle is equal to the componerjt in this direction of their resultant: and 
hence the work done by a force tJfiich acts on a particle in a given displace- 
ment is equal to the swm of the quojfttities of work done in the same displacement 
by any set of forces into which this force can be resolved. 

Suppose now that in the course of a motion of the system, the particle m 
is gradually displaced from any pohition (which we can call its initial position) 
to some other position at a finite distance from the first (which we can call 
the ^nai position). The work done on the particle by the forces which act on 
it during this finite displacement is| defined to be the sum of the quantities of 
work done in the successive infiritesimal displacements by which we can 
regard the finite displacement as 'achieved. The work done in a finite dis- 
placement is therefore represented by the integral 



(• 



ds ^ ds dsj 

where the integration is taken bet|ween the initial and final positions along 
the arc s described in space by the particle during the displacement. 

• These definitions can now be extended to the whole set of particles which 
form the system considered ; the system being initially in any given con- 
figuration, we consider any mode of displacing the various particles of the 
system which is not inconsistent with the connexions and constraints; the 
sum of the quantities of work performed on all the particles of the system in 
the displacement is called the total work done on the system in the displace- 
ment by the forces which act on it. 

23. Forces which do no work. 

There are certain classes of forces which frequently occur in djmamical 
systems, and which are characterised by the feature that during the motion 
they do no work on the system. 






22, 23] T%e Equatiom of Motion 31 

Among these may be meDtioned 

1^ The reactions of fixed smooth surfaces : the term smooth implies 
that the reaction is normal to the surface, and therefore in each infinitesimal 
displacement the point of application of the reaction is displaced in a direction 
perpendicular to the reaction, so that no work is done. 

2**. The reactions of fixed perfectly rough surfaces ; the term perfectly 
rough implies that the motion of any body in contact with the surface is one 
of pure rolling without sliding, and therefore the point of application of the 
reaction is (to the first order of small quantities) not displaced in each 
infinitesimal displacement, so that no work is done. * 

3^ The mutual reaction of two particles which are rigidly connected 
together: for if (a?i, yi, z^ and {x^, y,, z^ arjr the coordinates of the particles, 
and (X, F, Z) are the components of the forqe exerted by the first particle on 
the second, so that (— X, - Yf — Z) are the! components of the force exerted 
by the second particle on the first, the to^al work done by these forces in 
an arbitrary infinitesimal displacement is 

X (Sx^ - Sa?0 + F(Sya - Syl) + Z (Bz^ - Sz,). 

But since the distance between the particlei is invariable, we have 

S {{x, - x,y + (y, - y,y +L - z,y] = 0, 

or (x^ - a?i) (&ra - Sa?i) + (y^ - y,) (Sy^ - Sl) + (z^ - z^) (8z^ - Szi) = 0, 

and since the force acts in the direction of[ the line joining the particles, we 

have 

X :Y : Z^ix^-x^) : (yj-fyO : (z^-z^). 

Combining the last two equations, we have 

Z(&r,-&F0+y(%-Syi) + ^(S^«-S^i) = O, 

and therefore no work is done in the aggrerate by the mutual forces between 
the particles. t 

4^ A rigid body is regarded from t^ie dynamical point of view as an 
aggregate of particles, so connected together that their mutual distances are 
invariable. It follows from S° that the reactions between the particles which 
are called into play in order that this condition may be satisfied (or molecular 
forces as they are called, to distinguish them from external forces such as 
gravity) do, in the aggregate, no work in any displacement of the body. 

5^ The reaction at a fixed pivot about which a body of the system can 
turn, or at a fixed hinge, or at a joint between two bodies of the system, are 
similarly seen to belong to the category of forces which do no work. 

In estimating the total work done by the forces acting on a djmamical 
system in any displacement of the system, we can therefore neglect all forces 
of the above-mentioned types. 



82 The Equatums of Motion [ch. n 

24 The coordinates of a dynamical system. 

Any material system is regarded (Tom the dynamical point of view as 
constituted of a number of particles, subject to interconnexions and con- 
straints of various kinds; a rigid body being regarded as a collection of 
particles, which are kept at invariable distances from each other by means 
of suitable internal reactions. 

When the constitution of such a system (Le. the shape, size, and mass of 
the various parts of which it .is composed, and the constraints which act on 
them) is given, its configuration at any time can be speciBed in terms of a 
certain number of quantities which vary when the configuration is altered, 
and which will be called the coordinates of the system ; thus, the position of a 
single free panicle in space is completely defined by its three rectangular 
coordinates (x, y, z) with referenoe to some fixed set of axes ; the position of 
a single particle which is constrained to move in a fixed narrow tube, which has 
the form of a twisted curve in space, is completely specified by one coordinate, 
namely the distance s measured dlong the arc of the tube to the particle &om 
some fixed point in the tube whioh is taken as origin ; the position of a rigid 
body, one of whose points is fiyfed, is completely determined by three co- 
ordinates, namely the three Eult,rian angles 0, ^, i/c of § 10; the position of 
two particles which are connectea by a taut inestensible string can be defined 
by five coordinates, namely the ihree rectangular coordinates of one of the 
particles and two of the directiou-cosines of the string (since when these five 
quantities are known, the position of the second particle is uniquely deter- 
mined); and so on. j 

Example. State the number of independent coordinates required to specify the 
configuration at aoy instant of a rigid loodj which ia conBtrained to move in contact with 
a given fixed smooth surfece. 

We shall generally denote by, n the number of coordinates required to 
specify the configuration of a system, and shall suppose the systems con- 
sidered to be such that n is finite. ^ The coordinates will generally be denoted 
by gi, qt, ..-^n- If the system contains moving constraints (e.g. if it consists 
of a particle which is constrained to be in contact with a surface which in 
turn is made to rotate with constant angular velocity round a fixed axis), 
it may be necessary to specify the time t in addition to the coordinates 
9i> 9a> ■-- 9n> in order to define completely a configuration of the system. 

The quantities ji, j,, ... jn, are frequently called the velocities correspond- 
ing to the coordinates q„ q^, ... }„. 

A heavy flexible string, free to move in space, is an example of a dynamical system 
which is excluded by the limitation that n is to be finite; for the configuration of the 
string cannot be expressed in terms of a finite number of parameters. ■ 

26. Solonomic and non-liolonomic systems. 

It is now necessary to call attention to a distinction between two kinds 



24-26] The EqucUions of Motion 33 

of dynamical systems, which is of great impoi-tance in the analytical discussion 
of their motion : this distinction may be illustrated by a simple example. 

If we consider the motion of a sphere of given radius, which is constrained 
to move in contact with a given fixed plane, which we can take as the plane 
of xy, the configuration of the sphere at any instant is completely specified 
by five coordinates, namely the two rectangular coordinates (x, y) of the 
centre of the sphere and the three Eulerian angles ff, tf>,-^ of § 10, which 
specify the orientatioQ of the sphere about its centre. The sphere can take 
up any position whatever, so long as it is in contact wiih the plane ; the five 
coordinates (x, y, 0, (f>, ■^) can therefore have any arbitrary "alues. 

If now the plane is smooth, the displacement from any position, defined 
by the coordinates (x, y, 0, tf), ■^), to any adjacent position, defined by the 
coordinates (x+Sx, y + By. O + S0, tfi + S^,. if-+&f), where hx, Sy, 80, Sift. S^ 
are arbitrary independent infinitesimal quafitities, is a possible displacement, 
i.e. the sphere can perform it without violat|iog the constraints of the system. 
But if the plane is perfectly rough 
8^, Byjt, are arbitrary ; for now tJ 
point of contact is zero (to the 
satisfied, and this implies that t 
longer independent, but are muti 
as to satisfy two linear equations 
perfectly rough plane, a displace} 
changes in the coordinates is not ne 

A dynamical system for whicl 
infinitesimal changes in the coordi >fc*2v/ ^"b " 

(as in the case of the sphere on t fU^t^,,, f* J 

said to be noi 
,) are arbitran 
m, these will i 
for non-holon 
itisfied betwe* 
cement. The 
)f the system. 
that the nuni 
ient coordina 

form, of the equations of motion of a holonomic system, 
nsider the (notion of a holonomic system with n degrees 
'n <li,---w ^ ^^^ coordinates which specify the con- 
stem at tlhe time t. 



i-p-^ 



84 



The Equations of Motion 



[CH. n 



Let mi typify the mass of one of the particles of the system, and let 
{j^ii J/it ^i) ^ i^s coordinates, referred to some fixed set of rectangular axes. 
These coordinates of individual particles are (from our knowledge of the 
constitution of the system) known functions of the coordinates 9i, ^s, ... 9n of 
the system, and possibly of t also ; let this dependence be expressed by the 
equations 

{^i—fiifliy 921 •••» ?n, t\ 
yi'=^^i{<lit 52» •••! Jn, 0> 

Let (X,, Yiy Zi) be the components of the total force (external and 
molecular'^ ^vcling on the particle m^ ; then the equations of motion of this 

particle are 

rriiXi = Zf, , rmyi = Yu niiJii = Zi, 

j 

Multiply these equations by 

dqr ' I , dqr ' dqr ' 

respectively, add them, and sup for all the particles of the system. We 
thus have 

where the symbol 2 denotes summation over all the particles of the system ; 
this can be either an integ^lffon (if the particles are united into rigid bodies) 
or a summation over a discrete aggregate of particlea 

But we have 



dxi d f^/i.^dfi,/ dfi. dfi\ dfi 



so 



dq 
dqr 



. d_ (dxi\ 
^'"'dAdqJ 

dt r'dqr) * Wa?r * dq^qr^'^ ' " ^ dqndqr^"" ^ ^^J 
_ d f. d±i\ . d±i 



dqr) ^*dq, 
and therefore we have 



1 



-i^-^i\^^(-^^y^'+^-^^-^4^-^+y^'-^*^)- v 



1 ' 



# 




26] 



The Equations of Motion 



85 



Now the quantity 

represents the sum of the masses of the particles of the system, each 
multiplied by half the square of its velocity ; this is called the Kinetic 
Energy of the system. From our knowledge of the constitution of the 
system, the kinetic energy can be calculated* as a function of 

Jii 32> ••• ?n> ?i> 3s> ••• Jn» '5 

we shall denote it by 

T{q\y ft, ... gn, 9i, ?«. ••• ?n, 0* 

and shall suppose that T is a known Unction of its arguments. Since 

and y{ and i{ are likewise linear functions of ft, ft, ... qm we see that 7 is a 
quadratic function of ft, ft, ... ft ; if the functions /, ^, •^, do not involve the 
time explicitly (as is generally the case if there are no moving constraints 
in the system), the quantities x, y, i, are r^omogeneovs linear functions of 
ft, ft, ... fti £Lnd then 7 is a homogeneous quadratic function of ft, ft, ... q^. 

From the definition it follows that the kinetic energy of a system is essentially 
positive; ^is therefore a positive definite quadratic form in ^j, ^3, ... ^n, and so satisfies 
the > conditions that its discriminant and the principal minors of every order of its 
dif/briminant are positive. 

We have thus derived from the equations/ of motion the equation 

and the expression on the left-hand side of tliis equation does not involve the 
individual particles of the system, except in, so far as they contribute te the 
kinetic energy T, We have now te see if the right-hand side of the equation 
can also be brought te a form in which the individuality of the separate 
particles is lost. 

For this purpose, consider that displacement of the system in which ^the 
poordinate ft is changed to ft + hqr, while the coordinates 



ft, ft 



> • 



qr—it qr+it ••• 5n, 



and the time (so far as this is required for the specification of the system) are 
unaltered. Since the systerajSu-hokaiomiCj^ this can be effected without 
violating the constraints. ^In this displacement, the coordinates of the 
narticle nii are changed te 



yi + ||s3„ 



•'*^^--- 



The 



of performing this ealoiilation for rigid bodies are given in Chapter Y. 

3—2 






86 The Equations of Motion [ch. n 

and therefore the total work done in the displacement by all the forces which 
act on the particles of the system is 

Now of the forces which act on the system, there are several kinds which 
do no work. Among these are, as was seen in § 23, 

1°. The molecular forces which act between the particles of the rigid 
bodies contained in the system : 

2". The pressures of connecting-rods of invariable Iength^,the reactions 
at fixed pivots, and the tensions of taut inextensible strings : 

3". The reaction of any fixed smooth surfaces or curves with which 
bodies of the system are constrained to remain in contact ; or of perfectly 
rough surfaces, so far as these can enter into holonomic systems: 

i". The reactions of any smooth surjaces or, curves with which bodies 
of the system are constrained to remain in contact, when these surfaces or 
curves are forced to move in sopie prescribed maimer ; for the displacement 
considered above is made on the supposition that t, so far as it is required for 
the specification of the system, /is not varied, i.e. that such surfaces or cuives 
are not moved during the displacement ; so that this case reduces to the 
preceding. 

The forces acting on the syiitem, other than these which do no work,-*are 
called the external forces . It ft Hows that the quantity \ 

is the work done by the external forces in the displacement which correspondsk 
to a change of 5, to 9,+ 85,, the other coordinates being unaltered. This ie 
a quantity which (from our knowledge of the constitution of the system, and 
of the forces at work) is a known function of q,, 5,, ... q„, (; we shall denote 
it by 

Qriq„q„ ...qn,t)Sqr. 

We have therefore 

dt\dqr/ dqr 
This equation is true for all values of r frori Ito n inclusive ; we thus 
have n ordinary differential equations of the second order, in which ji, g,, ... q^ 
are the dependent variables and ( Ls the independent variable; as the number 
f differential equations is equal to the number of dependent variables, the 
quatioDs are theoretically sufficient to determine the motion when the 
litial circumstances are given. We have thus arrived at a result which maf 
e thus stated : 
Let T denote the kinetic energy of a dynamical system, arid let \ 



k 



26, 27] The Equatims of Motion 37 

denote the work done by the external farces in an arbitrary displacement 
(Bqi, Sq^, ... Sqn), so tiiat T, Qi, Qa. ••• Qn are, from our knowledge of ih^ 
constitution of the system, known functions of ji, q^, ... jn, ^i, ?2> ••• ?n, t\ 
then the equations which determine the motion of the system may be written 



dtW-d^r^^- (r«l,2,...n). 



These are known as Lagrange's equations of motion. It will be observed 
that the unknown reactions (e.g. of the constraints) do not enter into these 
equations. T he determination of the ae-JiBactions forms. a-Beparatrfi branch of 
mechanics/which is know njts Kineto-static s : so we can say that in Lagrange's 
equations the kineto-staticai relations of the problem are altogether eliminated. 

27. Conservative forces : the Kinetic Potential, 

Certain fields of force have the property that the work done by the forces 
of the field in a displacement of a dynamical system firom one configiiration 
to another depends only on the initial and final configurations of the system, 
being the same whatever be the sequence of infinitesimal displacements by 
which the finite displacement is effected. 

Gravity is a conspicuous example of a field of force of this character ; the work done 
hy gravity in the motion of. a particle of mass m from one position at a height h to 
another position at a height k above the earth's surface is mg{h—k), and this does not 
depend in any way on the path by which the particle is moved from one position to the 
other. ' 



Fields of force of this type are said to be conserwitive . 

Let the configuration of any dynamical system be specified by n 
coordinates q^, q^, ... }„. Choose some cjonfiguration of the system, say 
that for which 

qr = OLry j (r = 1, 2, . .. n), 

as a standard configuration ; then if the external forces acting on the system 
are conservative, the work done by these forces in a displacement of the 
system fi^m the configuration (51, jaj ••• qn) to the standard configuration is a 
definite function of ^i^g'a* ••• Jm not depending on the mode of displacement. 
Let this function be denoted by F(gj, gr,, ... g„); it is called the Potential 
Energy of the system in the configuration (ji, 92, ••• Jn). In this c€U3e the 
work done by the external forces in an arbitrary displacement 

(Sji, Sgrj, ... Sqn) 

is evidently equal to the infinitesimal decrease in the function V, corresponding 
to the displacement, i.e. is equal to the quantity 



88 The Equations of Motion [ch. n 

Lagrauge s equations of motion therefore take the form 

dfdT\ dT dV 



dfdT\ dT _ dV / -1 9 \ 

dt\dqr) dqr'^dq/ (r^ i, z, ... n). 



If we introduce a new function L of the variables qi, q^, ... qn, ?i> ••• 3n> t, 
defined by the equation 

then Lagrange's equations can be written 

l©-i = ^' (r = l,2,...n). 

The function L is called the Kinetic Potentia l/or Lo ffranffian Amctio ni 
this single function completely specifies, so far as dynamical investigations 
are concerned, a holonomic system for which the forces are conservative. 

28. The explicit form of Lagrange* a equations. 

We shall now shew how the second derivates of the coordinates with 
respect to the time can be found explicitly irom Lagrange's equations. 

Let the configuration of th^ dynamical system considered be specified by 
Gk)ordinates ^i, ?s, ... ^n; we shall suppose that the configuration can be 
completely specified in terms of these coordinates alone, without t, so that 
the kinetic energy of the system is a homogeneous quadratic function of 
q\f q%i ••• 9n* As was seen in § 26, this is always the case when the 
constraints are independent of the time, but not in general when the 
constraints have forced motions (as for instance in the case of a particle 
constrained to move on a wire which is made to rotate in a given way). 

Suppose then that the kinetic energy is 

n n 

where ata^ajj^, and where the coefficients aja are known functions of 

?i» ?J> ••• qn* 
The Lagrangian equations of motion for the system are 

d(dT\ dT ^ / 1 Q X 

dtW-d^r^'' (r = l,2,...n), 

JeC^-^')-^l|/g'** = ^ (^ = 1' 2, ... n). 



or 2ar,5f, + 2 2 Uj3m = 0r. (r = l, 2, ... n), 

«=l /ritual L ^ J 

y Mf^, /^^ ^-Z'^, ^- /-^'";/^-'^-^;/-*^r. ■ 



27-29] 



* The Equations of Motion 



39 



where the symbol , which is called a Christoffers symbol*, denotes the 



expression 



2 \dqm dqi dqr J ' 



These equations, being linear in the accelerations, can be solved for the 
quantities g,. In fact, let D denote the determinant 

(hi Oia au'"(hn , iyt^t\^'^oiLy^^' f^^ 

021 Cl^ On 



Ojl «82 



Oni 



Ct»n 



and let A^ be the minor of Ort in this determinant. Multiply the n equations 
of the above system by Ai,, A^,, ... An^y respectively, and add them: re- 

n 

membering that the quantity S Ar^ an is zero when 8 is different from v, and 
has the value D when 8 is equal to i/, we have 

«=l«=«lr=l L ^ J r«l 



or 



1 " »» * [l rri] 1 ** 

g; = -^ 2 2 2 -4^,. U«g«+n ^ ^rvOr. 

•^ 1^1 m-l r=l L ^ J -^ r=l 



This equation is true for all values of v from 1 to n inclusive ; and these 
n equations, in which qi, q^, ... qn are given explicitly as functions of gj, jj, 
••• 9*1) 9i> 9s> ••• 9ni can be regarded as replacing Lagrange's equations of 
motion. 

1 

29. Motion of a 8y8tem which is constrained to rotate uniformly round an 
axis. 

In many dynamical systems, some part of the system is compelled by an 
external agency to revolve with constant angular velocity q> round a given 
fixed axis; the motion of a bead on a wire which is made to rotate in this 
way is a simple example. There is, as we have seen, no objection to the 
direct application of Lagrange's equations to such cases, provided the system 
is holonomic; but it is often more convenient to use a theorem which we 
shall now obtain, and which reduces the consideration of systems of this kind 
to that of systems in which no forced rotation about the given axis takes 
plat 3. 



* It was introdnced by Christoffel, Journal fUr Math, lxx. (1869), and is of importanoe in the 
theory of qnadratie differential forms. 



40 The Equationa of Motion [ch. n 

Suppose that, independently of the prescribed motion roand the axis, the 
system has n degrees of freedom, so that if the given axis is taken as axis of 
z, and any plane through this* axis and turning with the prescribed angular 
velocity is taken as the plane from which the azimuth ^ is measured, the 
cylindrical coordinates of any particle m of the system can be expressed in 
terms of n coordinates g,,g], .-., ^m these expressions not involving the time t. 
Then ii the kinetic energy of the syBtmn in the actual motion be T, and if the 
work done ^the external forces in an arbitrary infinitesimal displacement 
be Q,S3, + Q,Sg,+ ... +Q„5g„, where ft, Q,, ..., Q„ will be supposed to 
depend only on the coordinates q,, qx, ..., qn, and if the kinetic energy of 
the system when the forced angular velocity is replaced by zero be denoted 
by 2*1, we have 

r=i2m{2' + f' + r'(^ + ffl)'), 

Ti = ^'2m{£* + r' + r'i>% 

Now the quantity ^tmr" will be a function of g,, ^j, ..., q„, which is 
determined by our knowledge of the constitution of the system : denote it by 
W. The quantity Smr*^ will also be a known function of g,, q,, ..., q^, 
jn ■•■I ?n. being linear in ^,, 5,, ..., j„; it will be zero if, when w is zero, the 
motion of every particle has no component in the direction of increasing ; 
while if n is equal to unity, so that there is only one coordinate q, it will be 
the perfect differential with respect to t of a function of q : these are the two 
cases of most frequent occurrence, and we shall include them both by as- 
suming that Smr"^ is of the form -,- , where F is a given function of the 
coordinates q^, g„ ,,,, g„. 

We have therefore 

and the LagraDgiaa equations 

S. J* -a7r-«" ('■-■■2 ») 



These equations shew that, subject to the assumption already mentioned, 
the motion is the mine as if the pt'esenbed angular velocity were zero,' and 
the potential energy were to contain an additional term — ^Smr'w'. In this 
way, by modifying the potential energy, we are enabled to pass from a 
system which is constrained to rotate about the given axis to a system for 



1 



29, 30] The liquations of Motion 41 

which this rotation does not take place. The term centrifugal forces is 
sometimes used of the imaginaiy forces introduced in this way to represent 
the effect of the enforced rotation. 

30. The Lagrangian equutions for qudsi-coordinates. 

In the form of Lagrange s equations given in § 26, the variables are n 
coordinates g^,, gr^, ..., }„, and the time t; the knowledge of these quantities, 
together with a knowledge of the constitution of the system, Is sufficient to 
determine the position of any particle in any configuration of the system, 
which may be expressed by saying that g/, g„ ..., jn, are true coordinates of 
the system. We shall now find the form which is taken by the equations 
when the variables used are no longer restricted to be true coordinates of 
the system*. 

Consider a system defined by n true coordinates ji, ja. •••» Jm the 
kinetic energy being T and the work done by the external forces in a 
displacement (8g„ Sjs, •.., Sg^n) being QiSgri + Qs^a+ ••• -^Qn^n* so that the 
Lagrangian equations of motion of the system are 

d (dT\ dT \. / 1 o X n\ 

diW-W^^^" (/. = !. 2, ...,n)...(l). 

Let (Oi, a>a, ..., oDn, be n independent linear combinations of the velocities 
?i> 321 ••.! ?», defined by relations 

o>r = airgi + a2r?j+... + flW?ni (^=1, 2, ..., n)...(2), 

where «„, On, ..., a„» are given functions of g,, q^, ..., JnJ and let dTTi, d7r„ 
..., diTn, be n linear combinations of the differentials dqi, dq^, ..., dqn, defined 
by the relations 

dTTr = a,y dgi 4- OarC^a + . . . + Ojirdg'n (^=1, 2, ..., n), 

where the coefficients a are the same as in the previous set of equations. 

These last equations would be imnmdiately integrable if the relations 
s — = -;r — were satisfied for all values of /c, ?•, and m, and in that case variables 

oqm 9g« 

TTr would exist which would be true coordinates; we shall not however 
suppose the equations to be necessarily integrable, so that diTi, dir^, ..., diTf^ 
will not necessarily be the differentials of coordinates ttj, ttj, ..., TTn; we shall 
call the quantities dTr,, dTTj, ..., dirn differentials of quasi-coordinates. 

Suppose that the relations (2), when solved for ji, ga, ..., jn. give the 
equations 

. ?« = ^«jWi + i8rta)a + ... 'fiS.nWn (^=1,2, ..., n) ...(3). 

* ParticaUur cafles of the theorem of this article were known to Lagrange and Ealer : the 
general form of the equations is due to Boltzmann (Witn. Sitzung»bericht€f 1902) and Hamel 
{ZeiUehnft fUr Math. u. Phyt. 1904). 



y 




1 



42 The Equations of Motion [oh. 

Multiplying the Lagrangian equations (1) by j3„, 0„, ..., /9„,, respective! 
and adding, we obtain the equation 

Now 2 Q.Sq,. is the work done by the external forces on the ayatem in i 
arbitrary displacement, so Xff„Q,Svr is the work done in a displaceme 
in which all the quantities Sir are zero except &trr. If therefore the wo 
done by the external forces on the system in an arbitrary infinitesimal di 
placement (Btti, Stt,, .,,, Sttb) is n,&7ri + II,&ir,+ ... + n„for», we have 

By means of equations (3) we can eliminate ^i, q^, .... q^, from t 
function T, so that T becomes a function of o},, o),, ..., &)„, qi, q^, ..., ^„ (^ 
suppose for simplicity that t is not contained explicitly in T); let this foi 
of r be denoted by f. 

Tu u dT ^df 

Then we have r^™S= — «-,, 

dq^ , da>, 
and therefore 

But S &rr^a ^ zBi^ or unity according as r is different irom, or equal 
8: so we have 

dt fej "^ t r ^" "d^' aV. " 7 ^" ai". ' • 

We also have 

a?, a?. 1 3<B. Sji 39i f m Swf 3?i 

Sr , 37 iq. 5 , . . 3f . 

coordinate; we shall denote it by the symbol -— whether iTr '" * t' 
coordinate or not. Also the expression 

depends only on the connexion between the true coord! ui' i.e^ i-^— i>be c 
ferentials of the quasi-coordinates, and is independent oi' the nature 
motion of the dynamical system considered : we shall denote this expressi 
by "iru- We have therefore 

,^/^f^,__ ,|?-|?.n, (.-1,2 »> 

OtO, OTTr 



•tf-'ised rn temw of the 
tr^e coordinates, the 
- ^r- are satisfied, and 



(r = l. 2, ....«). 

tints 0, which is fixed, so 
i Eulerian angles 0, tp, ^, 
and moving with if, with 
anient (Sfl, 80, if) of the 
1, iJiTj, round Or, 6^, Oj, 
ials of quasi -coordinates : 
ular velocity of the bodj 
' qnosi-coordinatea cotre- 
e equationa of motion of 



of -:■ "ii »i. *. *. +i 
ely of the external forces 



9ir, 
etion. 

igy functioD can be 
roes depend not only 
f the bodies. 
,tion is specified by 
tone by the external 



(r-1, 2, .... n) 
he Lagrangian equa- 

(r=l, 2, ...,n) 






44 



The Equations of Motion 



■ 

[ch; n 



(r=l, 2, ..., n). 



and if a kinetic pottyntial L be defined by the equation 

» L^T-V, 

the equations take \!txe customary form 

^ dt\dqj dqr 

The function V c^an be regarded as a generalised potential energy 
function. An examphe of suchitt system is furnished by the motion of a 
particle subject to We\ber's electrodynamic law of attraction to a fixed point, 
the force per unit mas:^ acting op the particle being 

where r is the distance drf the particle from the centre of force : in this case 
the function V is defined ^by thq equation 

Example, If the forces §U, Qg, ,,,1 Q^y of a dynamical system which is specified by 
coordinates ^d ^2) •••) ?n ^'^^ dekivable nrom a generalised potential-function 7, so that 



Qr 



87. d /8F\ 



(r=l, 2, ..., n), 



shew that ft, ft, ..., ft must 
relations 



dq^ dt 
linear jTanctions of ^'i, ^'2, ..., ^«, satisfying the n (2n - 1) 






On the general conditions for the existence of a kinetic potential of forces, reference 
may be made to 

Hebnholtz, Journal fUr Mluh., VoL 100 (1886). 
Mayer, Leipzig, Berickie^ WoL 48 (1896). 
Hirsch, Math, Annalm^ ViL 60 (1898). 



32. Initial motions. 



of Ai 



The differential equations of "^notion of a dynamical system cannot in 
general be solved in a finite iottfy ' i fcnns of known functions. It is how- 
ever always possible (except in th^- \» inity of certain singularities vhK'}» 
need not be considered here) to solv^ a set of differential equations hy p^r.cer- 
series, i.e. to obtain for the dependent variables q^, q^, ..., <t'„, e .^r^ jssions of 

the type - ? 

?i =0, +6, ^ + Ci t* + di^+ ... 



gn - On-f 6fi*-Vcn^*-f-dnt»+ ... ; 



\ 



LS 



31, 32] The Equations of Motion 45 

the coefficients a, 6, . . . can in fact be obtained by substituting these series in 
the differential equations, and equating to zero the coefficients of the various 
powers of t ; the expansions will converge in general for values of t within 
some definite circle of convergence in the ^-plane*. 

It is plain that these series will give any information which may be 
required about the initial character of the motion {t being measured from the 
commencement of the motion), since aj is the initial value of ^i, &i is the 
initial value of gi, and so on. This method of discussing the initial motion 
of a system is illustrated by the following example. 

Example, Conflider the motion of a particle of unit mass, which is free to move in a 
plane and initially at rest, and which ia acted on by a field of force whose components 
parallel to fixed rectangular axes at any point (x, y) are (iT, T) ;'and let it be required to 
determine the initial radius of curvattire of the path. 

Let (^+^, y+17) be the coordinates of any point adjacent to the initial point (x, y), 
so that ^, 17, may be regarded as small quantities ; then the equations of motion are 



•••••• 






If therefore we assume for { and i; the ezpaosions 






(it is not necessary to include terms of lower order than ^, since the quantities ^, *;, ^, 17, 
are initially zero), and substitute in these dififerential equations, we find, on comparing 
the coefficients of various powers of <, the relations 

a-iZ(*,y), .6-0, c^^Yzg+r^^), ^ 

The path of the particle near the point (x, y) ia therefore given by the series 

where u denotes the quantity \t\ 

Now if the coordinates ( and 17 of any curve are expressed in terms of a parameter u, 
the radius of curvature at the point u is known to be 

X^dii) "^ \dii) J 

du^ du du^ du 
* Whittaker, A Course of Modern AnalytUt § 21^ 



I ■ 



-- "V 



46 The Equations of Motion [gh. n 

80 the radius of curvature corresponding to the zero value of ic, for the curve given by the 
above expressions, is 

3(z«+r«)< ^ 

and this is the required radius of curvature of the path of the particle at the initial point. 

33. Similarity in dynamical systems. 

If any system of connected particles and rigid bodies is given, it is 
possible to construct another system exactly similar to it, but on a different 
scale. If now the masses and forces in the two systems, which we can call 
the paMern and model respectively, bear certain ratios to each other, the 
workings of the two systems will be similar, though possibly at speeds which 
are not the same but bear a constant ratio to each other. 

To find the relation between the various ratios involved, let the linear 
dimensions of the model and pattern be in the ratio x : 1, let the masses of 
corresponding particles be in the ratio y : 1, let the rates of working be in the 
ratio £: : 1, so that the times elapsed between corresponding phases are in the 
ratio 1 : z, and let the forces be in the ratio w : 1. Then for each particle we 
have an equation of motion of the form 

i mJc = X ; 

so if m is altered in the ratio yil^x is altered in the ratio xf^ : 1, and X is 
altered in the ratio w : 1, we mustj have 

I w = xyz^t 
and this is the required relation between the numbers x, y, z. in. 



V Example, If the forces acting are' those due to gravity, we have « — /a .ind conse- 

^ ;C ^ «- ' quently a^= 1, so that the rates of working are inversely as the square roots o^ the linear 

dimensions. ■ -' ^ 

' If the forces acting are the mutlial gravitation^ of the particles, every particle 

attracting every other particle with a force proportion^ to the product of the masses and 
the inverse square of the distance, we have w=^y^la^^ so the rates of working are in the 

ratio y^ : x^, 

I 

34. Motion with reversed forces, 

A special case of similarity is that in which the ratio w has the value — 1. 

We have seen that the motion of any dynamical sjrstem which is subjected 
to constraints independent of the time, and to forces which depend only on 
the positions of the particles, is expressed by the Lagrangian equations 

d (dT\ dT ^ / 1 o 

dtWJ'^r^'' (r=l,2,...,n) 

where the kinetic energy T ia a homogeneous quadratic function of the 
velocities gi, jj, ..., jn, involving the coordinates gi, g„ ..., g», in any way, 
and Q is a function of ji, ja, ..., gn only. 



■ 

I 



wmm 



32-35] The liquations of Motion 47 

Introduce a new independent variable defined by the equation 

T=si<, where i = v^ — l, 

and let acceni^s denote differentiations with regard to t. Then since 

•J- 1 r-r- j and ^ are homogeneous of degree — 2 in dt, the above equations 

become 

TrKd^O^Wr'^" (r=l,2,...,n) 

where t!C is the same function of qi\ q^', ..., jn, Ji, ..., Jm that T is of 

• • • 

?i> ?j> •••> ?n> ?i> S'a* •••> ?n« 

But if T (instead of t) be now interpreted as denoting the time, these last 
equations are the equations of motion of the same system when subjected to 
the same forces reversed in direction. Moreover, if «!, Og, ..., a,i, ^,, ^,, ..., 
fin are the initial values of ji, q^, ..., jni qu q^, ..., jn, respectively in any 
particular case of the motion of the original system, then ai, Og, ..., On, —ifii, 

— i^,, ..., —I fin will be the corresponding quantities in the transformed 
problem. We thus have the theorem that in any dynamical system subjected 
to constraints independent of the time and to forces which depend only on the 
position of the particles, the integrals of the equations of motion are still real 

if the repUiced by V— It and the initial velocities fii, fi^, ..., fin, by — V— Ifi^, 

— V— l/9a, ..., — v'— Ifin respectively ; and the expressions thus obtained repre- 
sent the motion which the same system would have if with the same initial , ^^c^) 
conditions, it were acted on by the same forces reversed in direction, ^^ '^' j^,y^^^ . 

36. Impulsive motion. 

In certain cases (e.g. in the collision of rigid bodies) the velocities of the 
particles in a dynamical system are changed so rapidly that the time occupied 
in the process may, for analytical purposes, be altogether neglected. 

The laws which govern the impulsive motion of a system bear a close 
analogy to those which apply in the case of motion under finite forces : they 
can be formulated in the following way. 

The number which represents the mass of a particle, multiplied by the 
vector which represents its velocity at any instant, is a vector quantity 
(localised in a line through the particle) which is called the momentum of 
the particle at that instant; the three components parallel to rectangular 
axes Oxyz of the momentum of a particle of mass m at the point (x, y, z) are 
therefore (mx, my, mi). If any number of particles form a dynamical system, 
the sum of the components in any given direction of the momenta of the 
particles is called the component in that direction of the mxymentum of the 
system. The impulsive changes of velocity in the various particles of a 
connected system can be regarded as the result of sudden communications 
of momentum to the particles. 

The effect of an agency which causes impulsive motion in the system 



48 The Equations of Motion [ch. n 

will be measured by the momentum which it would communicate to a single 
free particle. If therefore {u^, Vq, w^ are the components of velocity of a 
particle of mass m, referred to fixed axes in space, before the impulsive 
communication of momentum to the particle, and if {u, v, w) are the com- 
ponents of velocity of the particle after the impulse, then the vector quantity 
(localised in a line through the particle) whose components are 

m (w - tio), m(v- Vo), m(w — Wo\ 
represents the impulse acting on the particle. 

For the discussion of the impulsive motion of a connected system of 
particles, it is clearly necessary to have some experimental law analogous to 
the law of Action and Reaction of finite forces ; such a law is contained in 
the statement that the total impulse acting on a particle of a connected 
system is equal to the resultant of the external impulse on the particle (i.e. the 
impulse communicated by agencies external to the system, measured by the 
momentum which the particle would acquire if free) together with impulses 
directed along the lines which join this particle to the other particles which 
constrain its motion; and the mutvully induced impulses between two connected 
particles are equal in magnitude and opposite in sign. 

If we regard the components of an impulse as the time-integrals of the 
components of an ordinary finitdj force which is very large but acts only for 
a very short time, the law just stated agrees with the law of Action and 
Reaction for finite forces. 



Change of kinetic energy dite to imposes. 

The change in kinetic energy of a dynamical system whose particles are acted on by a 
given set of impulses may be determined in the following way. 

Let an impulse /, directed along a line whose direction-cosines ^referred to fixed axes of 
reference are (X, ^ v), be communicated to a particle of mass m, changing its velocity 
from Vo> ^'^ * direction whose directioi^cosines are (Zq, Mqj AW to v, in a direction whose 
direction-cosines are (Z, M, N). The equations of impulsive motion are /ynv-^^ ^^j "'• 

Multiplying these equations respectively by ^ '>*iAAKt/,»l-*^%''V/ 

i(t>Z-f-roZo), i (tjJ/-f- ro^o)> and HvN+VqNq), 
and adding, we have T^i/u'**') 

The change in kinetic energy of the particle is therefore equal to the product of the 
impulse and the mean of the components, before and after the impulse, of the velocity of 
the particle in the direction of the impulse. 

Now consider any dynamical system of connected particles and rigid bodies, to which 
given impulses are communicated ; applying this result to each particle of the system, and 
summing, we see that the change in the kinetic energy of the system is equal to the sum of the 
impulses applied to it, each multiplied by the mean of the components, before and after the 
communication of the impulse, of the velocity of its point of application in the direction of the 
impulse. In this result we can clearly neglect the impulsive forces between the molecules 
of any rigid body of the system. 



86, 86] The Equations of Motion 49 

36. The Lagrangian equations of impulsive motion. 

The equations of impulsive motion of a dynamical system can be 
expressed in a form* analogous to the Lagrangian equations of motion for 
finite forces, in the following way. 

Let {Xiy Yi, Zi) be the components of the total impulse (external and 
molecular) applied to a particle m^ of the system, situated at the point 
(^9 yi> ^i)' The equations of impulsive motion of the particle are 

mi(xi — iin)^Xi, nii(yi^yio)=^ Fi, mi{ii^ii^ — Zu 

where (i^io, y^, iio) and {xu yi%> ii) denote the components of velocity of the 
particle before and after the application of the impulse. 

If 9i> 9tf •••> 9n denote the n independent coordinates in terms of which 
the configuration of the system can be expressed, we have therefore 



2«^{(.,-x.)g-H(y.-y.)| + (i.-i.)|} 



i \ oqr oqr oqrJ 

where the summation is extended over all the particles of the system. 

Now in forming the summation on the right-hand side of this equation, 
it is seen as in § 26 that the molecular impulses between particles of the 
system can be omitted : the quantity 



i \ oqr 



doci _^Yi^+ Z' — '] 
dqr dqr * dqrJ 



can therefore readily be found when the external impulses are known: we 
shall denote it by the symbol Qr. We have consequently 

But as in § 26 we have 



and similarly 



dxi dxi . dxi d ., . „v 

, dXi 3 /I . a\ 



where qro and qr denote the velocities of the coordinate qr before and after 
the impulse respectively. Thus if 

r = i2m<(i?i« + yi« + ii») 
i 

* Due to Lagrange, M€e, Anal. (2 6d.), Vol. n. p. 188. 
W. D. 4 



,- / 



52 Principles available for the integration [ch. in 

by taking 
The form 

^•■ = X,<;r„a:„...,*t,i), (r=l,2, ...,fc), 

may therefore be regarded as the typical form for a set of differential 
equations of order k. 

If a function /{a^, x, xt, t) is such that -jj is zero when {x,,x,, ...,xi) 

are any functions of t whatever which satisfy these differential equations, the 



/{x„ Xj, ..., xt, t) = Constant 
is called an tntegrul of the system. The condition that a given function / 
may furnish an integral of the system is easily found ; for the equation 
d/jdt = gives 

|^z.+|^x.+... + |fz. + f-o, 

dxi cXi dxi at 

and this relation mnst be identically satisfied in order that the equation 

/(*ii X,, ■••, Xk, = Constant 
may be an integral of the system of differential equations. 

Sometiinee the Atnction / itself (aa distinct from the equation /= constant) is called an 
integral of the eyBtem. 

The complete solution of th3 set of differential equations of order k is 
furnished by k integrals 

/rix„x„....xt,t) = ar, ir^l,2....,k), 

where a,, a,, .... at, are arbitrary constants, provided these integrals are 
distinct, i.e. no one of them is algebraically deducible from the others. For . 
let the values of x,, x^, ..., x^, obtained from these equations as functions of 

t, Oi. Oi Qii be 

a;, = ^(ai, a a^.t), (r = l, 2, ..., i); 

then if (ir,, IT, xi^ are any particular set of fimctions of 2 which satisfy the 

differential equations, it follows from what has been said above that by giving 
to the arbitrary constants Or suitable constant values we can make the equations 

Mx„x^, ..., ict, 0=-Or ('• = 1,2, .... fc> 

true for this particular set of functions {x^, x,, ..., x^f', and therefore this set 
of functions {x^, x,, ..., o^) will be included among the functions defined by 
the equations Xr = ^r- The solution of a dynamical problem with n degrees 
of freedom may therefore be regarded as equivalent to the determination of 
2n integrals of a set of differential equations of order 2n. 



37, 38] Principles available for the integration 53 

Thus the differential equation 
which is of the second order, possesses the two integrals^ 






tan~*?-^«aa«,' 



where 04 and Oj are arbitrary constants. On solving these equations for q and q, we have 

rg-Oi* sin (^+02) 
[^=ai*cos(<+a2), 

and these equations constitute the solution of the differential equation. 



! 



The more elementary division of dynamics, with which this and the 
immediately succeeding chapters are concerned, is occupied with the dis- 
cussion of those dynamical problems which can be completely solved in terms 
of the known elementary functions or the indefinite integrals of such functions. 
These are generally referred to as problems soluble by quadratures. The 
problems of dynamics are not in general soluble by quadratures ; and in those 
cases in which a solution by quadratures can be eflfected, there must always 
be some special reason for it, — in fact the kinetic potential of the problem 
must have some special character. The object of the present chapter is to 
discuss those peculiarities of the kinetic potential which are most frequently 
found in problems soluble by quadratures, and which in fact are the ultimate 
explanation of the solubility. 

38. Systems with ignorahle coordinates. 

We have seen (§ 27) that the motion of a conservative holonomic dy- 
namical system with n degrees of freedom, for which the coordinates are 
9i) 991 ••• > 9n stnd the kinetic potential is Z, is determined by the differential 
equations 

d fdL\ dL ^ /TO \ 

dm)'wr^' (r=i.2,...,n). 

The quantity ^ is generally called the momentum corresponding to the ^JT^J^'^/^ 
coordinate 9,.. 

It may happen that some of the coordinates, say qi, q%, *>. ,qky &i*e not 
explicitly contained in Z, although the corresponding velocities ^1, 9a, ... , 9ik 
are so contained. Coordinates of this kind are said to be ignorahle or cyclic ; 
it will appear in the following chapters that the presence of ignorahle 
coordinates is the most frequently-occurring reason for the solubility of 
particular problems by quadratures. 

The Lagrangian equations of motion which correspond to the k ignorahle 

coordinates are 

d /dL 



(|^J = 0, (r = l,2, ...,A). 



dt \dq 



Principles avaiktble for the integration [ch. m 



and on integratiou, these can be written 

9i a 



(r-1, 2,....k), 



0t, ■■•, ffk ftfe constants of integration. These last equations are 
fc integrals of the Byatem. 

all now shew how these k integrals can be utilised to reduce the 
le set of L^[rangian differential equations of motion*. 

denote the function L— % q, ^ . By means of the k equations 

i"^" ('-i.^ *)■ 

[press the k quantities ji, 9,, ... , q^, which are the velocities cor- 

I to the ignorahle coordinates, in terms of 

fft+i, ?*+». ..- , ?n. 9*+i. ?*+». ■-■ . ?«. A. A. ■■■ 1 ^*; 

ippose that in this way the function R is expressed in terms of the 
of quantities. ' 

et Sf denote the increment produced in any function / of the 
qi+„qk+t, ... ,9b.9ii9i> ■■■ <4n (or of the quantities qt+,,qt+i, ■■■ ,?«. 

II A) Ai ■■■ I A) hy arbitrary infinitesimal changes Bqii+,, Sqt+i, ■■-, 
,. , hqn, in its arguments. Then we have 



SR.s{L-iy^). 



inition of R. But 





SL = 


11^ 


+ 


11^-, 


J 3i,. 




K 


i^''i> 


k 


i^^^i 


3,8/9-, 


ive 


therefore 






a?. 


-,8,. 






Sii = 


i 


9i, 


+ 


11^'- 


J_?,8A, 



the infinitesimal quantities occurring on the right-hand side of this 
ore arbitrary and independent, the equation is equivalent to the 

inEtonDatian whiob follow* ia r«All; k cam of the Hamiltouian tniiBfonattioD, which 
in Chftpler X; it waa however Grat sepantel; given by BoDth in 1876, and agmewfaat 
nhoItE. 



•I 



(U% 



88] Principles available for the inteffration 56 

system of equations 

(r = A; + l,A; + 2, ...,n), 



dL 


diz 


dqr' 


dqr' 


dL 


dR 


dqr' 


~dqr' 


3r = 


dR 



(r^k + l,k + 2, ...,n), 

Substituting these results in the Lagrangian equations of motion, we 
have 

(H fdR\ dR ^ / , , , « 

Now iZ is a function only of the variables j|.+,, y^+a, ...,?«, J*+i, ...,?•», 
and the constants )8i, /8a, ..., /8|.: so this is a new Lagrangian system of 
equations, which we can regard as defining a new d3mamical problem with 
only (n — k) degrees of freedom, the new coordinates being q^+i, qt+^y •••,?*»» 
and the new kinetic potential being R. When the variables gt+j, j^+j, ...,?«> 
have been obtained in terms of t by solving this new dynamical problem, the 
remainder of the original coordinates, namely ?i, 9a> ••• » ?*, can be obtained 
from the equations 

^r = -jg^d«, .(^=1, 2, ...,&). 

Hence a dynamical problem with n degrees of freedom, which has k ignorahle 
coordinates, can be reduced to a dynamical problem which ha£ only (n — k) 
degrees of freedom. This process is called the ignoration of coordinate . 

The essential basis of the ignoration of coordinates is in the theorem that when the 
kinetic potential does not contain one of the coordinates qr explicitly, although it involves 
the corresponding velocity q^, an integral of the motion can be at once written down, 

namely ^= constant This is a particular case of a much more general theorem which 

will be given later, to the effect that when a dynamical system admits a known infinit esi- 
mal con tact-transformation, an in tegral of the sy stem can be immediately obtained. 

If the original problem relates to the motion of a conservative dynamical 
system in which the constraints are independent of the time, we have seen 
that its kinetic potential L consists of a part (the kinetic energy) which is 
a homogeneous quadratic function of q^, 9,, ..., q^ and which involves 
?*+!> ?*-f8» •••, ?n in any way, together with a part (the potential energy with 
sign reversed) which involves jt+i, 3*+,, ..., q^ only. But in the new 
dynamical system which is obtained after the ignoration of coordinates, the 
kinetic potential 22 cannot be divided into two parts in this way : in fact, R 
will in general contain terms linear in the velocities. And more generally 
when (as happens very frequently in the more advanced parts of Dynamics) 
the solution of one set of Lagrangian differential equations is made to depend 



66 Principles availcMe for the integration [pa. 

on that of another set of Lagrangian differential equations with a smaller 
number of coordinates, the kinetic potential of this new system is not 
necessarily divisible into two groups of terms corresponding to a kinetic and 
a potential energy. We shall sometimes use the word natural to denote those 
systems of Lagrangian equations for which the kinetic potential contains 
only terms of degrees 2 and in the velocities, and non-natural to denote 
those systems for which this condition is not satisfied. 

As an el&mple of the ignoration of coordinates, consider a dynamical system with tsc^ 
degrees of freedom, for which the kinetic energy is 



n 2 



aad the potential energy is 

where a, 6, c, d, are given constants. 

It is evident that q^ is an ignorable coordinate, since it does not appear explicitly in T 
or 7. 



^=i.-L-b-2+W2»-^-^?2*, 



The kinetic potential of the system is 

and the integral corresponding to th(s ignorable coordinate is 

where /3 is a constant, whose value is determined by the initial circumstances of the motion. 
The kinetic potential of the new dynamical system obtained by ignoring the coordinate 



and the problem is now reduced to the solution of the single equation t *- 

or §j+(2rf+W3'2=0. 

As this is a linear differential equation with constant coefficients, its solution can be 
immediately written down : it is 

q^^A sin {(2rf + 6/3«)* t + f }, 

where A and c are constants of integration, to be determined by the initial circumstances 
of the motion. This equation gives the required expression of the coordinate qf in terms 
of the time : the value of q^ in terms of t can then be deduced from the equation 

qi^»\ifl'\-W)dt, 

which gives 

ji=(|3«+i/964«)t-— ^^t8in2{(2d+6/3«)*«+*}, 

4 (Za + 0/3*) 

and so completes the solution of the system. 



.if^^lV. ",■« 



^ 1 38, 39] Principles available /or the integration 67 

®^ I 39. Special cases of ignoroHon; integrals of momentum and angular 

^^^ I momentum. 

,nd I 

i We shall now consider specially the two commonest types of ignorable 

m coordinates in dynamical problems. 
)te . (i) Systems possessing an integral of mom^entum. 

Let the coordinates of a conservative holonomic dynamical system with 
n degrees of ifreedom be ji, g,, ..., $»; and let T be the kinetic energy of the 
system, and V the potential energy, so that the equations of motion of the 
system are 

d (dT\ dT dV , 1 o X 

dt \dqr/ dqr oqr 

Suppose that one of the coordinates, say gi, is ignorable, and moreover is 
such that an alteration of the value of qi by a quantity I, the remaining 
coordinates 9a> 9i> •••> 9n being unaltered, corresponds to a simple translation 
of the whole system through a distance I parallel to a certain fixed direction 
in space ; we shall take this to be the direction of the a7-axis in a system of 
fixed rectangular axes of coordinates. 

Since qi is an ignorable coordinate, we have the integral 

;rr- = Constant, 

and we shall now discuss the physical meaning of this equation. 
We have 

where the summation is extended over all the particles of the system, 

= ^miXi, since in this case ^ = 1, ;r^ = 0, ^ = 0. 

dqi ' dqi dqi 

Now XmiXi represents(§ 35) the component parallel to the a;-axis of the 
momentum of the system of particles m,-, and consequently this is the 

physical meaning of the quantity ^ in the present c€ise. 

()T 
The intecral ^rr = Constant 

can therefore be interpreted thus: When a dynamicai system can be 
translated as if rigid in a given direction tvithout violating the constraints, 



Prindjplea available for the integration [ch, m 

he poteriiMil energy is thereby unaUered (the way in which the kiaetic 
Y depends on the velocities is obviously unaltered by this translation, so 
irrespondiiig coordinate is ignorable), then the component parailel to this 
ion 0/ the momeHtum of the system is constant. 

lie result is called the law 0/ conservation of momentum, and systems to 
it applies are said to possess an integral 0/ momentum. 
(ii) Systems possessing an integral of angular momentum. 
gain taking a system with coordinates q,, 9,, ..., g„ and kinetic and 
tial energies T and V respectively, let us now suppose that the 
nate g, is ignorable, and moi'eover is such that an alteration of ji by 
intity a, the other coordinates remaining unchanged, corresponds to 
sle rotation of the whole system through an angle a round a given fixed 
n space : we shall take this line as the axis of z in a system of 
rectangular axes of coordinates, 
nee 9, is an ignorable coordinate, we have the integral 

;i-r = Constant (1), 

e have to determine the physical interpretation of this equation, 
e have as before 

i the summation is extended over all the particles of the system. But 
write 

Xi " r,- cos ij>i, yi = n sin ^, 

ve dif>i = dqi, 

dxi dxc . , 

d^rnr '"""'*'"'"■ 

dq, 
berefore 

^ ='^mi(-Xiyi + yia:i) (2). 

ow if r denote the distance of any particle of mass m from a given 
ht line at any instant, and if tt> denote the angular velocity of the 
le about the line, the product mr'm is called the angular momentuTn 
1 particle about the line. 

)t be any point, and let P, P", be two consecutive positions of the 
ig particle, the interval of time between them being dt. Then the 



39] Principles available for the integration 59 

angular momentum about any line OK through is clearly the limiting 
value of the ratio 

-y: X Twice the area of the projection of the triangle OPP' on 
a plane perpendicular to OK^ 

so if (Z, m, n) are the direction-cosines of OK and if (X,, /a, v) are the direction- 
cosines of the normal to the triangle OPP', we see that the angular 
momentum about OK is equal to the product of (tk+mfi +nv) into the 
angular momentum about the normal to the plane OPP\ It is eyident from 
this that if the angular momenta of a particle about any three rectangular 
axes Oxyz at any time ai-e K^K^K respectively, then the angular momentum 
about any line through whose direction-cosines referred to these axes are' 
(Z, w, n) is ZAi + mAj + nA, ; we may express this by saying that angular 
momenta about axes through a point are compounded according to the vectorial 
law. 

The angular momentum of a dynamical system about a given axis is 
defined to be the sum of the angular momenta of the separate particles of 
the system about the given axis ; in particular, the angular momentum of 
a system of particles typified by a particle of mass m, whose coordinates are 
(^> yu ^<)> about the axis of z is STn^r^**^, where 

Xi = r< cos 4>u Vi = ^» fiin ^i, 

and the summation is extended over all the particles of the system ; this 
expression for the angular momentum of a system can be written in the form 

Imi iyiXi - Xiyi\ 

i 

and on comparing this with equation (2) we have the result that the angular 
momentum of the system considered, about the axis of Zyis ^ . 

The equation (1) implies therefore that the angular momentum of the 
system about the axis of z is constant : and we have the following result : 
When a dynamical system can he rotated cw if rigid round a given axis vnthout 
violating the constraints, and the potential energy is thereby unaltered, the 
angular momentum of the system about this axis is constant. 

This result is known as the theorem of conservation of angvla/r 
momentum. 

Example. A system of n free particles is in motion under the influence of their 
mutual forces of attraction, these forces being derived from a kinetic potential F, which 
contains the coordinates and components of velocity of the particles, so that the equations 
of motion of the particles are 



^ 



.. dv d /8r\ ^ 



60 Principles available for the integration [ch. m 

shew that these equations possess the integrals 

2 ( nirXr + 5T ) ~ Constant, 

2 ( fnr^r + 5^ ) = Constant, 

2 I mr^r + ^ j ssConstant, 

r dV dV] ' 

2 hnr (yA - ^rifr) + y»- aJ" ~ **" 5^ J- =Constant, 

2 •! m^ (Zr^r - J^r ^). + «r 5^ - "^r o^ f = Constant, 

2 \nir {Xrifr ~ ^r^r) + J?r g^ - ^r gj | =• Constant, 

which may be regarded as generalisations of the integrals of momentum and angular 
momentuHL (I^vyO 

40. The general theorem of angular momentn/m. 

The integral of angular momentum is a special case of a more general 
result, which may be obtained in the following way. 

Consider a dynamical system formed of any number of free or connected 
and interacting particles : if they are subjected to any constraints other than 
the mutual reactions of the particles, we shall suppose the forces due to these 
constraints to be counted among the external forces. 

Take any line fixed in space, and choose one of the coordinates which 
specify the configuration of the system (say qi) to be such that a change in 
9i, unaccompanied by any change in the other coordinates, implies a simple 
rotation of the system as if rigid round the given line, through an angle 
equal to the change in 9,. Wb suppose the constraints to be such that this 
is a possible displacement of the system. 

The Lagrangian equation for the coordinate qi is 



dt 
and this reduces to 



d(dTy_dT_Q 
dt\^q^) dqr^'' 



since the value of 9, (as distinguished from qi) cannot have any effect on the 

kinetic energy, and therefore ^- must be zero. Now ^ is the angular 

momentum of the system about the given line; and Qi^i is the work done 
on the system by the external forces in a small displacement Bqi, i.e. a small 
rotation of the system about the given line through an angle Sqi, from which 
it is easily seen that Qi is the moment of the external forces about the given 
line. We have therefore the result that the rate of change of the angular 



39-41] Principles available for the integration 61 

momentwm of a dynamicqX system ahout any fixed line is equal to the moment 
of the external forces about this line. The law of conservation of angular 
momentum obviously follows from this when th6 moment of the external 
forces is zero. 

Similarly we can shew that the rate of change of the momentwm of a 
dynamical system parallel to any fixced direction is equal to the component, 
paraMel to this line, of the total external forces adding on the system. 

For impulsive motion it is easy to establish the following analogous 
results : 

The impulsive increment of the component of momentum of a system in any 
fixed direction is equal to the component in this direction of the total external 
imptUses applied to the system. 

The impulsive increment of the angular mom£ntum of a system round any 
axis is equal to the moment round thai aads of the external impulses applied 
to the system. 

41. The Energy equation. 

We shall now introduce an integral which plays a great part in dynamical 
investigations, and indeed in all physical questions. 

In a conservative dynamical system let 9,, 93, ... , jn be the coordinates 
and let L be the kinetic potential : we shall suppose that the constraints are 
independent of the time, so that £ is a given function of the variables 
9i» ?a> ••• > ?n> ?i> ?8> ••• > ?n Only, not involving t explicitly. We shall not, at 
first, restrict L by any further conditions, so that the discussion will apply to 
the non-natural systems obtained after ignoration of coordinates, as well as 
to natural systems. 

We have 

di « .. ax ^ . ax 



if'^) 



= ,1 ?^ af + i, *^ It §|) ' ^y ^^" Lagrangian equations 

_d_(^ . ax\ 
""dArti^^ajJ- 

Integratmg, we have 

where A is a constant. 

This equation is an integral of the system, and is called the integral of 
energy or law of conservation of energy. 

We have seen that in natural systems, in which the constraints do not 
involve the time, the kinetic potential X can be written in the form T— F, 



Principles avaUcMe for the integration [ch. in 

he kinetic energy of the system) is homogeneous and of degree 2 
lities, while F is a function of the coordinates only. In this case, 
he integral of enei^y becomes 



"3jr 

IT—T+V, since T \s homogeneous of degree 2 in j,, j„ .... g„, 

V+V. 

ws that in conservative natural systems, the sum of the kinetic and 

er^ies is coTistant. This constant value h is called the total energy 

n. 

tter result can also he obtained directly from the elementary 

f motion. For from the equations of motion of a siogle particle, 

THjii = Xi, TTiiji = Yi, mi'ii.= Zi, 

ttOi {XiXi + yiyi + Zi2i) = 2 (Xjij + Fiji + ^i«i). 

mmmation is extended over all the particles of the system, or 
d . Simi (ii" + y,-" + ii") =- 2 (Xdx + Ydy + Zdz), 

increment of the l;iaetic energy of the system, in any infinitesimal 
path, hs equal to the work done by the forces acting on the system 
; of the path, and therefore is equal to the decrease in the potential 
he system. The sum of the kinetic and potential energies of the 
herefore constant. 

tioD of energy 

implicity we suppose the eystem to consiat of a single particle) ia true not 
r, y, z) denote coordinatee referred to any filed axes, but also wbeu they 
inates referred to axes which are moving with any motion of translation 
rection with constant velocity. 

f< li <leiiote the coordinates of the particle referred to axes fixed in apace 

to the moving axes Oxyz, so that 

are the constant components of velocity of the origin of the moving axes, 
ndt already proved is that 

rf . jm (£* + ^> + ft - Jrff + Frf, + £af, 

n^{(,i+a)'>+(y'^b^+(,i+cy)=X^dx+adt)+r^dy+bdl)■K^^(L+cdl), 



41, 42] Principles avaUahle for the integrcUion 63 

Now we have 

=m{ai'\-blij+oC)dt 
and therefore 

which establiahee the theorem. 

It may be noted that from this result the three equations of motion of the particle 
can be derived, by taking x^^-aC etc., and subtracting the equation of energy in the 
coordinates {x, tf, z) firom the equation of energy in the coordinates ((, 17, (), 

42. Reduction of a dynamical problem to a problem with fewer degrees of 
freedom, by means of the energy-eqiuition. 

When a conservative dynamical system has only one degree of freedom, 
the integral of energy is alone sufficient to give the solution by quadrature& 
For if 3 be the coordinate, the integral of energy 

is a relation between q and q ; if therefore q be found explicitly in terms of q 
from this equation, so that it takes the form 

we can integrate again and obtain the equation 

t = I ^^rx + constant, 
Jf(9) 

which constitutes the solution of the problem. 

When the system has more than one degree of freedom, the integral of 
energy is not in itself sufficient for the solution ; but we shall now shew that 
it can be used for the same purpose as the integrals corresponding to ignor- 
able coordinates were used, namely to reduce the system to another dynamical 
system with a smaller number of degrees of freedom*. 

In the function i, replace the quantities g^, g,, ... , jn, by jija', q^q^', ..., 
qiqn, respectively, where g/ denotes ^ : and denote the resulting function 
by ft (ji, g/, g,', ... , gn, gi, ga> ••• , 9n). Then diflFerentiating the equation 

L(qi, gj, ... , gn, gi, ga, ••• , ?n) = ft(gi, gg', gs', ... , g/, ?i» ?a, ••• , gn), 

, az aft » Or aft 

we have 5^ = 5-^-" ^ t^^— > (1), 

agi agi r=2?i'ag/ ^ 

az 1 aft , « « 

dql^ld^ (r = 2,3,...,n) (2), 

a| = i (r=l,2,3,...,n) (3). 

* Whittaker, Mest. of Math. xxz« (1900). 



Principles available for the integration [oh. 



OUB (1) and (2) give 




312 3i . i q, 3i 


(♦)■ 


1 the iategral of energy 




1^-1— ■ 




by gij/ for all values of r from 2 to n inclnaive, and then from this 


btain j, aa a function of the quantities (j,', j,', , . 
ng this expression for 9,, express the fiiDCtion 


?■'.?..?. ?")i 






)f (?.'. ?.'. - . ?»'. ?., ?.. - . 9«)- Let the function thus obtained 


J by £'; then from (4) we see that L' is the 


an , , 

same as -, but 
85, 


expressed. 


ntiating the equation of energy, which by (4) 


can be written in 


'.|-"-*. 




ling it as a relation which implicitly determinea 


5i as a function of 


les (?,', ?.' 3-'. '?■. * ?■). "« '»'« 




. ^n 85, dn . d-n 


(5), 


I'di^dq; dq/ ^'iq,dq, 




. 8"n ftj, an . s-n 

*35.-a?,-3s, ''8,\35, 


(«). 


,, an 




6 an identity in the variables (}/. q,' q„', tfi. 


9i ?„),wehave 


SL' a-n a-na?, 

a?,'"3iV dq,'dq; 


('). 


3i' a-n j^a-na?, 

3,, 85.35, + a5.'a5, 


(8> 


ring equations (5) and (7), we have 




w _ 1 an 
a?/ 5, a?' 


(i- = 2,3 »). 


Lring equations (6) and (8), we have 




ar 1 an 
as, s.3i,' 


('•-1,2 «)■ 



42] Principles available for the integration 65 

Combining these with equations (2) and (3), we have 

— = — and ?^' = iM 

Substituting from these equations in the Lagrangian equations of motion, 
we obtain the system 

d (dL'\ . aX' / Q Q X 

dAd^r^'Wr''' (r«2,3,...,n), 

or finally . 

Now these may be regarded as the equations of motion of a new dynamical 
system in which L' is the kinetic potential, (q^, q^, ••• , qn) ore the coordinates, 
and qi plays the part of the time as the independent variable. The new system 
will, like the systems obtained by ignoration of coordinates, be in general 
non-natural, i.e. i' will not consist solely of terms of degrees 2 and in the ( (^-^y 
velocities (q^, q$, ... , qn); but on account of its possession of the Lagrangian 
form, most of the theorems relating to djniamical systems will be applicable 
to it. The integral of energy thus enables us to reduce a giveru\dyncmiical 
system with n degrees of freedom to another dynamical system with only (n — 1) 
degrees of freedom. 

The new dynamical system will npt in general possess an integral of 
energy, since the independent variable qi occurs explicitly in the new kinetic 
potential L\ But if qi is an ignorable coordinate in the original system, 
then ji will not occur explicitly in any stage of the above process, and there- 
fore will not occur explicitly in L\ From this it follows that the new system 
will also possess an integral of energy, namely 

2 g/ ;5—> — i' = constant, 
r^2 dqr 

and this can in its turn be used to reduce further the number of degrees of 
freedom of the system. 

The preceding theorems shew that any conservative dynamical system with 
n degrees of freedom and (n — 1) ignorable coordinates can be completely 
integrated by quadratures ; we can proceed either (a) by first performing the 
process of ignoration of the coordinates, so arriving at a system with only one 
degree of freedom, which possesses an integral of energy and can therefore be 
solved in the manner indicated at the beginning of the present article ; or 
(13) we can first use the integral of energy to lower the number of degrees of 
freedom by unity, then use the integral of energy of the new system to lower 
the number of degrees of freedom again by unity, and so on, obtaining finally 
a system with one degree of freedom which again can be solved in the manner 
indicated. 

w. D. 5 



6 Principles availabU for the integration [ch. m 

Example. The kinetic potential of a dy oamical syBtem is 

i-i/(ft)?i'+i?,'-*C?.). 

ition between the variables j, and q^ is given by the differential 

lere L' ie defined by the equation 

Q-natural dynamical system repre8ent«d by the last differential 
integral of energy, and hence Botve the system by quadratures. 

of the variables ; dynamicaX systems of LimtviUe's type. 
aical equations which are obviously aoluble by quadratures 
e equations of those systems for which the kinetic energy 

i fli (?i) 9i' + i »» (9.) 3.' + • ■ • + i v„ (}„) j„S 

lergy ie of the tbrm 

r= w, (9.) + w,(s,) + ... +«'»(9n), 

itf,, w, Wn are arbitrary fiiDCtions of their respective 

the kinetic potential breaks up into a sum of parts, each 

ily one of the variables. 

the Lagrangian equations of motion are 

k(9.)-9rl-i''r'(9r)9r' = -«'/{9r). (r = 1. 2, ... , «), 

«r (9r) 9r + i w; (9,) g." = - wj (Vr), (r = 1 . 2 n). 

, be immediately integrated, and give 

H(9r).9r'+«'--(9r) = c„ (r=l. 2. ...,«), 

are constaats of integration ; these equations can be 
since the variables qr and t are separable, and we thus 



-/k 



',(g.) 



■ d?r + 7r. {»- = l,2 n), 



I2c-2w,(s.)j 

are new constants of integration. These last equations 
on of the problem. 

EtensioD of this class of dynamical systems was made by 
3d that all dynamical problems for which the kinetic and 
in respectively be put in the forms 

;g,)+ ... +Wn(yn) 

g.) + -.+«»(9»)* 
idratures. 



V 



42, 43] Principles available for the inteffration 67 

For by taking 

J Vtv(g7) dqr = g/, (r = 1, 2, ... , n), 

where j/, Ja', ... , 9n' are new variables, we can replace all the functions 
Vi (qi)y Va (?j), • • • , Vn ( Jn) by unity ; we shall suppose this done, so that the 
kinetic and potential energies take the form 

F= - K(3i) + Wj(3a)+ ... + Wniqn)}, 
u 

where u stands for the expression 

The Lagrangian equation for the coordinate qi is 



dtKdqJ 9?i"" dqi' 



(U 

5^(t«Z.)-ig^/g.'+g.'+...+g,») = -g^^. 

Multipljdng this equation throughout by 2uqi, we have 

But from the integral of energy of the system, vre have 

where A is a constant. The equation for the coordinate qi can therefore be 
written in the form 

|(.V)-2(»-'')i.|_-2»j.| 

=a,,|_((»-F)») 



Integrating, we have 






iw« ji» = Aii, (g,) - Wi (gr,) + 7i , 



where 71 is a constant of integration. We obtain similar equations for each 
of the coordinates (ji, Ja, ... , ?«); the corresponding constants (71, 7a, ..., 7n) 
must satisfy the relation 

7i+7a+ ... +7n = 0, 

in virtue of the integral of energy of the system. 

5—2 



/ 



pies availcible for the integrcUion [oh. m 

give 

) + y,}-*dq, - (Am, (3,) - w,{q^) + y,]-idq, = ... 

^ {hUn{qn)-W«(qn) + yn]-*dqn, 

lations, which can be immediately integrated since the 
ated, furnishes the solution of the system. 



Miscellaneous Examples. 

Qta {J£, F) of the force actii^ on » particto of unit mass at the 
do Dot involve the time t, shew that b; elimination of I from the 
he solutioD of the problem ia mode to depend on the differeatiiil 



r-x 



e particles ie in motion, and their potential energy, which depends 
as, ia imaltered when the system in an; configuration is translated 
J distance in anj direction. What int^rals of the motion can 

systemwith two degrees of freedom the kinetic energy is 

y is 

istants. Shew that the value of q^ in terms of the time is given by 

Dnetants. 

ential of a dynamical system is 

constants : shew that q^ is given in terms of t by the equation 

constant and ^ denotes a Weierstraasian elUptic function. 

system with ignorable coordinates the kinetic energy is the sum of 

7* of the velocities of the non-ignored coordinates and a quadratic 



here are three coordinates x, y, ip and one coordinate <jt is ignored 

ns of motion of the type 

7"\ 37" Sir 3F , . , f 3 /3A\ 5 /Zi\) „ 



43] Principles available for (he integration 69 

where V is the potential energy, k is the cyclic momentum, and the differential coefficients 
of ^ with respect to x and y are calculated from the linear equation by which k is 
expressed in terms of k, y, ^. (Camb. Math. Tripos, 1904.) 

6. The kinetic potential of a dynamical system with two degrees of freedom is 

By using the integral of energy, shew that the solution depends on the solution of the 
problem for which the kinetic potential is 



x-(^.,...y, 



and by using the integral of energy of this latter system, shew that the relation between 
qi and q^ is of the form 

where c and c are constants of integration, and ^ denotes the Weierstrassian elliptic 
function. 

7. The kinetic energy of a dynamical system is 

and the potential eneigy is 

1 



F. 



«j-/»_«* 



2i^+qi 



Shew (by use of Liouville's theorem, or otherwise) that the relation between ^i 
and q^ is 

^i + ^2% + 2**^i ^8 cos y « sin* y, 
where a, 6, y are constants of integration. 

8. The kinetic energy of a particle whose rectangular coordinates are (^, y) is 
i(^^+J^')} and its potential energy is 

» 

where (il, A\ ^, jB', (7) are constants and where (r, r^ are the distances of the particle 
from the points whose coordinates are (c, 0) and ( — c, OX where c is a constant. Shew that 
when the quantities \{r-\-r^) and ^{r-r') are taken as new variables, the system is 
of Liouville's type, and hence obtain its solution. 



CHAPTER IV. 

THE SOLUBLE PROBLEMS OF PARTICLE DYNAMICa 

44. The particle vrUh one degree of freedom : the pendulum. 

As examples of the methods described in the foregoing chapters, we shall 
now discuss those cases of the motion of a single particle which can be solved 
by quadratures. 

We shall consider first the motion of a particle of mass m, which is free to 
more in the interior of a given fixed smooth tube of small bore, under the 
action of forces which depend only on the position of the particle in the tube. 
The tube can in the most general case be supposed to have the form of a 
twisted curve in space. 

Let s be the distance of the particle at time t from some fixed point of the 
tube, measured along the arc of the curve formed by the tube : and let f{s) 
be the component of the external forces acting on the particle, in the direc- 
tion of the tangent to the tube. 

The kinetic energy of the pai-ticle is 
J mi* 
and its potential energy is evidently 

- f/WA, 
where «, is a constant. The equation of energy is therefore 

i mi' =/'/(«)* + <=. 
where c is a constant. 

Integrating this equation, we have 

ds + l, 

where I ie another constant of integration. This equation represents the 
solution of the problem, since it is an integral relation between 8 and t, 
involving two constants of integration. 



=(i)7:i/>>-^ 



^>_ 



t y 



'^^ 



/ 



44] The SohMe Problems of Partide Dynamics 



71 






The two constants c and I can be physically interpreted in terms of the 
initial circumstances of the particle's motion ; thus if the particle starts at 
time ^ = <o from the point 8 = So, with velocity % then on substituting these 
values in the equation of energy, we have 

c = i mtt*, 

and on substituting the same values in the final equation connecting s and t, 
we have i = f^. 

The most famous problem of this type is that of the simple pendulum ; in 
this case the tube is supposed to be in the form of a circle of radius a whose 
plane is vertidal, and the only external force acting on the particle is gravity*. 
Using to denote the angle made with the downward vertical by the radius 
vector from the centre of the circle to the particle, we have 

8 — a0 and f(s) = — m,g sin ; 
so the equation of energy is 

ad^ = 2g cos + constant 

= — 4gr sin' ^ + constant. 

Suppose that when the particle is at the lowest point of the circle, the 
quantity -^^ has the value h. Then this last equation can be written 



,1 



>- 



.••*i 



2^ 







o»^ = 2gh - 4ga ain* | . 



Taking sin ^ = y, this becomes 




Now in the pendulum-problem there are two distinct types of motion, 
namely the " oscillatory," in which the particle swings to and fro about the 
lowest point of the circle, and' the "circulating," in which the velocity of the 
particle is large enough to carry it over the highest point of the circle, so 
that it moves round and round the circle, always in the same sense. We 
shall consider these cases separately. 

(i) In the oscillatory type of motion, since the particle comes to rest 
before attaining the highest point of the circle, y must be zero for some value 
of y less than unity, and therefore h/2a must be less than unity. Writing 

h = 2aJfc», 

where A; is a new positive constant less than unity, the equation becomes 



^-"^{^-^■m-th 



■A 






* In actual pendulums, the ttibe is replaced by a rigid bar connecting the particle to the 
centre of the circle, which serves the same purpose of constraining the particle to describe 
the circle. 



i 

A 



i 

-I 



72 The SohMe Problems of Particle Dynamics [ch. iv 

the solution of this is* 

where ^ is an arbitrary constant. 

This equation represents the solution of the pendulum-problem in the 
oscillatory case : the two arbitrary constants of the solution are ^ and k, and 
these must be determined from the initial conditions. From the known 
properties of the elliptic function sn, we see that the motion is periodic, its 
period (i.e. the interval -of time between two consecutive occasions on which 
the pendulum is in the same configuration with the same velocity) being 

. Jo 

(ii) Next, suppose that the motion is of the circulating type; in this 
case h is greater than 2a, so if we write 2a » hk^y the quantity k will be less 
than unity. 

The differential equation now becomes 

y'=^(i-y*)(i-*'y'). 

the solution of which is 



4 (-1 K, where 
<9 



''"{'JI'-t'-''}- 



and in this ^ and k are the two constants which must be determined in 
accordance with the initial conditions. 

(iii) Lastly, let h be equal to 2a, so that the particle just reaches the 
vertex of the circle. The equation now becomes 

or y^'Vfa-n 

the solution of which is 



y = tanh 



{v/f(e-0}. 



It was remarked by Appellf that an insight into the meaning of the imaginary period 
of the elliptic functions which occur in the solution of the pendulum-problem is afibrded 
by the theorem of § 34. For we have seen that if the particle is set free with no initial 
velocity at a point of the circle which is at a vertical height k above the lowest point, 
the motion is given by 

y « A sn |^(« - «,), *} , where *»- A ; 

* Gf. the aathor'8 Course of Modem AnalyiU, § 189. \ 

t CompUt Bendus, Vol. 87 (1878). \ 



V 






44,46] The Soluble Problems of Particle Dynamics' 78 

and therefore by § 34, if, with the same initial conditions, gravity were supposed to act 
vpwardsy the motion would be given by 

y^JcNi |i ^1 (t-to), ir| . 

But the period of this motion is the same as if the initial position were at a height 
(2a— A), gravity acting downwards: and the solution of this is 

y=iPsn|^(r-ro),iP|, where it^=l-ife«. 

The latter motion has a real period 4 (-) ^ ; and therefore the function 

must have a period 4 [ - ) K\ so the function sn (t<, k) must have a period AiK', The 
double periodicity of the elliptic function sn is thus inferred from, dynamical considerations. 

Example, A particle of unit mass moves on an epicycloid, traced by a point on the 
circumference of a circ}e of radius h which rolls on a fixed circle of radius a. The particle 
is acted on by a repulsive force fir directed from the centre of the fixed circle, where r is 
the distance from this centre. Shew that the motion is periodic, its period being 

'(a+26)>-a«U 



sn 



%ir 



f (a+26)'-a« )J 
I M«' J 



[This result is most easily obtained when the equation of the epicycloid is taken in the 
form 

4 being the arc measured horn, the vertex of the epicycloid.^ 

46. Motion in a moving tube. 

We shall now discuss some cases of the motion of a particle which is free 
to move in a given smooth tube, when the tube is itself constrained to move 
in a given manner. 

(i) Ttibe rotating uniformly. 

Suppose first that the tube is constrained to rotate with uniform velocity 
«) about a fixed axis in space. We shall suppose that the particle is of unit 
mass, as this involves no real loss of generality. 

We shall moreover suppose that the field of external force acting on the 

particle is derivable from a potential-energy function which is symmetrical 

with respect to the fixed axis, and so can be expressed in terms of the 

cylindrical coordinates z and r, where z is measured parallel to the fixed 

axis and r is the perpendicular distance from the fixed axis ; for a particle 

in the tube, this potential energy can therefore be expressed in terms of 

the arc 8 : we shall denote it by F(«), and the equation of the tube will be 

written in the form 

r=g(8). 



e Problems of Partide Dynamics [ch, iv 

»f the particle Ib the same as if the prescribed angular 
[ the potential eoei^ were to contain an additional 
i can at once write down the equation of energy in the 

iJ--i«.-|y(.)l' + r{.)-o, 

i have 

+ afl{g («)[« - 2F (s)]-* ds + constant, 

n t and s represents the solution of the problem. 

ating tube is plane, and the particle can describe it witb 
Lied axis in vertical and in the plane of the tube, and the leld 
ty, ehew that the tube must be in the form of a paiabola with 
[ownwards. 

moves under gravity in a circular tube of radius a which 
ixed vertical axis inclined at an angle a to its plane ; if be 
xuticle from the lowest point of the circle, shew that 

led with the roots 



'ih constant acceleration parallel to a fixed direction. 
)tion of a particle in a straight tube, inclined at an 
al, which is constrained to move in its own vertical 
izoDtal acceleration/ 

horizontal and that of y vertically upwards, with the 
tion of the particle, we have for the kinetic energy 

: y cot o + \ft\ 

= Ji/* cosec* a + y cot a .yi + i/'P, 

MS 



dt\dyj dy ' 



I 46, 46] The Soluble Problems of Particle Dynamics 76 

I gives therefore 

-^ (y cosec' a -vft cot «) = — 5^, 

or y = (— ^— /cota)8in*a. 

Integrating, we have, supposing the particle to be initially at rest, 

y = ^^« (— ^ sin a —/cos a) sin a, 
and therefore 

a? = i ^' (— flf cos a +/sin a) sin a. 

These equations constitute the solution of the problem: it will be observed 
that in this system thg kj^ifttif* ft^f^^gY ipv^l^p^-^ the time explicitly, so no 
integ ^l of energy exists . 

46. Motion of two interacting free particles. 

We shall next consider the motion of two particles, of masses roi and m^ 
respectively, which are free to move in space under the influence of mutual 
forces of attraction or repulsion, acting in the line joining the particles and 
dependent on their distance from each other. 

The system has six degrees of freedom, since the three rectangular coordi- 
nates of either particle can have any values whatever. We shall take, as the 
six coordinates defining the position of the system, the coordinates (X, F, Z) 
of the centre of gravity of the particles, referred to any fixed axes, and the 
coordinates (a:, y, z) of the particle m, referred to moving axes whose origin is 
at the particle ttii and which are parallel to the fixed axes. 

The coordinates of ?ni, referred to the fixed axes, are 

\ mi + Wa ' Wi -h r/ia ' m^ + rn^j ' 

and those of 77^, referred to the fixed axes, are 

\ mj + TWj mi + TTia 7?ii + mj/ 

The kinetic energy of the system is therefore 

*V mi + ?nB/^*V mi + mj/ *^V m^-\-niJ 

* \ mi + m,/ * \ mi + m^J * \ mi + mj 
or T^\{m,-¥m,){X'-\-Y^ + t)^-^—'^{a^^'y^^'Z''). 

^ ^ ^ ^^ '^mi + 7Wa ^ 

The potential energy of the system depends only on the position of the 
particles relative to each other, so can be expressed in terms of (x, y, z) ; let 
it be V(x, y, z). 



ble Problems of Particle ]>ynamic8 [ch. if 
juations'of motion of the system are 

7 = 0, ^ = 0, 

x' TOi + m, 3y ' ni,+m, 3« ' 

these equations shew that Oie centre of gravity moves in 
wfoTrm, velocity, and the other three equations shew that 
Hve to m^ia the mme as if nii were Jioced and m, viere 

8 fane derived from the potential energy V. 

e porttclea move in space under vay law of mutual attraction, 
bo their paths meet an arbitrary fixed plane in two points, the 
I through a fixed point. (Mehmke.) 

s tn general : Hamilton' a theorem. 

lewB that the problem of two interacting &ee particles 
iblem of the motion of a single fi-ee particle acted on by 
ds or from a fixed centra. This is known aa the prtHem 
lere is clearly no loss of generality if we suppose the 
[> be unity. 

projected in any way, it will always remain in the plane 
the centre of force and the initial direction of projec- 
loes any force act to remove it from this plane. We can 
osition of the particle by polar coordinates (r, 6f) in this 
force being the origin. Let P denote the acceleration 
! of force. We shall for the present not suppose that P 
on of r alone, 
y of the particle is 

)y the force in an arbitrary infinitesimal displacement 

-P&r. 
quations of motion of the particle are therefore 

)n gives on integration 

7^6 = h, where A is a constant ; 

)rreBponding to the ignorable coordinate 6, and can be^ 
1 as the integral of angular momentum of the particle 
>rce. 



> 



46, 47] The Solvble Problems of Particle Dynamics 77 

To find the differential equation of the path described (which is generally 
called the orbit or trajectory), we eliminate dt fix)m the first equation by using 

the relation 

d h d ^ 

dt^r^dO' 
we thus obtain the equation 

r^ de\r^ d0) 7^" ' 
Gty writing u for 1/r, 

d^u P 



-Tn^ +W = 



This is the differential equation of the orbit, in polar coordinates; its 
integration will introduce two new arbitrary constants in addition to the 
constant A, and a fourth arbitrary constant will occur in the determination of 
t by the equation 

^ = T i'l^dO + constant. 

The differential equation of the orbit in (r, p) coordinates, (where p 
denotes the perpendicular from the centre of force on the tangent to the 
orbit), is often of use : it may be obtained directly from Siacci's theorem 
Tj.fc.to (§ IS), which (since h is now constant) gives at once 

p* dr* 
which is the differential equation of the orbit. 

Since h^vp, where v is the velocity in the orbit, we have from this equation 

r ' 
which may be written in the form 

where q is the chord of curvature of the orbit through the centre of force. 

We frequently require to know the law ofjorce which must act towards a 
given point in order that a given curve may be described ; this is given at once 
by the equs^tion 

if the equation of. the curve is given in polar coordinates ; while if the equa- 
tion is given in (r, p) coordinates, the force is given by the equation 

p^dr' 



78 The Soluble Problems of Partide Dynamics [oh. nr 

If the equation of the curve is given in rectaugular coordinates, we pro- 
ceed as follows : 

Take the centre of force aa origiD, and let f{x, y) = be the equation of 
the given curve. The equation of angular momentum is 

Differentiating the equation of the curve, we have 

fx ■ ^ +/y • y = 0, where /, stands for ^ . 

ox 

From these two equations we obtain 

-V, . _ kf, 

*/« + «/■/ ^ ^A + Sfy' 

Differentiating again, we have 

Perfonning the differentiationB, thia gives 

_ i<-''<.-f,'U+V.U„-f.'i„ ) 

But the required force ia P, where S = — i* - ; and therefore we have 

p . tV (/,-/_ -2/././, +/.■/„) . 
{'f. + nf.f 
this equation gives the required central force. 

The moat important case of this result is that in which the curve 
f(x, y) = is a conic, 

y(x, y) = aa?-v Zkx;/ + fcy" + 25a; + 2/y + c = 0. 
In this case we find at once that the expression 

/«/,--2/-™/./,+/»/.' 
has, for points on the conic, the constant value 

- (ahc + 2/gh -ap-bg'- cA'), 
while the quantity 

has the value 

and so is a constant multiple of the perpendicular from the point (x, y)on the 
polar of the origin with respect to the conic. We thus obtain, for the force 
under which a given conic can be described, an elegant espresaion due to 
Hamil ton *, namely that the /orce acting on the particle tft ^ position {x, y) 
' Proe. Boy- Iriib Acad. IM6, 



f Particle Dynamics 79 

mire of force to ifu poivi (a:, y), and 
from {w, if) on the polar of the centre 



lich is left to the student, maj together be 

F & force directed to a fixed point, vaiTing 
uid ioveraely aa the cube of the distance 

n of a foroe directed to the ongtn, of 

a, &,y are constants, the orbits are conies 

w"-o. 

B shewn that these two laws of force are 



ExampU 1. If a conic be deacribed under the force ^ given 1^ Hamilton's theorem, 
shew that the periodic time is -^ p^, where p^ is the perpendicular from the oen^ of i 
the conic on the polar of the centre of force, (Olaisher.) 

Example 8. Shew that if thp force be i 



/ 



a particle will deacribe a conic having its asTmptotea parallel to the lines 

if properly projected. (Glaisher.) 

48. The integrable cases of central forces; problems soluble in terms of 
cireidar and eltipUe functions. 

• The most important case of motion under central forces is that in which 
the magnitude of the force depends only on the distance r. Denoting the 
force by/(r), the differential equation of the orbit ia 

Integrating, we have / 



integrating this equ 



where c ia a constant : integrating this equation again, we have 
l)-idr 



JO The Soluble Prdblemg <^ Partide Dynamics [ce. it 

LTirl thia in tht^ equation of the orbit in polar c^^rtfioatee. When r has been 
I equation in terms of 6, the time ib given by the integral 



'ir 



r*d5 + cc«iatanfc. 



t of motion under central forces ia tkertfot%^ always soluble by 
en the force is a fimction only of the distant We shall now 
is in which the quadrature can be effected iit-ffflTUB of known 
central force being supposed to vary as tnoo^ positive or 
:al power, — say the nth, — of the distance. \ '•- 
find those problems for which the integration can be effected 
ular functions. The above integral for the determination of 
in the form 



0= l(a + bu* + cu""*-')"* du. 



■e constants ; except when n ™ — 1, when a logarithm replaces 
'~'. If the problem is to be soluble in terms of circular funo- 
omial under the radical in the integrand must be at most of the 
this gives 

-n-l = 0,l, or 2, 

ly 

» = -l,-2, or -3. 
= — 1 is hovever excluded by what has already been said, and 
lias to be added, since in this case the irrationality becomes 

u' is taken as a new variable. 

find the cases in which the integration can be effected by the 
unctions. For this it is necessaiy that the irrationality to be 
lid be of the third or fourth degree* in the variable with 
\i the integration is taken. But this condition is fulfilled if 
!, or — 5, when u is taken as the independent variable ; 
, or — 7, when u* is taken as the independent variable. 
lat the problein of motion under a central force which varies as 
\f the distance is soluble by circular or elliptic functions in the 

n = 5, 3, 1, 0,-2, -3, -4, -5, -7. 

;w that the problem is soluble by elliptic fanotions when n has the 
ll values : 

»--3. -i. -J. -S. -i- 

1 which motion under a central force varying as a power of 
soluble by means of circular functions are of special interest, 
d, as shewn above, to the values 1, — 2, — 3, of n ; the case 
■ Whittaker, Hodtm Analytu, gg 164—186. 



\ 






48] TAc Soluble Problems of Particle Dynamics 81 

n = — 2 will be considered in the next article : the cases n = 1 and n = - 3 
can be treated in the following way. 

(i) n=l. 

In this case the attractive force is 

/(>•)-/"■, 

so the equation of the orbit becomes 

du, 



--i ['(»»-*-»■) dv. 



2(^-7) = oos- 






This is tbe equation of an ellipse (when ^ > 0) or hyperbola (when /* < 0) 
referred to its centre. The orbits are there/ore conic* whose centre w at 
the centre offeree. 

(ii) n = -3. 

the attractive force ia 

of the orbit becomes 

we have 

= 4co8 {k6+e), where A"=l-^,, when fi.<h\ 

= ^cosh(fe5+e), where 4* = ^ — 1, when >t>A', 

= A8 + e, when m"^*i 

,8e A and e are constants of integration. 

s are sometimes known as Cotes' smralf ; the . last is the 



I Problems 0/ Particle Dynamics [oh. it 

uying aa the inTsrse cube of the distance, it ma; be obeerred 

central force P(r) to the origin, then the orbit 

r./(M), 
a be dmcribed under a central force P(y)+-^, where e ia 

time between corresponding pointa, L& points for which the 
lue, in the two orbita being the same. 
iter to the second orbit, we have 



P-h" 



'"""("^S'*) 



=A'V+^(P-AV). 
le new constant of momentum h' so that 

le intervals of time between corresponding points in the two 
aa be wntten ;^ = ;^ )) we have 

This ia sometimes known aa Nmeton't theorem of rwolving 

notion corresponding to 

n = 5, 3, 0, - 4, - 5, - 7 

to elliptic integrals ; on inverting the integials, we 

srms of elliptic functions. As an example we shall 

towards the centre of attraction ; we shall snppoee 
ected with a velocity less than that which would be 
»t at an iofiDite distance to the point of projection, 

intity — J7. Then the equation of energy 
an 



48] The Soluble Problems of Particle Dynamics 83 

Introducing in place of r a new variable p defined by the equation 






the differential equation becomes 

The roots of the quadratic 

'^ 3 9 2A* 

are real when 7 is positive ; theii* sum is ^, and the smaller of them is less 
than — ^. Hence if the greater and less of the roots be denoted by ei and €% 
respectively, and if 0, denotes — ^, we have the relations 

^ + ^ + ei=0, 
6i>e,>ei, 

(^)"-4(p-e,)(p-e.)0>-^), 

so p = jp(^-€), 

where e is a constant of integration, and the function ^ is formed with the 
roots Sit 6^, ^. Thus we have 






Now r is real and positive, and, as we see from the equation of energy, 

cannot be greater than a/^* So f^(^— €) + ^ is real and positive and 

has a finite lower limit; but when ei>^9>^, the function ff{0 — €) is real 
and has a finite lower limit for all real values of only when e is real ; 
so e is purely real, and by measuring from a suitable initial line we can take 
€ to be zero. We have therefore 

1 



-gy 



h {«> {6) + 41* ' 
and this is the equation of the orbit in polar coordinates. 

The time can now be determined from the equation 



t 



= Ih'' 



Mf I dd 
or t 



ih'Jt 



6—2 



■'^■^, 



•'^ 



.»" 



11 



*/ 



M 



^1 



t 



/ 



"! 



^ luble Problems of Particle Dynamics [ch. iv 
integration, we have 

P'eierstraseian zeta-fuoction*. This equation determinea t 

that the equation of the orbit of a particle which movee under the 
attractive force ulr* can be writteti in the form 

?-'"('-3ra'')' 

, where A ia the angular momentum round the origin and E in the 
irg7 over the potential energ; at infinitj. 

(Cambridge Math. Tripos, Part I, 1894.) 
tide u attracted to the origin with constant acceleration ft; shew 
, vectorial angle, and time, are given in tenns of a real auxiliary 
of the type 

-i>(™+-i)-i'(-i+«). 

r-iC(«t+i«)+«|»c-.+<i)-iCC,,). 

,_ -^t^^, »{»,+iu+t+<')''(»i-»t'<^) (Schoute.) . 

<T(«,+.u-«,-a)<T(«,+«,+a) 

its of special interest oq an orbit are the points at which 
fter having increased for some time, begins to decrease : 
tcreased for some time, begins to increase. A point 
mer of these classes is called an apocerUre, while points 
re called periceatrea ; both classes are included under the 
At an apse, if the apse is not a singularity of the orbit 
e 

%-"■ 

ngent to the orbit is perpendicular to the radius vector. 
ion and perihelion are generally used instead of apocentre 
I the centre of force is supposed to be the Sun. 
le movea under an attraction 



that the angle subtended at the centre of force by two consecutive 



^T. 



it of angular momentum. 

• Cf. Whittokar, Modent Analsii$, SS 209, 314. 



i^^^'l) 



48, 49] The Solvble Problems of Particle Dynamics 86 

49. Motion under the Newtonian law. 

The remaining case in which motion under a central force varying as an 
integral power of the distance can be solved in terms of circular functions is 
that in which the force varies as the inverse square of the distance. This 
case is of great importance in Celestial Mechanics, since the mutual attractions 
of the heavenly bodies vary as the inverse squares of their distances apart, in 
accordance with the Newtonian law of universal gravitation. 

(i) The orbits. 

Consider then the motion of a particle which is acted on by a force 

directed to a fixed point (which we cau take as the origin of coordinates), of 

magnitude fiu\ where u is the reciprocal of the distance from the fixed point. 

Let the particle be projected from the point whose polar coordinates are 

(c, a) with velocity Vq in a direction making an angle y with c; sd that the 

angular momentum is 

h s cvo sin 7. 

The diflferential equation of the orbit is 

d^u P fi 



d^ AV VoV 8in« 7 ' 

this is a linear differential equation with constant coefficients, and its 
integral is 

^- o ^^' « (1 -f e cos (g - w)l, 

where e and v are constants of integration. This is the equation, in polar 
coordinates, of a conic whose focus is at the origin, whose eccentricity is e, 
and whose semi-latus rectum I is given by the equation 

I tyo*c'sin*7 
A* 
the constant w determines the position of the apse-line, and is called the 
perthelion-constant 

The circuinBtanoe that the focus of the conic is at the centre of force b in accord with 
Hamilton's theorem ; for if the centre of force is at the focus of the conic the perpen- 
dicular on the polar of the centre of force is the perpendicular on the directrix, which is 
proportional to r, as by Hamilton's theorem the force must be proportional to l/f>. 

To determine the constants e and cr in terms of the initial data c, a, 7, Vq, 

we observe that initially 

zi I du I . 

substituting these values in the equation of the orbit and the equation 
obtained by differentiating it with respect to 6, we have 

Vo*c sin' 7 = ^ + /Lt6 cos (a — «•), 

Vo'c sin 7 cos 7 = /Lt6 sin (a — «•). 



i 



I%e Solidtle Problems of Particle Dynamics [ch. it 

; these equations for e and w, we obtaiD 

i. , Vt*(? sin' 1 2vJc sin' y 
/*' M 

cot (a — «r) = — ;-^ h tan 7. 
CO,' 8in ly cos 7 ' 

ui-niajor axis, when the conic is an ellipse, is generally called the 
nee of the particle ; denoting it by a, we have 
I 

uting the values of I and e* already found, we have 

"•'-"(!-;)■ 

>n determines a in terms of the initial data. 

le occupied in describing the whole circumference of the ellipse, 

nerally called the periodic time, is 

r X area of ellipse, 
«Bent6 twice the rate at which the area is swept out by the radius 
t periodic time is therefore — ,— , where b is the semi-minor axis. 
ve 

A = tJjC sin 7 = V^Z = 6 a/ ^ . 

xiic tiTne it Sir a/ — . It is usual to denote the quantity /«'a~* 
leriodic time can then be written 



he mean mation, being the mean value of 6 for a complete period. 

1 shewQ bj ^rtrand apd Koeni ga that of all laws of force which give a zero 
iiiito dietADce, the Newtonian law ia the only odb for which all the orbits are 
es, and also Uie only one for which all the orbits are closed curves. 

Shew that if a centre of force repels a particle with a foroe varying as the 
9 of the distanoe, the orbit is a branch of a hyperbola, described about 



1 vdociiy. 

• now the case in which the orbit Is an ellipse ; the equation 



---0-S- 



49] The Soluble Problems of Partide Dynamics 87 

establishes a connexion between the mean distance a and the velocity Vq and 
radius vector c at the initial point of the path. Since any point of the orbit 
can be taken as initial point, we can write this equation 

where v is the velocity of the particle at the point whose radius vector is r. 
Similarly if the orbit is a hyperbola, whose semi-major axis is a, we find 

and if the orbit is a pambola, the relation becomes 

r 

It is clear from this that the orbit is an ellipse, parabola, or hyperbola, 

2u 
according as V ^ — , i.e. according as the initial velocity of the particle is 

c 

less than, equal to, or greater than, the velocity which the particle would 
acquire in failing from a position of rest at an infinite distance from the 
centre offeree to the initial position. 

It can further be shewn that the velocity at any point can be resolved into 
a component r perpendicular to the radius vector and a component ^ perpen- 
dicular to the axis of the conic ; each of these components being constant. 

For if /S be the centre of force, P the position of the moving particle, 

the intersection of the normal at P to the conic with the major axis, OL 

the perpendicular on 8P from 0, and /SFthe perpendicular on the tangent at 

P from S, it is known that the sides of the triangle SPO are respectively 

perpendicular to the velocity and to the components of the velocity in the 

two specified directions ; and therefore we have 

,. , ,. J- ^ v.SP h.SP h 

Component perpendicular to the radius vector = ~pfr = ay PQ ~ PL 

h fi 

SO 
and Component perpendiculai' to the axis = ^ x Component perpendicular 

to the radius vector 

which establishes the result stated. 

Example 1. Shew that in elliptic motion under Newton's law, the projections, on the 
external bisector of two radii, of the velocities corresponding to these radii, are equaL 
Shew also that the sum of the projections on the inner bisector is equal to the projection 
of a line constant in magnitude and direction. (Cailler.) 



e Solvble Froblems of Partide Dynamics [cel it 

Shew that in elliptic motion under Newton's law, the qii&ntitj I Tdt, 

the kinetic enei^, iDtegTat«d orer a complete period, depends only od 
Be and not on the eccentricity. (Qrinwis.) 

At a oert^ point in an elliptic orbit described under a force fi/f^, the 
denlf changed by a small amount. If the eccentricities of the former and 
ual, shew that the point is an eitremity of the minor axis. 

momaliex in elliptic motion. 

i is describing an ellipse under a centre of force in the focus S, 
igle ASP of the point P at which the particle is situated on 
isured from the apse A which is nearer to the focus, is called 
jly of the particle and will be denoted by 6 ; the eccentric 
nding to the point P is called the eccentric anomaXy of the 
ill be denoted by u : and the quantity nt, where n is the mean 
is the time of describing the arc AP, is called the mean 
B particle. We shall now find the connexion between the 



1 between and u is found thus : 



r = a—ex, where x is the rectangular coordinate of P referred 
to the centre of the ellipse as origin, 
r = a(l — ecoBii). 
(1 — ecosu)(l + ecoa ff) = 1 — c", 
ich can also be written in the forms 
. « /l-e\t e 

sm u = V; H- ■ 

I + e cos p 

X between u and nt can be obtained in the following way : 

3P = — T . - X Area ASQ, where Q is the point on the auxiliary 

circle corresponding to the point P on the ellipse 
2 
= — J (Area JCQ-AreaSCQ}, where C ia the centre of 

the ellipse 

2 fa' a*e . \ 



49] The Soluble Problems of Particle Dynamics 89 

Lastly, the relation between and nt can be found as follows : 

We have 

nt^U'-e&mfL 

Replacing u by its value in terms of 0, this becomes 

( 1 + ecostf J 1 +6C0S^ ' 

which is the required relation ; this equation gives the time in terms of the 
vectorial angle of the moving particle. 

Example 1. Shew that 






OS I 

u=nt'\-2'% '■Jr(re)Bmmtj 



cohere the tymbols J denote Beesd coefficients. 

For we have 
1 du 1 



n dt 1 -ecoau 



_ I fu d{ru) 5 oo8«U f».ooem<.rf(«<) j.^^^^ ^^^^^^ 
2iryo \-eco%u r=i ir yo l-ecosw "^ 

~5~ / ^**+ ^ / cos {r (« - « sin t«)} (iw 

*fr y r-i IT y 



= 1+2 2 Jr{re)QOAmti. 

lDt^i;rating, we have the required result. 
Example 2. Shew that 



50* 
B'^nt^^ sin nt+-r «ixi 2nt+... . 

4 



.^cample 3. In hyperbolic motion under the Newtonian law, shew that 



B . ,a . B^ 



(.+ l)*co8^-(.-l)*sin^ 

fi V<-log) f rl+«(c8-i)* ""'^ 

% A t Al 1 



(e+l)*C08^+(«-l)*8in|j 



+0COB^' 



and in parabolic motion, shew that 

where p is the distance from the focus to the vertex. 

Example 4. In elliptic motion under Newton's law, shew that the sum of the four 
times (counted from perihelion) to the intersections of a circle with the ellipse is the same 
for all concentric circles, and remains constant when the centre of the circle moves 
parallel to the major axis. (Oekinghaus.) 

* Whittaker, Modem AnalytU, § 82. 
t Ibid., § 158. 



90 The Soluble Problems of Particle Dynamics [ch. iv 

(iv) Lamberfs theorem. 

Suppose Qow that it is required to express the time of describiDg any arc 
of an ellipse under the Newtonian law, in terms of the focal distances of the 
initial and final points, and the length of the chord joining them. 

Let u and u' be the eccentric anomalies of the points ; then we have 
n X the required time = u' — e sin u' — (u — e sin it) 

= (u' — w) — Zesin — 5 — cos -^ — , 

Now if c be the length of the chord, and r and r' be the radii vectorea, we 
have 



d c' = a'(cosM'-coau)' + 6'(sin«'-8inw)' 

. , . , u' - u /, , ,u + u'\ 
= ia' sin' — s— 1 1 — e* cos' — ^ — I , 

-.2sm- ^-(l-e'cos'-.g- j. 

Hence we have 

r + r' + e 



= 2 - 2 cos |- g- + cos-' (e cos " tJi^l . 

J r + r'-c c> a ("'-", .(' M + u'\l 

and ■ — = 2 - 2 COS j s— + cos"' I « cos —3— If ■ 

and therefore* 

a ■ _,l/»- + / + c\* u'-u ,/ u + u'\ 

^'"^ 2[—^~) ^ + cos-'(ecoB-2-). 

and 2«n'g^— ^J ^+cos'(^*co8 ^-j. 

Thus if quantities a and are defined by the equations 



2 
the last equations give 



l/r + r' + c\i . 8 l/r + Z-cV 



_o , J a + yS u + m' 

a - p = « — «, and cos — „— = e cos — a— . 

Thus finally we have 

nxthe required time = a— j9 — 2co8 — „-- sin —^ , 
-(o- sin a) -09 -sin ,8). 
This result is known as Lambert's theorem. 

' It will be notioed th»t owing to the preaenoe of (he ndioals, Lambert'B theorem U not frea 
from Bmbignitj of Bign. The reader will be able to determine without diffloalt; the inteipretatioi) 
of Btgn oorresponding to txij given position of the initiki ftnd final pointa. 



49, 50] The Soluble Problems of Particle Dynamics 91 

Example 1. To obtain the form o/Lamberfs theorem applicable to parabolic motion. 

If we suppose the mean distance a to become large, the angles a and ^ become veiy 
small, so Lambert's theorem can be written in the approximate form 

Required time = — ^-^ > 

-©'U(^")*-C-^)'}' 

and this is the required form. 

Example 2. Establish Lambert's theorem for parabolic motion directly from the 
formulae of parabolic motion. 

Example 3. Apply Lambert's theorem to prove that the time of falling vertically 
under gravity through a distance c is 



^©'H-(e)'*?^'}' 



where a is the distance from the centre of the earth of the starting-point and g the value 
of gravity at this point (ColL Exam.) 

60. The mutual transformation of fields of central force and fields of 
parallel force. 

If in the general problem of central forces we suppose the centre of force 
to be at a very great distance from the part of the field considered, the lines 
of action of the force in different positions of the particle will be almost 
parallel to each other; and on passing to the limiting case in which the 
centre of force is regarded as being at an infinite distance, we arrive at the 
problem of the motion of a particle under the influence of a force which is 
always parallel to a given fixed direction. 

For the discussion of this problem, take rectangular axes Ox, Oy in the 
plane of the motion, Ox being parallel to the direction of the force ; and let 
X (x) be the magnitude of the force, which will be supposed to be independent 
of the coordinate y. The equations of motion are 

x=^X(x), y = 0, 

and the motion is therefore expressed by the equations 



« = ay + 6=| {2jX{x)dx'\'c]-^dx-^l, 



where a, 6, c, I are the constants of integration ; the values of these are 
determined by the circumstances of projection, i.e. by the initial values of 
ar, y, X, y. 

While the problem of motion in a parallel field of force is a limiting case 
of the problem of motion under central forces, it is not difficult to reduce the 
latter more general problem to the former more special one. 



^ 



!fte 8(dv^le Problems of Particle Dynamics [oh. it 

Mrticle is in motion under a force of magnitude P directed to a 
(which we can take as origin of coordinates), the equations of 



liar momentum of the particle round the origin (which is constant) 
et this be denoted hy h. Introduce new coordinates X, Y, defined 
'graphic trausformatioQ 



a new variable defined by the et]uation 



have 



2*- I s 



dX 

We 


4© 


dt 


dT 

ST' 


4© 


dt 
dt 


irx 
dp' 


■0, 


dry 
dP 



[dt 



=(f- 



y 

[uatioas shew that a particle whose coordinates are (Z, 7) would, 
iterpreted as the time, move as if acted on by a force parallel to 

r and of magnitude ^ . As the solution of this transformed 

1 yield the solution of the original problem, it follows that the 
item of motion under central forces is reducible to the problem of 
parallel field of force. 

[. Shew th&t tb« path of a fre« particle movii^ under the influence of 
is a parabola with its axis vertical nod vertei^ upwards. 
i. Shew that the magnitude of the force parallel to the axis of x under 
rve/(x,y)=0 can be deacribed is a constant multiple of 

3. If a parallel field of force is such that the path described bj a free 
conic whatever be the initial oonditiona, shew that the force varies as the 
>f the distance from some line perpendicular to the direction of the force. 

nvefs theorem. 

proceed to discuss the motion of a particle which is simultaneously 
r more than one centre of force. An indefinite number of parti- 
of motion of this kind can be obtained by means of a theorem due 
which may be stated thus : 



60-62] The Soluble Problems of Particle Dynamics 93 

If a given orbit can he described in each of n given fields of force, taJcen 
separately, the velocities at any point P of the orbit being Vi, v^, ... Vn, 
respectively, then the same orbit can be described in the fisld of force which 
is obtained by superposing all these fields, the velocity at the point P being 

(V + V+... + V)*. 

For suppose that in the field of force which is obtained by superposing 
the original fields, an additional normal force 22 is required in order to make 
the particle move on the curve in question; and let it be projected fix>m 
a point A so that the square of its velocity at A is equal to the sum of the 
squares of its velocities at A in the original fields of force. Then on adding 
the equations of energy corresponding to the original motions, and comparing 
with the equation of energy for the motion in question, we see that the 
kinetic energy of the motion in question is the sum of the kinetic energies 
of the original motions, i.e. that the velocity at any point P is 

Hence, resolving along the normal to the orbit, we have 

P 

where m is the mass of the particle, p the radius of curvature of the orbit, 
and Fx, F^, ... Fn are the normal components of the original fields of force 
at P. 

J.. mvi^ „ mvf „ mvn* j. 

P P P 

and therefore 22 is zero ; the given orbit is therefore a free path in the field 
of force which is obtained by superposing the original fields. 

Example. Shew that an ellipae can be described if fwoee 

reepecHvely act in the directions of thefocL 

This result follows at once from Bonnet's theorem when it is observed that the given 
forces are equivalent to forces ^ and ^ acting in the directions of the foci, together with 

a force -^ x distance, acting in the direction of the centre of the ellipse. 

62. Determination of the most general field offeree under which a given 
curve or family of curves can be described. 

m 

Let ^{x, y)^c 

be the equation of a curve ; on varying the constant c, this equation will 
represent a family of curves. We shall now find an expression for the 
most general field of force (the force being supposed to depend only on the 



ble ProMema of Particle Dynamics [ch. iv 

:le on which it acts) for which this fomilj of curves 
of a particle. 

velocity of the particle, and {X, Y) the components of 
parallel to the coordinate axes. The tangential and 

acceleration being = -7- and - reepectively, we have 

^ *•(+."+ w - 1 ^ *,(*••+ *.■)-». 
^ *, (*^ + w)-' + 1 ^ *.(*••+ *.")-*■ 

its value, namely 

(■!>.■ + «' 

!' = -»(*.' + *»'). 

^«' + <^')~*(^ar~^ a;)« *''*'^ equation becomes 

, since it depends on the velocity with which the given 
and as X and Y are to be functions of the position 
an take u to be an arbitrary function of x and y; 

" («MVi' - «/v*w) + i ^ C**"!- - «/v«-). 

ry (unction of w and y. These expressions' for the field 
he curves of the given family are orbits were first given 



( a partidt can deteribe a ifiven curve under any arbUrary fitrctt 
ifixadpoinU, provided theteforcti tatufj/ the relation* 

kpk^dt\ r^ ) 
pi the perpendicular on the tangent, from the t* of the given fixed 
■adiui of eurvalure of tha given carve. 
aono&l componeDts of force on the particle ai« 



-sp.g .nd ir-ip,^. 



62, 53] The Soluble Problems of Particle Dynamics 95 

80 from the equation 
we have 



OP 






Example 2. A particle can describe a given curve under the single action of any one 
of the forces ^n ^, ...| acting in given (variable) directions. Shew that the condition to 
be satisfied in order that the same curve may be described under the joint action of forces 
^19 ^ti •••! acting in the directions of ^i, <^, ..., respectively, is 



s,^M(g)=o. 



where cii is the chord of curvature of the curve in the direction of <l>k. (Curtis.) 

Example 3. A point moves in a field of force in two dimensions of which the work 
function is V; shew that an equipotential curve is a possible path, provided F satisfy the 
equation 

«-/<') 0&7-'i5 Ti^'^m* m'^m'- «- --' 

53. The problem of two centres of gravitation. 

The equations of motion of a pai'ticle moving in a plane under arbitrary 
forces cannot be integrated by quadratures in the general case. The most 
famous of the known soluble problems of this class, other than problems of 
central motion, is the problem of two centres of gravitation, i.e. the problem of 
determining the motion of a free particle in a plane, attracted by two fixed 
Newtonian centres of force in the plane ; its integrability was discovered by 
Enler *. 

Let 2c denote the distance between the two centres of force ; and take 
the point midway between them as origin, and the line joining them as axis 
of Xj so that their coordinates can be taken to be (c, 0) and (— c, 0). The 
potential energy of the particle (whose mass is taken to be unity) is therefore 

where /a and jjf are constants depending on the strength of the centres of 
attraction. 

Now any ellipse or hyperbola with the two centres of force as foci is a 

possible orbit when either centre of force acts alone, and therefore by Bonnet's 

theorem it is a possible orbit when both centres of force are acting. It 

is therefore natural, in defining the position of the particle, to replace the 

rectangular coordinates (x, y) by elliptic coordinates (^, 97), defined by the 

equations 

a? = c cosh f cos 17, y = c sinh f sin 17. 

* Mimoires de Berlin, 1760. 



*Me PrfMems of ParUde Dynamics [oh. I7 

= Constant and 17 = Conatant then represent respectively 
\B whose foci are at the centres of force ; and these are 
' orbite. 

'fgy. when expressed in terms of { and ti, becomes . 



c (cosh f — coa 17) c (cosh f + ooa ij) ' 
jy T is given by the equations 

' (cosh' f - cos' 1j)(f + f). 

evidently of Liouville's type (§ AS), and can therefore 
3 method applicable to this class of questiona The 
for the coordinate ^ is 

'f-co8»,)^l-c»co8hfsinhf(^+^).-|?, 

• ,)■ I*} _ 2c» coflh f sink f («wh» f - coB» ij) f (P + ^) 

= - 2(c08h°f -C08*1j)f ^ , 

n of energy T + V = h, 

;08',)|||^+2(A-F)|^(cosh-f-co8'i7) 
(oosh'f — cos*j;)] 

f - cos' )7) + ^ (cosh f + COB ^) + ^ (cosh f - COS »;)} 

+ ^coshf). 
ave 

■ — cos' 17)' f* = A cosh' f + cosh f — 7, 

t of integration. 

rom the equation of enet^, which can be written 

•)'<P + ^) 

ih' f - cos' 7j) + - (cosh f + cos ij) + — (cosh f — cos tj). 



68, 64] The SoltMe Problems of Particle Dynamics 97 



we have 



^ (cosh* f — cos' 17)" ^ = — A cos' ff — 



fi ^fi 



Eliminating dt between these equations, we have 

(d|)» (dvY 



cos 17 + 7. 



h cosh' f + - — — cosh f — 7 — A cos' 17 — ^- — - cos 17 + 7 

c c 

Introducing an auxiliary variable u, we have therefore 

u = jih cosh' f + '^^tA cosh f - 7! df , 



w = I -!— A cos* fj — ^- — cos 17 + 7 



-i 



c2i7. 



These are elliptic integrals, and we can therefore express f and 17 as 
elliptic functions of the parameter u, say 

These equations determine the orbit of the particle, the elliptic coordinates 
(^> v) being expressed in terms of the parameter u. 

64. Motion on a surface. 

We shall next proceed to consider the motion of a particle which is free 
to move on a smooth surface, and is acted on by any forces. 

Let (Xy F, Z) be the components, parallel to fixed rectangular axes, 
of the external force on the particle, not including the pressure of the surface : 
let (Xy y, z) be the coordinates and v the velocity of the particle, 8 the arc and 
p the radius of curvature of its path, ;^ the angle between the principal 
normal to the path and the normal to the surface, and (X, /a, v) the direction- 
cosines of the line which lies in the tangent-plane to the surface and is 
perpendicular to the path at time t\ the mass of the particle is taken as 
unity. 

The acceleration of the particle consists of components v-t- along the 

tangent to the path and - along the principal normal ; the latter component 

can be resolved into — sin x along the line whose direction-cosines are (\, /a, v) 

and — cos ^ along the normal to the surface. We have therefore the equations 
of motion 



CL8 as d>8 (18 



t; 



- sin Y = X\ + Yfi + Zv 
P 



W. D. 



(A), 

(B). 
7 



uHe Problems of Particle Dynamics [oh. iv 

rith the equation of the surface, are sufficient to determiDe 
equatioD of the surface may be regarded as giving z in 
and by using this value for z we can express all the 
; in equations (A) and (B) io terms of x, y, x, y, x, y: 
B) thus become a system of differential equations of the 
determination of x and y in terms of t. 

conservative, the expression 

-Xda-Ydy-Zdn 

al of a potential-energy function V(a;, y, z) ; equation (A) 
^grated, and gives on integration the equation of energy 

ifl*+F(*, y, z) = c, 

■„ Substituting the value of «" given by this equation in 

P 

inating z by means of the equation to the surface) a 
of the second order between x and y, and is in bet the 
of the orbits on the surface. 

equations of motion on a surface are not integrable by 
^neral case : there are however two cases in which the 
mulated in such a way as to utilise results obtained 



r no forces. 

1 forces act on the particle, equation (B) gives ;t- = 0, so 
! on the surface ; the integral of energy shews that this 
with constant velocity. 

t movei tinder no force* on the Jixed smooth nil«i tur/aee wkote lint 
1, the direction-cotinet of the generator at the point i being 



Uince of the point on the surface whose coordiust«a are («, ;/, t) 
, measured along the generator, and let (0, 0, be the coordinates 
s generator meeta the line of striction. Then we have 



64] The SdtMe Problems of Particle Dynamica 

The lanetic eoergf of the particle ta 

We eaa talta v and f as the two ooordinatea which define the positioa of the p 
it is evident that the coordinate f is ignorable, and the correaponding Integml is 



The inters! of energy is 

3*— A, where A is a const 

Eliminating ^ between these two integrals, we have 

**(e»+m«)=ai»*+(2A-**))»i*co9ec«a. 

If i is initially suiBcientlj large compared with f, the quantity {ih-i^ia positi 
shall anppoae this to be the case, and shall write 

(2A-i«)m*cosec*a-2AX», where X is a new cons 

the equation thus becomes 

The integration of this equation can be efiected by introducing a teal ai 
Tsriable x, defined by the equation 

Writing vXiax'', this becomes 

sad this is equivalent to the equation 

where the roots «,, a,, «j, of the function ff (u) are real and are defined by the equat 

The connexion between the Tarisbles v and u is therefore expressed by the equation 

Substituting this value of v in the equation which connects v and t, we have 



(2A)* < = J ''' " gU!*!"' "^ du + Constant 
~ I (-«)+!> {u+«,)|rf«+Const«nt» 
= -«,M-f {w+B>i) + Constant. 
* CI. Wbittaker, Modtm Analy$U, § 183. 



lie Soluble Problems of Particle Dynamics [ch. it 

eipreascB tlie time ( in terms of the auxiliuy Tuiable u, and thus in 
ith the equation 

..-.Xm{|»(»)-.,}-*, 
9iioii between f and (. 

ion on a developable surface. 

irface on which the particle moves is developable, we can utilise 

heoretDS that the arc s and the quantity ^ are unaltered by 

he surface on a plane : these results, applied to the equations of 
a above, shew that if in the motion of a particle on a developable 
r any forces the surface is developed on a plane, the particle will 
plane curve thus derived from its orbit with the same velocity as 
ded the force acting in the plane motion is the same in amount 
n relative to the curve as the component of force tn the tangent- 
iurface in the surface-motion. 

. A tmoolh partidt it projeeUd along the rtu-faet of a right circular cotu, 
erliaU and vertex nptparda, vni/i the velocity doe to the depth beloa the vertex, 
path traced oat on the ante, when developed into a plane, will beoftht form 

r*ainfrf-o*. (CoU. Eiam.) 

eloping tbe cone, the problem becomes the same as that of motion in a plane 
uit repulsive force from tbe origiu, and with the velocity compatible with 
igio. We therefore have the integrals 

f> + r'tf' = CV, where (7 is a constant, 

t*8~h, where A is a constant 




Talent to the equatii 



. If in the motion of a point P oa& developable surface the tangent IP to 
gression deacribeu areas proportional to tbe times, shew that the component 

ndicular to IP and in the tangent-plane is proportiooal to ^, where p is 

curvature of tbe edge of regression. (Hszzidakis.) 



64, 66] The Soluble Problems of Partide Dynamics 101 

55. Motion on a surface of revolution ; cases soluble in terms of circular 
and elliptic functions. 

The most important case of surface-motion which is soluble by quadra- 
ture is the motion of a particle on a smooth surface of revolution, under 
forces derivable from a potential-enerory function which is symmetrical with 
respect to the axis of revolution of the surface. 

Let the position of a point in space be defined by cylindrical coordinates 
(^1 ^1 4>), where ^ is a coordinate measured parallel to the axis of the surface, 
r is the perpendicular distance of the point from this axis, and ^ is the 
azimuthal angle made by r with a fixed plane through the axis. The 
equation of the surface will be a relation between z and r, say 

r=f{z\ 

and the potential energy will be a function of z and r (it cannot involve <f>, 
since it is symmetrical with respect to the axis), which for points on the 
surface can, on replacing r by its value f(z\ be expressed as a function of z 
only, say V(z); the mass of the particle can be taken as unity. 

The kinetic energy is, by § 18, 

5r = ^(i« + ;4 + r»(^«) 

The coordinate ^ is evidently ignorable ; the corresponding integral is 

—T'^k, where & is a constant, 

or {/iz)}*4 = k; 

this equation can be interpreted as the integral of angular momentum about 
the axis of the surface. 

The equation of energy is 

T+ V^h, where A is a constant, 

and substituting for <^ in this equation from the preceding, we have 

{[f'iz)Y + 1] i* + *» [f{z)\-* + 2F (z) = 2A ; 
integrating this equation, we have 

t^{[[f' {z)Y + 1]* [2A - 2 F {z) - Jfc» [f{z)]-^y^ dz + Constant. 

The relation between t and z is thus given by a quadrature ; the values 
of r and 4> ^^^ ^^en obtained from the equation of the surface and the 
equation 

. {f{z)Y^^k, 

respectively. 



Problems of Partide DynamicB [cH. iv 

le motion on those surfaces for which this qaad- 
means of known functioDS, when the axis of the 
measured positively upwards) and gravity is the 

sr. 

the circular cylinder r = a, the above integral 

ites is so chosen that 2^* = J:*, we have 
•^ — ^{t — t^, whore fg is a constant 



where 0^ is a constant. 



lur&ce is the sphere 

'lie spherical pendulum, and can be realised by 
) attached to a fixed point by a light ri^d wire 
ibout the point. 

tnre for t becomes 

'.h-1ge){l}-ii')-i?\-^dz. 

right-hand side of this equation is an elliptic 
>w reduce to Weieretrass' canonical form. Denote 
the cubic 

! and —loiz, and positive for very large positive 
values of z which occur in the problem considered 



65] The Soluble Problems of Fartide Dynamics 108 

(which must necessarily lie between — Z and + i, since the particle is on the 

sphere) we see that one of the roots (say Zi) is greater than I and the other 

two (say 5j and z^ where z^ > z^) are between I and — I. The values of z in 

the actual motion will lie between z^ and z^, since for them the cubic must be 

positive. 

h 2Z* 

Write -? = o- H ?> where f is a new variable^ 

^9 9 

and ^""Sa"*" a ^''^ (r= 1,2,3) 

so that ei, e^, e^y are new constants, which satisfy the relation 

«i + «» + e«= ^ (^1 + -^8 + -^'"^j^ ^> 
and also satisfy the inequalities ^i > e, > 6s. 
The relation between t and z now becomes 

^=/i4(r-«i)(?-«i)(?-a-*rfr, 

or (r=i>(^ + €), 

where € is a constant of integration and the function fp is formed with the 
roots ej, e,, 6s. 

Now when ei, e^, e^, are real and in descending order of magnitude, 
fp (u) and fp^ (u) are both real when u is real, in which case fp (u) is greater 
than 6i, and also when u is of the form a>s + a real quantity, where a>s is 
the half-period corresponding to the root 6,; in this latter case, fp(u) lies 
between e^ and e^. Since in the actual motion z lies between z^ and ^, it 
follows that ^ lies between 6, and e^, and therefore the constant € must con- 
sist of an imaginary part o), and a real part. depending on the instant from 
which time is measured : by a suitable choice of the origin of time, we can 
take this real part of € to be zero, and we then have 

This equation gives the connexion between z and t We have now to 
determine the azimuth <^. For this we have the equation 

80 9 — 9o = 






where ^o isSa constant of integration. 






IvHe Problems of Particle Dynamics [ch. iv 

begiatioD, we take X and fitohn the (imaginary) values of 
; to the values I aud —I of z respectively; so that X and fi 
lefined by the equations 



I^W-y'W-^. 






;ii>(i + <.,)-i>(x))(j>(< + «.)-i)0»)j 

kg n dt dt ) 



i!\—!L 









-f(«-X.)-f(. + X) + 2r(X.). 



I(<+»J-|)(».) »i(< + «, + X) 

w _ e-BM- (Ml <'(' + ".-<')^(' + ". + >-) . 

saes the aogle as a function of t, and so completes the 
>lem. 

lie bob of tbe epberioal pendulum is executing periodic MCillstions . 
on tbe apbere, shew tbat one of tbe points reacbed on tbe bigber 
on tbe lower parallel at wbicb the bob arriree after a balf-period 
Lziroutb wbicb always lies between one and two right anglea 

(Puiseux and Halphen.) 

holmd. 

he ptflblein of motion on the pantboloid, whose equation is 

r-2ols». 

quadrature for t becomes 

t-Uu*z^{ut-ig^-^ *dz. 

* Cf. Wbittaker, Hodem AtuU^lU, S 31fi, Ex. I. 



^ 






he Soluble Problems of Particle Dynaimu 

the solution of the problem in terms of elliptic f 
Lusiliary quantity v, defiQed by the equation 

1,- ['(a +«)-* (2A*- 23^- ^)'*de. 

(where a > j3) denote the roots of the quadratic 



this integral in the form 

»-(-|)"'fi«<'t'"><'-*<'-""'''''- 

ew variable J^ by the equation 

et, be the values of (; corresponding to the valut 
Fz; then the integral becomes 

{2^^}'— f^|4(f-e,)<f-».)(r-«.)!-'i*t 
J proved that the quantities e,, e,, e,, satisfy the reli 

ei + e,+ e, = 0, ei>e,>(i,. 
ary quantity v can now be replaced by an auxiliary 
! equation 



inversion of the integral gives 

constant of integration and the function p is form 
which are given by the equations 



l(a + a) ■ '^~ 3(rt + a) ' ^ 3(a + a) 
ictual motion 2 evidently lies between a and /8, it 
between e, and eg, and therefore (as we wish u to 
t of the constant e must be the half-period w,; the r 
i zero, since it depends merely on the lower limit of 



herefore 

h — na . ^ h 

smce o + yS = - . 



^ ■ 



Problems of Particle Dynamu 

ne t is 



?<^>}'/(y(.+..)-M.i» 



terms of the auxiliary variable «. 

mine the azimuth ^ ; for this we hat 

= *^ 
iaz 

p(M + o>.)-g, 



itegratioD, and / is an auxiliary com 



e written 

fa* i r g' (0 du 

Sr(a + a))» 2j (?{« + «.)- jj(0' 
id (as in the problem of the spberici 

n terms of the auxiliaiy variable u, 



rhose equation is 
r=z tan a, 
angle. 



I%e Soluble Problems of Particle Dynamics 107 

this is a developable sur&ce, we can apply the tbeorem of | 64, and 
lat the orbit of a particle on the cone under gravity becomes, when 
! is developed on a plane, the same as the orbit of a particle 
laas in the plane under a force of constant magnitude g cos a acting 
a filed centre of force (namely the point on the plane which corre- 
i the vertex of the cone). This (§ 48) is one of the known cases 

the problem of central motion can be solved in terms of elliptic 
, and this solution furnishes at once the solution of the problem of 
n the cone. 

<U 1, Shew that the motion of a particle under gravity on a atirface of revoluUoQ 
I ia vertical can also be solved in terms of elliptic functions when the surface ia 
mj one of the following equntiona 

(H -tu-JaV-a"^ (Kobb and StackeL) 

it 2. Shew that if an algebraic surface of revolution ia such that the equations 
le«ica can be eipresaed in terms of elliptic functions of a parameter, the surface 
ich that r* and i can be eipressed as rational functions of a parameter, i.e. the 
if the surface regarded as an equation between r* and i is the equation of 
tl curve ; where t, r, <f> are the cylindrical coordinates of a point on the 

(Kobb.) 
■Is 3. Shew that in the following cases of the motion of a particle on a surface 
ion, the trajectories are all closed curves : 

Iten the surface is a sphere, and the force is directed along the tangent to the 
uid proportional to cosec' 6, where S is the angular distance from the particle to 
(The trajectoriee are in this cose sphero-conics having one focus in the pola) 
lien the surface is a sphere, and the force is directed along the tangent to the 
tnd proportional to tan 6 sec* 6. (The trajectories in this case are sphero-conics 
e pole as centre*.) 

Joukovskya theorem. 

hall now shew how to determine the potential-energy function under 
given family of curves on a surface can be described as the orbits of 
e constrained to move on the surface. 

three rectangular coordinates of a point on the surface can be 
d in terms of two parameters, say u and o, so that an element of arc 
; surface is given in terms of the increments of u and v to which it 
ids by an equation of the form 

dif " E du^ + 2F dudv -If Odii', 
, F, G are known functions of u and v. 

ODZ has examined the posaibilit; of other caees, In Ball, de la Soc. Math, dt France, v. 



'.e Problems of Pt^tide DynamUx [ 

irvcB which are to he the orbita under the re 
led by an equation 

q{u, o) = CODStBDt, 

p (u, v) = Constant 
Tes which is orthogonal to these. 
!ind t; we can take p and q as the two para 
ion of a point on the surface; let the line-e 
eters be expressed by the equation 

iing absent, because the curves p = Constui 
ight angles : E' and G' being known functj 

>f a particle which moves ou the surface is 

T=\{E'f+Q'p'); 
ns of motion are therefore 

iknown potentiat-eoei^ function, which it is r 

to be BatiaBed by the value j » ; they then 
130'.,_3y 
2iq'' dq' 

IS,'""' — ^- 

liave 

[uation, we have 

-sw^ + V^'/iq), where/is an arbitrary fum 



The Soluble Problems of Partide Dynamiea 109 



therefore 

e g denotes an arbitrar; functioo. 

fow ^ is A, {p), the diflFereotial parameter • of the first order of the 

ion p ; and thus we have a theorem enunciated by Joukovsky in 189U, 
if q= Constant is the equation of a family of curves on a surface, and 
Constant denotes the family of curves orthogonal to these, then the curves 
'onstant can be freely described by a particle under the influence of forces 
edfrom the potential-energy function 

>'-i,(P)j(;.)+4,(p)//w||^}.i,, 

! / and g are arbitrary functions, and A, denotes the first differential 

he above equations give 

*'' dq/ 85 8''^+ O' ■ 

lence the equation of energy in the motiuo is 
iO'p^+V-fiq). 

MiSCELLANEOUH EXAMPLES. 
A paj^icle movea under gravity on the Bmooth cjcloid wboxe equation iu 

t denotea the uc aud i^ the angle made hj the tangent to the curve with the 

Dtal: shew that the motioo ia periodic, the period being 4n^ -. 

A particle movee in a etnooth circular tube under the inBuence of a force directed 
x«d point and proportional to the distance from the point. Shew that the motion is 
I same character as in the pendulum-probleni. 

( the liae-element od a aurface is given bj the equation 

dt' = Ed>,' + 2Fdudv + Q(bi\ 
It differential parametei of a tunotioD ^ («, c) ie given b; the tbrmola 



A, (*) 



'^.\^m-''i^i^''(Mf\- 



e differential paiametiir ia a deformation-covariant of (be mrfaee, i.e. when a ohaage of 
lei it made from (u, e) to (u', v'), the differential parameter tranaferniB into the eipreuion 
1 ia the eame »a; vith the neo variable* (u', v') and the oorieaponding new ooeffioientt 

■,a-). 



110 The Soluble Problems of P article Dynamics [ch. iv 

3. A particle moves in a straight line under the action of two centres of repulsive 
force of equal strength fi, each varying as the inverse square of the distance. Shew that^ 
if the centres of force are at a distance Sc apart and the particle starts from rest at 
a distance kc, where ^ < 1, from the middle point of the line joining them, it will perform 
oscillations of period 

IT 

2^/?(r=I«)//i j^ (1 -ifc»8in« B)^ dB. 

(Camb. Math. Tripos, Part I, 1899.) 

% 

4. A particle under the action of gravity travels in a smooth curved tube, starting 
from rest at a given point of the tube. If the particle describes every arc OP in 
the same time that would be taken to slide down the corresponding chord OP, shew that 
the tube has the form of a lemniscate. 

5. A particle is projected downwards along the concave side of the c:urve ^-Ho^^O 

with a velocity § (2a^)^ from the origin, the axis of x being horizontal ; shew that the 
vertical component of the velocity is constant. (Nicomedi.) 

6. A particle moves in a smooth tube in the form of the curve f^ss2a^ cos 26, under 
the action of two attractive forces, varying inversely as the cube of the distance, towards 
the two points on the initial line which are at a distance a frx)m the pole. Prove that if the 

absolute force is ^ and the velocity at the node 2fiVa, the time of describing one loop of 

the curve is ira'/Sfi*. (Camb. Math. Tripos, Part I, 1898.) 

7. A particle describes a space-curve under the influence of a force whose direction 
always intersects a given straight line. Shew that its velocity is inversely proportional to 
the distance of the particle from the line and to the cosine of the angle which the 
plane through the particle and the line makes with the normal plane to the orbit. 

(Dainelli.) 

8. A heavy particle is constrained to move on a straight line, which is made to 
rotate with constant angular velocity a> round a fixed vertical axis at given distance frx>m 
it. Shew that the motion is given by the equation 

r = ^c"^ cos a + -fl« " *' cos a, 

where r is the distance of the particle frx)m a fixed point on the line, a is the angle made 
by the line with the horizontal, and A, B are constants. (H. am Ende.) 

9. A heavy particle is constrained to move on a straight line, which is made to 
rotate with given variable angular velocity round a fixpd horizontal axis. Shew that the 
equation of motion is 

r = +^ sin a sin ^ - rrf* sin' a + ad sin a, 

where a is the angle between the line and the axis of rotation, 3 the angle made with 
the vertical by the shortest distance a between the lines, and r the distance of the 
particle fr^m the intersection of this shortest distance with the moving line. 

(VoUhering.) 

10. A particle slides in a smooth straight tube which is made to rotate with uniform 
angular velocity a> about a vertical axis : shew that, if the particle starts frt>m relative 
rest frt>m the point where the shortest distance between the axis and the tube meets the 
tube, the distance through which the particle moves along the tube in time t is 

-f cot a cosec a sinh' (i at sin a), 

or 

where a is the inclination of the tube to the vertical 

(Camb. Math. Tripos, Part I, 1899.) 



CH, iv] The Soluble Problems of Particle Dynamics 111 

11. A particle is constrained to move under no external forces in a plane circular tube 
which is constrained to rotate uniformly about any point in its plana Shew that the 
motion of the particle in the tube is similar to that in the pendulum-problem. 

12. A small bead is strung upon a smooth circular wire of radius a, which is con- 
strained to rotate with imiform angular velocity «a about a point on itself. The bead is 
initially at the extremity of the diameter through the centre of rotation, and is projected 
with velocity 2»6 relative to the wire : shew that the ix>sition of the bead at time t 
is given by the equation 

Bin<^=:8n&«»^/a (modulus ajh) 

or 

sin<^8(6/a)8n»;, (modulus hja) 

according as a < or > 6, <^ being the angle which the radius vector to the bead makes 
with the diameter of the circle through the centre of rotation. 

(Camb. Math. Tripos, Part I, 1900.) 

13. Shew that the force perpendicular to the asymptote imder which the curve 

can be described is proportional to 

^(«*+y*)"'. 

14. A particle is acted on by a force whose components (X, T) parallel to fixed axes 
are conjugate functions of the coordinates {x, y). Shew that the problem of its motion is 
always soluble by quadratures. 

15. If ((7) be a closed orbit described by a particle under the action of a central force, 
8 the centre of force, the centre of gravity of the cmre (O), O the centre of gravity of 
the curve (C) on the supposition that the density at each point varies inversely as 
the velocity, shew that the points Sy 0, O^ are coUinear and that 2SO=ZS0, 

(Laisant.) 

16. Shew that the motion of a particle which is constrained to move in a plane, 
under a constant force directed to a point out of the plane, can be expressed by means of 
elliptic functions. 

17. Shew that the curves 

whete OihyC are arbitrary constants and/ is a given function, can be described under the 
same law of central force to the origin. 

18. Shew that when a circle is described under a central attraction direqjbed to 
a point in its circtLmference, the law of force is the inverse fifth power of the distance. 

19. A particle describes the pedal of a circle, taken with respect to any point in 
its plane, under the influence of a centre of force at this point. Shew that the law 
of force is of the form 



4+^ 



where A and B are constants. 



Shew that the law of force is also of this form when the inverse of an ellipse with 
respect to a focus is described under a centre of force in the focus. (Curtis.) 



I 



112 The Soluble Problems of Particle Dynamics [ch. iv 

20. Prove that, if when projected from r«/2, ^=0 with a velocity Fin a direction 
making an angle a with the radius vector the path of a particle be/(r, ^, R^ V, sin a)— 0, 
the path with the same initial conditions but under the action of an additional central 

force ^ is 

/(r, n$f iZ, r(n*sin*a+cos«o)*, »sina(n*sin»o+cos'o)"'*)=0, 
where 

^^^f' rsy/ain'a ' ^^^^ Exam.) 

21. A particle of imit mass describes an orbit under an attractive foit» P to the 
origin and a transverse force T perpendicular to the radius vector. Prove that the 
differential equation of the orbit is given by 

If the attractive force is always zero, and the particle moves in an equiangular spiral 
of angle a, prove that 

T^fjo^'^''-^ and A«(,isinacoso)*r^**. 

(Camb. Math. Tripos, Part I, 1901.) 

22. A particle, acted on by a central force towards a point varying as the distance, 
is projected from a point P so as to pass through a point Q such that OP is equal to OQ ; 

shew that the least possible velocity of projection is OP (fi sin POQ)^, where /i . OP b the 
force per unit mass at P. (Camb. Math. Tripos, Part I, 1901.) 

23. Find a plane curve such that the curve and its pedal, with regard to some point 
in the plane, can be simultaneously described by particles under central forces to that 
point, in such a manner that the moving particles are always at corresponding points 
of the curve and the pedal ; and find the law of force for the pedal curve. 

(Camb. Math. Tripos, Part I, 1897.) 

24. If /(a?, y) be a homogeneous function of one dimension, then the necessary 
and sufiBicient condition that the curve /{x, y)aBl be capable of description under accele> 
ration tending to the origin and varying with the distance alone, is that / be subject 
to a condition of the form 

Hence shew that the only curves of this class are necessarily included in the equation 

r(^+58in^ + Ccos^)"»l. 

Proceed to the discussion of the case wherein f{x, y) is homogeneous and of n 
dimensions. (ColL Exam.) 

25. An ellipse of centre C is described under the influence of a centre of force 
at a point on the major axis of the ellipse ; shew that 

n^sstf-tfsini^ 

where Zv/n is the periodic time, e is the ratio of CO to the semi-migor axis, and u ia the 
eccentric angle of the point reached by the particle in time t from the vertex. 

26. Two free particles /i and M move in a plane under the influence of a central force 
to a fixed point 0. Shew that the ratio of the velocity of the particle fi at an arbitrary 
point m of its path, to the velocity which is possessed at m by the central projection of If 
on the orbit of /i, is equal to the constant ratio of the areas described in unit time by the 
radii 0/i, OM, multiplied by the square of a certain function / of the coordinates of my 
which expresses th^ ratio of OM, Om. (Dainelli.) 



CH. rv] The Soluble Problems of Particle Dynamics 118 

27. A particle is moving freely in a parabola under an attraction to the focus. Shew 
that, if at every instant a point be taken on the tangent through the particle, at distance 
4acoB^^/(^+8intf) from the particle, this point will describe a central orbit about 
the focus, and the rate of description of areas will be the same as in the parabola ; where 
4a is the latus rectum, and 3 the vectorial angle of the particle measured from the apse 
line. (Camb. Math. Tripos, Part I, 1896.) 

28. When a periodic comet is at its greatest distance from the sun, its velocity 
receives a small increment dv. Shew that the comet's least distance from the sun 
will be increased by the quantity 

4«t; . {a»(l -c)//i(l +«)}*. (Coa Exam.) 

29. If POP' is a focal chord of an elliptic path described round the sim, shew that 
the time from P' to P through perihelion is equal to the time of falling towards the 
sun frt>m a distance 2a to a distance a (1 +cosa), where a=2fr ~(k'- u), and u'- u is the 
difference of the eccentric anomalies of the points P, P'. (Cayley.) 

30. A particle moves in a plane under attractive forces fi/r^r^j m/^*^* along the 
radii r, / drawn to two fixed points at distance 2d apart. Shew that, if it is projected 
with the velocity frx)m infinity, a possible path is a circle with regard to which the 
two fixed centres are inverse points, and that, if the radius of this circle is a, the periodic 
time is 

4iraV*(a*+rf")*. (CoU. Exam.) 

31. A heavy particle is projected horizontally with a velocity v inside a smooth 
sphere at an angular distance a frt)m the vertical diameter drawn downwards : shew that 
it will never fall below or never rise above its initial level according as 

i;* > or < o^ sin a tan a. (Coll. Exam.) 

32. A particle is projected horizontally with velocity F along the interior of a smooth 

sphere of radius a from a point whose angular distance from the lowest point is a. Shew 

that the highest point of the spherical 8iu*face attained is at an angular distance ff from 

the lowest point, where fi is the smaller of the values of ^, x given respectively by 

the equations 

(3^V-2^-W+I^=0l (ColLExam.) 

(C0S;^+C0Sa) 7^-20^ 8m';^-«0J ^ 

33. If the motion of a spherical pendulum of length a be wholly between the levels 

^iij ^a below the point of support, shew that at a time t after passing a point of greatest 

depth, the depth of the bob is 

^a{4-sn«fV(I%/14a)} (mod. ^/(7/65).) 

and that a horizontal coordinate referred to the point of support as origin is determined 

by the equation 

i?-(5r*/a) {- J^+fsn«* V(l%^/14«)}, 

which is a case of Lamp's equation. (Coll. Exam.) 

34. A particle is constrained to move on the surface of a sphere, and is attracted to a 

fixed point M on the surface of the sphere with a force that varies as r"' (<i*- r*)"*, where 
d IB the diameter of the sphere and r is the rectilinear distance frt)m the particle to M. 
If the position of the particle on the sphere be defined by its coiatitude B and longitude ^, 
with M as pole, shew that the equations of motion furnish the differential equation 



1 /dey , 1 ..,, 

-^-TTi ( jt) +-:--«n,*=acottf+6, 



sm' 

where a and h are constants ; and integrate this equation, shewing that the orbit is 
a sphero-conic. 

W. D. 8 



114 The Soluble Problems of Particle Dynamics [ch. iv 

35. A particle of mass m moves on the inner surface of a cone of revolution whose 
semi-vertical angle is a, imder the action of a repulsive force rtifi/r^ from the axis; the 

angular momentum of the particle about the axis being m Jyk tan a, shew that the path is 
an arc of a hyperbola whose eccentricity is sec a. (Camb. Math. Tripos, Part I, 1897.) 

36. Shew that the necessary central force to the vertex of a circular cone in order 
that the path on the cone may be a plane section is 

A B 

"5 - -3 . (ColL Exam.) 

37. A particle of unit mass moves on the inner surface of a paraboloid of revolution, 
latus rectum 4a, under the action of a repulsive force iir from the axis, where r is the 
distance from the axis; shew that, if the particle is projected along the surface in a 

direction perpendicular to the axis with velocity 2afi^, it will describe a parabola. 

(ColL Exam.) 

38. A smooth surface of revolution is formed by rotating the catenary «Bctan^ 
about its axis of symmetry, and a particle is projected along its surface from a point 

distant h from that axis with velocity h{a^-\'l^)^ll^. The direction of projection is such 
that the component velocity perpendicular to the axis is hjh and the particle moves in 
contact with the surface, under the influence of a force of attraction A^(r*+2a')/r* in the 
direction of the perpendicular r to the axis. Shew that, if gravity be neglected, the 
projection of the path on a plane at right angles to the axis will have a polar equation 

T 

c sinh - » €tB. (Coll. Exanr. ) 

39. A particle moves on a smooth helicoid, z^a<fi^ under the action of a force ^r 
per unit mass directed at each point along the generator inwards, r being the distance 
from the axis of z. The particle is projected along the surface perpendicularly to the 
generator at a point where the tangent plane makes an angle a with the plane of jry, its 

velocity of projection being fj^a. Shew that the equation of the projection of its path on 
the plane of ^ is 

a^/r^=aGC^ a cosh' {<f> cos a)— 1. 

(Camb. Math. Tripos, Part 1, 1896.) 

40. Shew that the problem of the motion of a particle under no forces on a ruled 
surfiaoe, whose generators cut the line of striction at a constant angle, and for which the 
ratio of the length of the common perpendicular to two adjacent generators to the angle 
between these generators is constant, can be solved by quadratures. (Astor.) 



CHAPTER V. 

THE DYNAMICAL SPECIFICATION OF BODIES. 

67. Definitions, 

Before proceeding to discuss those problems in the dynamics of rigid 
bodies which can be solved by quadratures, it is convenient to introduce and 
calculate a number of constants which can be assigned to a rigid body, and 
which depend on its constitution ; it will be found that these constants 
determine the dynamical behaviour of the body. 

Let any ripd b ody be considered ; and let the particles of which (from 
the dynamical point of view) it is constituted be tjrpified by a particle of 
mass m situated at a point whose coordinates referred to fixed rectangular 
axes are (Xy y, z). 

The quantity 2m(y* + -g*), 

where the symbol X denotes a summation extended over all the particles of 
the system, is called the moment of inertia of the body about the axis Ox. 
Similarly the moment of inertia about any other line is defined to be the sum 
of the masses of the particles of the body, each multiplied by the square of 
its perpendicular distance from the line. These summations are evidently in 
the case of ordinary rigid bodies equivalent to integrations ; thus 2m (^ + z*) 

is equivalent to III (j/^ + z^)pda!dydz, where p is the density, or mass per 

unit volume, of the body at the point (x, y, z). 

The quantity Xmxy 

is called the product of inertia of the body about the axes Ox, Oy ; and 
. similarly the quantities "S^myz and 'S,mzx are the products of inertia about 
the other pairs of axes. 

For the moments and products of inertia with reference to the coordinate 
axes, the notation 

A = Xm{f + z% B = 2m (-«• + a:»), G=«2m(a^ + y"), 

F = Xmyz, G = Xmzx, H =« 2ma^ 

will be generally used. 

Two bodies whose moments of inertia about every line in space are equal 
to each other are said to be eqaiTnomental, It will be seen later^that this 
involves also the equality of the products of inertia of the bodies with respect 

to any ])air of orthogonal lines. 

8—2 



/ 



116 The dynamical speciJiccUion of bodies [ch. v 

If M denotes the mass of a body and if £ is a quantity such that Mk^ is 
equal to the moment of inertia of the body about a given line, the quantity 
k is called the radius of gyration of the body about the line. 

In the case of a plane body, the moment of inertia about a line perpen- 
dicular to its plane is often spoken of as the moment of inertia about the 
point in which this line meets the plane. 

68. The moments of inertia of some simple bodies*. 
(i) The rectangle. 

Let it be required to find the moment of inertia of a uniform rectangular 
plate, whose sides are of lengths 2a and 26 respectively, about a line through 
its centre parallel to the sides of length 2a. Taking this line as axis Ox^ 
and a line through parallel to the other sides as axis Oy, the required 
moment of inertia is 

2m^, or I I ay^dxdy, 

where a- is the mass per unit area of the plate, or the surface-density as it is. 
frequently called ; evaluating the integral, we have for the required moment 
of inertia 

— ^ — , or Mass of rectangle x J6*. 

The moment of inertia of a uniform rod, about a line through its middle- 
point perpendicular to the rod, can be deduced from this result by regarding^ 
the rod as the limiting form of a rectangle in which the length of one pair 
of sides is indefinitely small. It follows that the moment of inertia ia 

question is 

Mass of rod x J6^ 

where 26 is the length of the rod. 

(ii) The rectangular block. 

Consider next a uniform rectangular block whose edges are of lengths 2a^ 
26, 2o; let it be required to find the moment of inertia about an axis Ox 
passing through the centre and parallel to the edges of length 2a. This 
moment of inertia is 

2m(y'4-'8^), or I I I p(y' + z^)dzdydx, 

where p is the density. Evaluating the integral, we have for the moment ot 
inertia 

?^(6» + c»), or Mass of block xi(6« + c»). 

* For practical porposes the moments of inertia of a body are determined experimentally ; a. 
convenient apparatus is described by W. H. Derriman, PhiL Mag, v. (1903), p. 648. 



r- 



o7, ^8] The dynamuxd specification of bodies 117 

« 

(iii) The ellipse and the circle. 

Let it now be required to find the moment of inertia of a uniform elliptic 
plate whose equation is 

about the axis of x. It is 

I I a-jf'dydx, where <r is the surface-density. 

Evaluating the integral, we have for the required moment of inertia 

iiral^a, or Mass of ellipse x J6'. 
The moment of inertia of a circle of radius 6 about a diameter is therefore 

Mass of circle x ^6*. 

(iv) TTie ellipsoid cmd the sphere. 

The moment of inertia of a uniform solid ellipsoid of density p, whose 
equation is 

about the axis of x is similarly 

1 1 / P (^ + '^*) dxdydz, integrated throughout the ellipsoid. 
To evaluate this integral, write 

where f , 17, ^, are new variables : the integral becomes 

pab^c jjjv^ d^dvd^ + pahc^ jjj^d^dvd^, 

where the integration is now taken throughout a sphere whose equation is 

P + i7"+r = l. 
Since the integrals 

Jjjpdfd^df, jJlv-dSclvdi. and jjj^d^dvd^, 

are evidently equal, the required moment of inertia can be written in the 
form 

pabo (b* + c^)jjj^dSdtid^, 

« 

or -j^ irpabc (ft* + c*), 

or Mass of ellipsoid x J (ft* + c*). 



or TT, 



118 The dynamical specificcUion of bodies [ch. v 

The moment of inertia of a uniform sphere of radius a about a diameter 
is therefore 

Mass of sphere x fa*. 

(v) The triangle. 

Let it now be required to find the moment of inertia of a uniform 
triangular plate of surface-density a, with respect to any line in its plane ; 
the position of the line can be specified by the lengths a, /8, y, of the per- 
pendiculars drawn to it from the vertices of the triangle. 

Taking {x, y, z) to be the areal coordinates of a point of the plate ; the 
perpendicular distance from this point to the given line is (aa? + ^Sy + 72^), and 
the required moment of inertia is therefore 



\{{ax + /8y + yzy dS, 



where dS denotes an element of area of the plate. 

Now if F denotes the length of the perpendicular from the point (x, y, z) 
on the side c of the triangle, and if X denotes the length intercepted on the 
side c between the vertex A and the foot of this perpendicular, we have 

Y=zb sin A 

and Z sin -4. — Fcos A = perpendicular from (x, y, z) on the side b 

= yc sin J.. 
We have therefore 

dydz^P^^dXdY=,-4—. dXdY^^da, 
^ d {X, Y) be sm A 2 A 

where A denotes the area of the triangle. Hence the integral jlj/*dS, where 
the integration is extended over the area of the triangle, can be written in 
the form 2A 1 1 t/^dydz, where the integration is extended over all positive 
values of y and z whose sum is less than unity : this is equal to 



2AfV(i-y)dy, 

Jo 



or iA. By symmetry, the integrals llaf^dS and jjz^dS have the same value, 
and a similar calculation shews that the integrals 

jjyzdS, jjzxdS, jjxydS, 
each have the value ^^A. 



N 



68, 69] The dynamical apecificcUion of bodies 119 

Substituting these values in the integral cr li(aur + /3y + 7'e^)^cZiS, the 
moment of inertia of the triangle about the given line becomes 

i (T A ( a« + i8« + 7* + /87 + 7« + a/8), 



or 



i X Mass of triangle x [[^) + {^-^) + («-±^] . 



But this expression evidently represents the moment of inertia about the 
given line of three particles situated respectively at the middle points of the 
sides of the triangle, the mass of each particle being one-third the mass of 
the triangle; the triangle is therefore equimomental to this set of three 
particles. 

Example, Shew that a uniform solid tetrahedron of mass M is equimomental to a set 
of five particles, four of which are each of mass -^M and are situated at the vertices 
of the tetrahedron, while the fifth particle is at the centre of gravity of the tetrahedron, 
and is of mass ^M, 

69. Derivation of the moment of inertia about any ojds when the moment 
of inertia about a parallel axis through the centre of gravity is knotvn. 

The moments of inertia found in the preceding article were for the most 
part taken with respect to lines specially related to the bodies concerned : 
these results can however be applied to determine the moments of inertia of 
the same bodies with respect to other lines, by means of a theorem which will 
now be given. 

Letf(x, y, z, X, y, i, x, y, '£) be any polynomial (not necessarily homogene- 
ous) of the second degree in the coordinates and the components of velocity 
and acceleration of a particle of mass m. Let (r^, y, z) denote the coordinates 
of the centre of gravity of a body which is formed of such particles, and write 

x^x^-xu y=y+yi, z = z+z^. 

If now we substitute these values for x, y, z, respectively in the function /, 
we obtain the follovdng classes of terms : 

(1) Terms which do not involve dq, yi, z^ : these terms together evidently 

give 

f(x, y, i, X, y, I, S, y, z). 

(2) Terms which do not involve x,y,z: these terms give 

fi^u Vu ^i» ^> yi» ^i» ^i» Vu i?i). 

(3) Terms which are linear in a?i, yi, Zi^ x^, yi, ii, a?i, j/i, Zi\ when the 
expression ^mf{x, y, z, x, y, i, x, y, z) is formed, the summation being taken 
over all the particles of the body, these terms will vanish in consequence o 
the relations ,^e 

Imx^^O, 2myi = 0, 2m^i = 0. u. jiixam.) 



120 



TKe dynamical specifiecUion of bodies 



[oh. V 



We have therefore the equation 



2w/(a?, y, z, X, y, i, x, y, ^)= 2m/(a?x, ^1,-^1, ^1, yi, ii, a?i, Ji, l?i) 

+/(^, 3?, ^, S, y, ^, S. % z) . 2m, 
and consequently the value of the expression Xmf, taken with respect to 
any system of coordinate axes, is equal to its value taken Mrith respect to a 
parallel set of axes through the centre of gravity of the body, together with 
the mass of the body multiplied by the value of the function / at the centre 
of gravity, taken with respect to the original system of axes. 

From this it immediately follows that the moments and products 0/ inertia 
of a body with respect to any axes' are equal to the corresponding m^m^nts and 
products of inertia with respect to a set of parallel a^es through the centre of 
gravity of the body, together with the corresponding moments and products of 
inertia, 'with respect to the original axes, of a particle of mass equul to thai of 
the body and placed at the centre of gravity, 

Aa au example of this result, let it be required to determine the moment of inertia 
of a straight uniform rod of mass J£ and length I about a line through one extremity 
perpendicular to the rod. It follows from the last article that the moment of inertia 

about a parallel line through the centre of the rod is ](M(^j ; and hence, applying the 

above result, we see that the required moment of inertia is 

or IMl*. 

60. Connexion between moments of inertia with respect to different sets of 
aoces through the sams origin. 

The result of the last article enables us to find the moments of inertia of 
a given body with respect to any set of axes, when the moments of inertia are 
already known with respect to a set of axes parallel to these. We shall now 
shew how the moments of inertia of a body with respect to any set of 
rectangular axes can be found when the moments of inertia are known with 
respect to another set of rectangular axes having the same origin. 

Let A, B, C, F, 0, H be the moments and products of inertia with respect 
to a set of axes Oocyz, and let Oxj/gf be another set of rectangular axes having 
the same origin ; the direction-cosines of either set of axes with respect to 
the other will be supposed to be given by the scheme 



z 



eav 



'l 


h 


h 


mj 


thj 




«i 


«, 



69, 60] The dynamical specification of bodies 121 

If the moments and products of inertia with respect to the axes Oafi/z' 
are denoted by A\ B, G\ F\ Q\ H\ we have 

A' =s ^m{y^ + /'), where the summation is extended over all the particles of 

the body, ' 

= 2m [x" (y + /,») + y* {m^ + m,*) + 2r« (w,« + n,') + 2y^ (m^w, + m^n;) 

+ 2za: (njis + nj^ + 2a:y (^2^ + ^wi,)} 
= 2m {a:" (mi* + n^) + y« (nj* + 1^) + -e« (Zi*+ mi»)- 277iiniy-^ - 2niZi^a? - 2iimia^} 
= 2m [k^ (y» + ^) + TTii' (-?» + a?)'\- n^^ {a^ + y«) - ZntjU^z - 2wiZi;?a; - 2limixy} 
= ^ii« + 5mi» + Crh» - 2iHni - ^Gn^li - ^Ekm^ 
and similarly 

J5' = ^Za* + JBm,* + Cn,« - iFrn^n^ - 2(?n,Za - ZHkm^, 
C = 4/,« + Brn^ + (7na» - 2^7^,71, - 2(?n,Z, - 2ffZ,m,. 

We have also 

r = 2my V 

= 2m (Zjic + mjy + n^) {l^ + m,y + rttz) 

= 2,Z, . 2??wj" + m,m, . 2my' + w jn, . 2m^' + {m^n^ + m,?!,) . 2my^ 

+ {^z + Wjia) • ^razx + (Zjmj + Z,m,) . tinxy 

= iWaC^ + C- 4) + imjm,(C+ A - B) + inan,(4 + B - C) 

+ (man, + THj^a) i^ + {nj^ + n^) G^ + (4i^ + ism,) JST, 
or 

— -P' = iiy, + Bm^mt + CniTii - F(mtrh + m,?!,) — ff (Z,Wa + Za^) — H{l^nii + ZjiTia), 
and similarly 

— G' = iiy^ + BmiTni + Cn^Ui — F(m^ni + miW,) — 6? (ZiW, + Z,ni) — fl'(Z,mi + Zim,), 
—H' = AIJ^ + Bmim^ + Cn^n^ — F{'min^ + marii) — ff (Zj^ + Zi^O — if (Zim, + iami). 

The quantities A\ B, G\ F', 0\ H\ are thus determined; these results, 
combined with those of the last article, are sufScient to determine the 
moments and products of inertia of a given body with respect to any set of 
rectangular axes when the moments and products of inertia with respect to 
aiiy other set of rectangular axes are known. 

' Example. If the origin of coordinates is at the centre of gravity of the body, prove 
tb^t the moments and products of inertia with respect to three mutually orthogonal 
an^ intersecting lines whose coordinates are 

*f (hi ^> »i> ^i> f*i> ''i)* (^2* ^> ^9 ^a» fhi ^i\ (hi ^i ^> ^»> Ms^ "s)* 

^'+ir(Xi«+MiH»'i*) etc. and /" - if (XjX8+fis/is+ vji^s) etc., 

whJire A\ B\ C\ F\ G\ H\ have the same values as above and M is the mass of the 
ly. (ColL Exam.) 



122 The dynamical specification of bodies [ch. y 

61. The principal axes of inertia; Cauchy's momentai ellipsoid. 

If now we consider the quadric surface whose equation is 

il a^ + By« + Ci* - 2 Fyz - ^ G^^a? - 2 Hxy = 1 , 

where A, B, C, F, Q, H, are the moments and products of inertia of a given 
body with respect to the axes of reference OxyZy it follows from the equation 

that the reciprocal of the square of any radius vector of the quadric is equal 
to the moment of inertia of the body about this radius. The quadric is 
therefore the same whatever be the axes of reference provided the origin is 
unchanged, and consequently its equation referred to any other rectangular 
axes Oxy'gf having the same origin is 

il V + By^ + C V - 2F'y'z' - 2G^'^V - 2H'xy' = 1 ; 

where A\ B, C\ F\ G\ H' are the moments and products of inertia with "l 

respect to these axes. 

This quadric is called the momenta! ellipsoid of the body at the point ; 
its principal axes are called the principal axes of inertia of the body at 0; the 
equation of the quadric referred to these axes contains no product-terms, and 
therefore the products of inertia with respect to them are zero : and the 
moments of inertia with respect to these axes are called the principal 
moments of inertia of the body at the point 0. 

The momental ellipsoid is also called the elliptoid of inertia ; its polar reciprocal with 
regard to its centre is another ellipsoid, which is sometimes called the ellipsoid of gyration. 

Example, The height of a solid homogeneous right circular cone is half the radius 
of its base. Shew that its momental ellipsoid at the vertex is a sphere. 

62. Calculation of the a/navlar momentum of a moving rigid body. 

We shall now shew how the angular momentum of a moving rigid body 
about any line, at any instant of its motion, can be determined. 

Let M be the mass of the body, {x, y, z) the coordinates of its centre of 
gravity ff, and (u, w, w) the components of velocity of the point 6?, at the 
instant £, resolved along any (fixed or moving) rectangular axes Oxyz whose 
origin is fixed; and let ((Uj, o),, a>s) be the components of the angular 
velocity of the body about G, resolved along axes Ox^yiZi^ parallel to the axes 
Oxyz and passing through 0. Let m denote a typical particle of the body, 
and let {x, y, z) be its coordinates and {u, v, w) its components of velocity ht 
the instant t ; and write ; 

x=:x+xi, y^y+yu z^z + z^, 



r 



61, 62] The dynamical specification of . bodies 125 

80 in virtue of the properties of the centre of gravity wn.:xe8— eav the axis of 

Xmxt = <i, 2my, = 0, Smx, s- tte kiuetic energy is 
moreover since (§17) we have "J^ te arbitrarily chosen, 

-X.S points, which is fixed, 
u,=t,tD,- y,«„ th = *,«, - ^,w,. w, = a.tout the instantaneous 
it follows that V>ody about this axis. 

Smu, = 0, 'S.mv, = 0, Xmw, «.-» ; ■ ■■ 

•**.is in its own plane, and the 

If hf denotes the angular momentum of the bo( J^f be the azimuth of the 

have therefore ^>Oa to the vertical, shew that 

ht=%m(xv — yii) "^et +, 

= tm {{x + IF,) (p + 1),) - (t/ + y,) (it + tr^« lamina about the horiaontal 
= S7«(S»-i/u) + 2m(x,«.-y.iO »*•*» the vertical. (CoU. Exam.) 

= jW"(^-p«) + Sm(a:,*<a,-ir,Zi(B,-y °f gravity and the motwn 
= M (xv — yu) — Gwi — FtOf + Cwj, 
where A, B, C. F.O,Hsx% the moments and prod^^^f *^^ ®''^^«' "' * '"^""g 
with respect to the axes G<r,y,t,. ^ ''^'«*>. «»>« '*epe°d8 on the 

»8 the kinetic energy of the 
Similarly the angular momenta about the axiaU now shew that these two 

"" /■ A,=Jtf(p;-«)+^«.-^«, ^"i*e independently of each 

J 1 under the influence of any 

The «nstlar momentum about any other lin ^^ t^^ ^^^ ^^^^ rectaniuta 

found (S 3^ by resolving these angular moment, rel,ti,o to axce fi«d in space, 

Goto*.™. If the body is constrained to to.-r" '^'!? ""'. '""•''"'• ■*'"''« 
is fixed in space, it is unnecessary to introduce 8?" }'"?■ ""e"»!ting in O, 
(», , .., »,) be the components of the angular ^' '"»''"° '""Sy » theH,fore 
fixed point with respect to any rectangular axe*' t' "> ^> y)> 
the fixed point as origin, and let A.B.G, F, fie energy of the motion relative 
prdduQts of inertia with respect to these axes 
{uiv, w) with respect to these axes of a P* -t ^g^ + ii'Jrtt. 
(4^y, «)are(5l7) ^ 

; xternal forces in an arbitrary dis- 

, . ,.-.«,-»„., ,.,«,-.«,4^ The Lag,«ngi.n equations of 

^od the angular momentum about the axis 
therefore be written in the form Ml=iZ, 

• 2fM (iE'w, - xeent - ya = © 

or I ■ - Omi-F<at + 

Similariy the angular momenta of th& =$, 
respectively are ^ 

/ ; ■ Aati — Hmt-^f 

and \ - Ha^-i- B(„J-^■^^■ 



122 The ^iyntifnieeU f^ecifieation of bodies [ch, v 

61. The principatf ^^ kinetic energy of a moving rigid body. 

If now we conaidef*' **^ * "K"*^ body which is in motion can be calculated 

': angular momenta. If the general theorem obtained 

Aai' + ^ case in which the polynomial /{x, y, z, x, y, i, x, y, 2) 

where A B C F ff ^')' ** immediately obtain the result that the kiiietic 

body with res^'t to thV^ ^'^^ of mass M is equal to the kinetic energy of a 

i moves with the centre of gravity of the body, together 

■^ — -^^ "^ of the motion of the body relative to its centre of 



63, 64] The dynamical speci^cation of bodies 125 

From this it foUows that if one of the coordinate axes — say the axis of x 
— is the instantaneous axis of rotation of the body, the kinetic energy is 
^Atoi*; and hence, since the directions of the axes can be arbitrarily chosen, 
the kinetic energy of any body moving about one of its points, which is fixed, 
is ^I(o\ where / is the moment of inertia of the body about the instantaneous 
axis of rotation, and cd is the angular velocity of the body about this axis. 

Example, A lamina can turn freely about a horizontal axis in its own plane, and the 
axis turns ahout a fixed vertical line, which it intersects. If ^ be the azimuth of the 
horizontal axis, and ^ the inclination of the plane of the lamina to the vertical, shew that 
the kinetic energy is 

where J, B^ H are the moments and product of inertia of the lamina about the horizontal 
axis and a perpendicular to it at the point of intersection with the vertical (Coll. Exam.) 

/ 64. Independence of the motion of the centre of gravity and the motion 

relative to it 

The result of the last article shews that the kinetic energy of a moving 
body can be regarded as consisting of two parts, of which one depends on the 
motion of the centre of gravity and the other is the kinetic energy of the 
motion relative to the centre of gravity. We shall now shew that these two 
parts of the motion of the body can be treated quite independeutly of each 
other. 

Let a rigid body of mass JIf be in motion under the influence of any 
forces. As coordinates defining its position we can take the three rectangular 
coordinates (j?, y, z) of its centre of gravity G, relative to axes fixed in space, 
and the three Eulerian angles {6, 4>t '^) which specify the position, relative 
to axes fixed in direction, of any three orthogonal lines, intersecting in 0, 
which are fixed in the body and move with it. The kinetic energy is therefore 

where f{d, <f>, yp-, 6, ^, '^) denotes the kinetic energy of the motion relative 
to G, 

Let XBx+YSy + ZSz + SS0 + ^S<f> + '9Sylr 

denote the work done on the body by the external forces in an arbitrary dis- 
placement (&v, By, Sz, BO, B<f>, S^) of the body. The Lagrangian equations of 

motion are 

Mx^X, My^Y, M'z^Z, 

dt \dd) d0 " ^' 
dt\d^J d<t> ^' 
dt \dyfrJ dyfr ' 



f ' 



I 



■ 



1 26 The dynamical specification of bodies [oh. v 

The first three of these equations shew that the motion of the centre of 
gravity of the body is the same as that of a particle of mass equal to the whole 
mass of the body, under the influence of forces equivalent to the total external 
forces acting on the body, applied to the particle parallel to their actual 
directions ; since the work done on such a particle in an arbitrary displace- 
ment would evidently be Xix + Yiy + Zhz, 

The second three equations shew that the motion of the body about its 
centre of gravity is the same as if the centre of gravity were flexed and the body 
subjected to the action of the same forces ; for in the motion relative to the 
centre of gravity, the kinetic energy of the body is f{0, 4>t '^> 6, ^, yjt), and 
the work done by the forces in an arbitrary displacement is 

These results are evidently true also for impulsive motion. 

Corollary. If a plane rigid body (e.g. a disc of any shape) is in motion in 
its plane, and if (x, y) are the coordinates of its centre of gravity, M its mass, 
the angle made by a line fixed in the body with a line fixed in the plane, 
Mk? the moment of inertia of the body about its centre of gravity, and if 
{X, Y) are the total components parallel to the axes of the external forces 
acting on the body, and L the moment of the external forces about the centre 
of gravity, then the kinetic energy is 

and the work done by the external forces in a displacement {Bx, Sy, SO) is 

X8x + YSy + LB0, 
and therefore the equations of motion of the body are ^ 

Mx^X, My=Y, Mm^L. 

Example. Obtain one of the equations of motion of a rigid body in two dimensions in 

the form 

M(j>f-{'k^'6)^L, 

where M is the mass of the body, / is the* acceleration of its centre of gravity, p is the 
perpendicular from the origin upon this vector, 211^ is the moment of inertia ahout the 
origin, 6 is the angle made by a line fixed in the hody with a line fixed in its plane, and L 
is the moment about the origin of the external forces. (ColL Exam.) 







1.^ 



OH. V] 



The dynamicol specification of bodies 



127 



Miscellaneous Examples. 

1. A homogeneous right circular cone is of mass M; its semi-vertical angle is /3, and 
the length of a slant side is l. Shew that its moment of inertia about its axis is 

^irPsin»A 

and that its moment of inertia about a line through its vertex perpendicular to its axis is 

ii^(l-i8in«/3), 

and its moment of inertia about a generator is 

iirPsin«i8(cos»/3+i). 

2. Shew that the moment of inertia of the area enclosed by the two loops of the 

lemniscate 

r»=a« cos 2^, 

about the axis of the curve is 

(3ir - 8) a» . 

^ — 7o X mass of area. 

3. Any number of particles are in one plane, if the masses are m^, m,, ..., their 
distances apart cfjj, ..., the relative descriptions of area hy^t •.•! and the relative irelocities 
^u> •••; prove that 

{2mim^di^)/2m, {2miin2hi^)/2m, (2fn|fi4t;i2')/22m, 

ore respectively the moment of inertia about the centre of inertia, the angular momentum 
about the centre of inertia, and the kinetic energy relative to the centre of inertia. 

(Coll. Exam.) 

4. Prove that the moment of inertia of a hollow cubical box about an axis through 
the centre of gravity of the box and perpendicular to one of the faces is 

where M is the mass of the box and 2a the length of an edge. The sides of the box are 
supposed to be Ihin. (ColL Exam.) 

Shew that the moment of inertia of an anchor-ring about its axis is 

2irp«a*c(c*4-ia»), 

a is the radius of the generating circle, c is the distance of its centre from the axis 
anchor-ring, and p is the density. 

Shew how to find at what point, if any, a given straight line is a principal axis of a 
and if there is such a point find the other two principal axes through it. 

uniform square lamina is bounded by the axes of a; a^^d y and the lines a;~2c, y=2c, 

comer is cut off it by the line a;/a+y/6«2. Shew ti?at the two principal axes at 

itre of the square which are in its own plane are inclined to the axis of x at angles 

«6-2.(a+&)+3c. (CoU.E.am.) 



tan 2^= 



(a-6)(a-*-6~2c) * 



\ 



Shew that the envelope of lines in the plane of an area about wfii^ that area has a 
itant moment of inertia is a set of confocal ellipses and hyperbolas. N^ence find the 



ion of the principal axes at any point. 



(Coll* Exam.) 



\ 



\ 



^ 






\ 



128 



The dynamical specification of bodies 



[CH. V 



tan 2^= 



8. Find the principal moments of inertia at the vertex of a parabolic lamina, latus 
rectum 4a, bounded bj a line perpendicular to the axis at a distance h from the vertex. 

Prove that, if 15A>28a, two principal axes at the point on the parabola whose abscissa 
is -a-^-ia^- 4ah/6 + 3A'/7)^ are the tangent and normaL (CoU. Exam.) 

9. Find how the principal axes of inertia are arranged in a plane body. Write down 
the conditions that particles nii at (Xi, yi), where t's^l, 2, ..., may be equimomental to & 
given plate. -Shew that the six quantities m|, 914, or^, x^t y^ y% can be eliminated from 
these conditions. 

If three equal particles are equimomental to a given plate, the area of the triangle 

formed by them is 3 >/3/2 times the product of the principal radii of gyration at the 
centre of gravity. (Coll. Exam.) 

10. A uniform lamina bounded by the ellipse b^a^-\-ahf^^a^b^ has an elliptic hole 
(semi-axes c, d) in it whose major axis lies in the line ^— y, the centre being at a 
distcmce r from the origin ; prove that if one of the principal axes at the point (x, y\ 
makes an angle B with the axis of jp, then 

Habxy-od [4 (sf V2 ~ r) (y ^/2- r) - (c«-d«)] 

a6 [4 («« -y>) +a» - 62] - erf [2 (a? V2 - r)« - 2 (y \/2 - r)«] ' 

(ColL Exam.) 

11. If a system of bodies or particles is moved or deformed in any way, shew that 
' the sum of the products of the mass of each particle into the square of its displacement 

is equal to the product of the mass of the system into the square of the projection in any 
given direction of the displacement of the centre of gravity, together with the sum of the 
products of the masses of the particles into the squares of the distances through which 
they must be moved in order to bring them to their final positions after communicating 
to them a displacement equal to the projection in the given direction of the displacement 
of the centre of gravity. (Fouret.) 

12. The principal moments of inertia of a body at its centre of gravity are (ii, B^ C) ; 
if a small mass, whose moments of inertia referred to these axes are (A\ E^ C'\ be added 
to the body, shew that the moments of inertia of the compound body about its new 
principal axes at its new centre of gravity are ' 

A'\'A\ B-\-B\ C+C\ 

accurately to the first order of small quantities. (Hoppe.) 

13. Shew that the principal axes of a given material system at any point are the 
normals to the three quadrics which pass through the point and belong to a certain 
confocal system. 

If (^, m, n, X, ft, v) be the six coordinates of a principal axis and the associated 
Cartesian system be the principal axes at the centre of gravity, then shew that 

Alk+Bm/jk-k-Cnv'^Oy 

and therefore all principal axes of a given system belong to a quadratic complex. 

(Coll Exam.) 

14. A smoothly jointed framework is in the form of a parallelogram formed by 
attaching the ends of a pair of rods of mass m and length 2a to those of a pair of rods of 
mass m! and length 26. Masses if are attached to each of the four comers. Express iAxe 
angular momentum of the system about the origin of coordinates, in terms of the 
coordinates (x, y) of the centre of gravity and the angles 6 and <f> between the two pairs of 
sides and the axis of x, (Coll. Exam.) 



/ 



/ 



/ 






■Jretdom: motion round 

have been developed in 
the foregoing chapters in order to determine the motion of holonomic systenia 
of rigid bodies in those cases which admit of solution by quadratures. 

It ia natural to consider first those systems which have only one degree of 
freedom. We have seen (§ 42) that such a system is immediately soluble \jj 
quadratures when it possesBes an integral of energy : and this principle is 
sufficient for the integration in most cases. Sometimes, however (ag. when 
we are dealing with systems in which one of the surfaces or curve» of con- 
straint is forced to move in a given manner), the problem as originally fonna- 
lated does not possess an integral of energy, but can be reduced (e.g. by the 
theorem of § 29) to another problem for which the integral of energy holda ; 
when this reduction has been performed, the problem can be integrated aa 
before. 

The following examples will illustrate the application of the^priqciples. 

(i) Motion of a rigid body round afixtd axtt. 

Consider the motion of a single rigid body which is free to turn elaovj an axis, fixed in 
the bod; and in apace. Let / be the moment of inertia of the body 'fbout the uis, so 
th»t its kinetic energy is \I6\ where 6 ia the angle made by a mav'^lb piano, passing 
through the oiis and fixed in the body, with a plane passing tl^ugh the axis and 
fixed iu apace. Let e be the moment round the axis of all theeitemal forcea acting on 
the body, ea that eS0 is the work done by theae forces in the infinitMBimal displacement 
which changes 6to6+i8. The Lagrangian equation of motion 

dt \Se/ 3d iMveitf 

then gives I'S-O, ^''W 

Uit terl _fM 

which is ft differential equation of the second order for the mnation of 0. 



^ 



,• % 



130 The Soluble Problems of Rigid I>gnomic9 [ch. vi 

If the forces are conservative, and V{6) denotes the puteti/tial energy, this equatioik 
becomes J 

which on integration gives the equation of energy ^ 

^/^+ V($!ji^Cy where c is a constant. 

Integrating again, we have 

^=/*f(2(c- r)}-*rf^+constant, 

and this relation between B and t determines the motion, the two constants of integration 
being determined by the initial condltiona 

The most important case is that ia which gravity is the only external force, and the 
axis is horizontal. In this case let be the centre of gravity of the body, C the foot of 
the perpendicular drawn from to the axis, and let CO^^h. The potential energy is 
- Mffh cos By where M is. the mass of the body and B is the angle made by CG with the 
downward verticaj^^-^d the equation of motion is 

I 

This is the same as the equation of motion of a simple pendulum of length I/Mh, and 
the motion can therefore be expressed in terms of elliptic functions as in § 44, the solution 
being of the form 

in the oscillatory case, and of the form 



■*■!-» {l(^-)'<-«^ 4 



in the circulating case. The quantity I/I£h is called the length of the equivalent simple 
pendtUnnL 

If be a point on the line CO such that OC^I/Afh^ the points and C are called 
respectively the centre of oscillation and the centre of suspension, A curious result in this- 
connexion is that the centre of oscillation and the centre of suspcTision are convertible^ 
Le. if is th&^ntre of oscillation when C is the centre of suspension, then C will be the 
centre of o^^ation when is the centre of suspension. To prove this result, we 
have by § 59 

Moment of inertia of the body about 0= Moment of inertia about 0+M, 00^ 

^I-M.CO*+M,00\ 
and therefore we Vjve 

Moment ? inertia of body about ^ 1- Mh^-^-M (JjMh - A)« 
Distance orcentre of gravity from "" IjMh—h 

^Mh^-M^IIMh-h) 

If therefore the bocQr were suspended from 0, the equation of motion would still be 

V^asSM' .. Mgh . . ^ 

yhe sy. ^+-f-8m^=0, 

which establishes the res* *> is evident that the period of an oscillation would be the 

same about either of the \ 7 and 0. 



\ 

1 



t 

I 



I v 



66] The Soluble Problems of Rigid Dynamics 181 

(ii) Motion of a rod w\ which an insect is crawling. 

We shall next study the motion of a straight uniform rod, of mass m and length 2ay 
whose extremities can -slide on the circumference of a smooth fixed horizontal circle of 
radius c ; an insect of mass equal to that of the rod is supposed to crawl along the rod 
at a constant rate v relative to the rod. 

Let 6 be the angle made by the rod at time t with some fixed direction, and let x be 
the distance traversod by the insect from the middle point of the rod. The kinetic 

energy of the rod is ^w ( c* — o" ) ^*> ^^^ *^® kinetic energy of the insect is due to a 

component of velocity {ir-(c*— a*)*^} along the rod and a component of velocity x6 
perpendicular to the rod, so the total kinetic energy of the system is 

there is no potential energy. 

Since x^vt^ (t being measured from the epoch when x is zero), we have 

The coordinate ^, which is now the only coordinate, is ignorable, and we have ther^ore 



or m 



(c«- y*) rf- w(c>-a«)* {t;-(c>-a«)*^*}+«i»*<*^*=«>n8tant, 

or ^ {2c« - J a« + v^fi) = constant 

Integrating this equation, we have 

^-^o=* tan"* {^^ (2c*- Ja«)-*}, 
where Bq and k are constants. This formula determines the position of the rod at any time. 

(iii) Motion of a cone on a perfectly rough inclined plane. 

Consider now the motion of a homogeneous solid right circular cone, of mass M and 
semi-vertical angle ^3, which moves on a perfectly rough plane (i.e. a plane on which only 
rolling without sliding can take place) inclined at an angle a to the horizon. Let I be the 
length of a slant side of the cone, and let B be the angle between the generator which is in 
contact with the plane at time t and the line of greatest slope downwards in the plane. 
Then if ^ be the angle made by the axis of the cone with the upward vertical, x ^ one 
side of a spherical triangle whose vertices represent respectively the normal to the plane, 
the upward vertical, and the axis of the cone ; the other two sides are a and (^tr - /3), the 
angle included by these sides being {ir — B). We have therefore 

cos X =■ <^os a sin j8 - sin a cos /3 cos B ; 

but the vertical height of the centre of gravity of the cone above its vertex is \l cos 0cos ;(, 
and the potential energy of the cone is Mg x this height ; if therefore we denote by V the 
potential energy of the cone, we have (disregarding a constant term) 

F= - \Mgl sin a cos* /3 cos B, 

9—2 



132 The Solvble Problems of Rigid Dynamics [ch. vi 

We have next to calculate the kinetic energy of the cone; for this the moments of 
inertia of the cone about its axis and about a line through the vertex perpendicular to the 
axis are required: these are easily foimd (by direct integration, regarding the cone as 
composed of discs perpendicular to its axis) to be ^J/^sin'^S and }ifZ2(co8'0+^sin'/9) 
respectively, and so the moment of inertia about a generator is, by the theorem of § 60 
(since the direction-cosines of the generator can be taken to be sin /S, 0, cos/3, with respect 
to rectangular axes at the vertex, of which the axis of z is the axis of the cone), 

Ji/?«(cos«/3+isin»0)sin«^+AJr^«sin«/3cos«0, 
or JJtr^«sin8i3(cos«0+}). 

Now all points of that generator which is in contact with the plane are instantaneously 
at rest, since the motion is one of pure rolling, and therefore this generator is the 
instantaneous axis of rotation of the cone. If « denotes the angular velocity of the cone 
about this generator, the kinetic energy of the cone is therefore (§63, Corollary) 

fi/Psin«0(oos«0+J)««. 
But (§ 15) we have 

a>s^cot/3, 

and substituting this value for », we have finally for the kinetic energy T of the cone 

the value 

T^-f i/^/«cos»i3 (co8«/3+i)rf«. 

The Lagrangian equation of motion 

becomes therefore in this case 

} if?« cos« i3 (cos« /3+i) ^+1%^ sin a cos« i3 sin ^-0, 

•• 7 sin a . ^ _ 

/(cos«/3+i) 

This is the same as the equation of motion of a simple pendulum of length 

/ cosec a (cos* ^ + }) ; 

the integration can therefore be effected in terms of elliptic functions, as in § 44. 

(iv) Motion of a rod on a rotating frame. 

Consider next the motion of a heavy uniform rod, whose ends are constrained to move 
in horizontal and vertical grooves respectively, when the framework containing the grooves 
is made to rotate with constant angular velocity a> about the line of the vertical groove. 

Let 2a be the length of the rod, M its mass, and 6 its inclination to the vertical. 
By § 29, the effect of the rotation may be allowed for by adding to the potential energy 
a term 

— ^ a)*p / 0?* sin' ^ cLp, 

where p is the density of the rod and x denotes distance measured from the end of the 
rod which is in the vertical groove ; integrating, this term can be written 

-Jir«)«a«sin«^. 

The term in the potential energy due to gravity is 

- Mffa cos $j 

and the total potential energy V is therefore given by the equation 

V= -Mffacose-^MaMsm^ e. 



V. 




65] The Soltm^Problems of Rigid Dynamics 133 

The horizontal and vertical com^nents of velocity of the centre of gravity of the rod 
are a sin . 6 and a cos ^ . ^, so the pif t of the kinetic enei^ due to the motion of the 
centre of gravity is ^Ma^if^ ; and since^^jbhe moment of inertia of the rod ahout its centre 
is iMa^y the part of the kinetic energ^^due to the rotation of the rod about its centre 
is \Ma^6^\ we have therefore for the total tdnetic energy jTthe equation 

The integral of enei^ is therefore 

\Ma^$^ - Mga cos ^ - \MvM sin' $ = constant, 
or, writing cos 6»mXy 

i.=(i-*.){..-(^-^y}. 

where c denotes a constant ; this constant must evidently be positive, since d^ and (1 - ^') 
are positive. We shall suppose for definiteness that f is not very large and that 3^/4a«o' 
is less than unity, so that x oscillates between the values 3g/4€ua*±9l». 

To integrate this equation, we write* 






^^Sa 12 64a«««^12 

where £ is a new dependent variable. Substituting this value for x in the differential 
equation, we have 

where the values 

correspond respectively to the values 

■■r ''~"^' *""45^«"«' ^""4^«'*'«' 

J ^1+^+^3 is zero and that ei>e^>e^. 

/ore (-i|f^ (^+y), whefe the function p is formed with the roots e^, e,, e,, 

otes a constant. Since ^i > ^g > «3, and (P (^+y) lies between e^ and ^3 for 

^since. x lies between 3^/4aa>' - c/<o and 3^/4a«i»' +«/«»), the imaginary part of 

/ must be the half-period u, ; the real part of y can then be taken as zero, 

ids only on the choice of the origin of time. We have therefore 

»'C^+«3)+8^ 12 Q4aW^l2 
equation determines $ in terms of L 

(v) Motion of a diac^ one of whose points is forced to move in a given manner. 

Consider next the motion of a disc of mass My resting on a perfectly smooth horizontal 
plane, when one of the points A of the disc is constrained to describe a circle of radius c 
in the horizontal plane, with uniform angular velocity «. 

* Cf. Whittaker, A Course of Modem AnalysU, § 185. 




134 The Soltible Problems of Rifid Dynamics [ch. vi 

Let O be the centre of gravity of the disc, and let il(? be of length a. The acceleration 
of the point A is of magnitude a^^ and is directed along the inward normal to the circle : 
if therefore we impress an acceleration co>*, directed along the outward normal to the 
circle, on all the particles of the body and supfK)se that A is at rest, we shall obtain the 
motion relative to A, The resultant force actif^g on the body in this motion relative to A 
is therefore Mc<o\ acting at 6^ in a direction ytd-allel to the outward normal to the circle. 

Let 6 and <^ be the angles made witl^^zed direction in the plane by the line AG and 

the outward normal to the circle respf^Ttively ; then the work done by this force in a small 

displacement b6 is /^ 

Mc^i^a sin (<^ - 6) M, 

and the kinetic energy of the body is ^Mi^B*^ where Mi^ is the moment of inertia of the 
body about the point A. The La^angian equation of motion is therefore 

jiii^B = i/cK^s sin {ip - 6), 

But since <^=a>, we have ^»0 ; so if ^ be written for {B- <f>\ we have 

This is the same as the equation of motion of a simple pendulmn of length J^glacts!^ ; 
the integration can therefore be performed by means of elliptic functions as in § 44. 

(vi) Motion of a disc rolling on a constrained disc and linked to it. 

Consider the motion of two equal circular discs, of radius a and mass My with edges 
perfectly rough, which are kept in contact in a vertical plane by means of a link (in the 
form of a uniform bar of mass m) which joins their centres : the centre of one disc is fixed, 
and this disc A is constrained to rotate with uniform angular acceleration a ; it is required 
to determine the motion of the other disc B and the link. 

Let (f> be the angle which the link makes with the downward vertical at time <, and 
let B be the angle turned through at time t by the disc A. The angular velocity of disc A 

m 

is By and the velocities of the points of the discs which are instantaneously in contact are 
therefore each aB. Since the velocity of the centre of the disc B is 2a^, it follows that the 
angular velocity of the disc B about its centre is 2^ - B. Since the moment of inertia of 
each disc about its centre is ^Ma^, the kinetic energy of the system is 

T^iM,^d'+iM.^{24>-i)^+iM.(2ay^*-\-im,'^4>*; 

and d=at+*y where f is a constant. 
The potential energy of the system is 

r=a - (^M-^ m) ag cos <^, 
and the Lagrangian equation of motion is 



dt\zx) a* "a<^' 



or ^ {(fiM-^^m) a^ - Ma^} = - (2i/^+ m) ag sin <^. 

Since B^Oy this equation gives 

(62/*+ |m} a^ - Ma^a + {2M ■\-m) ageiu fft^O, 
Integrating, we have 

(3if+$n») a^-Ma^aafi - (2M'^m)agQ08<fi^Cy 



66, 66] The Soluble Problems of Rigid Dynamics 136 

where c is a constant depending qd t^ initial conditions : and as the variables t and <f> are 
separable^ tHlS" equation can again be integrated by a quadrature : this final integral 
determines the motion. 

Example, If the system is initially at rest with the bar vertically downwards, sh^ 
that the bar will reach the horizontal position if 



a> 






66. The motion of systems with two degrees of freedom. 

In the dynamics of rigid bodies, as in the dynamics of a particle, the 
possibility of solving by quadratures a problem with two degrees of freedom 
generally depends on the presence of an ignorable coordinate. The integral 
corresponding to the ignorable coordinate can often be interpreted physically 
as an integral of momentum or angular momentum. The formation and 
solution of the differential equations is effected by application of the 
principles developed in the preceding chapters : this will be sh^wn by the 
following illustrative examples. 

(i) Bod passing through ring. 

Consider, as a first example, the motion of a uniform straight rod which passes through 
a small fixed ring on a horizontal plane, being able to slide through the ring or turn in any 
way about it in the plane. 

Let the distance from the ring to the middle point of the rod at time ^ be r, and let the 
rod make an angle 6 with a fixed line in the plane; let 2Z be the length of the rod, and M 
its mass. 

The moment of inertia of the rod about its middle point is ^M^^ and the kinetic energy 
is therefore 

there is no potential energy. 

The coordinate $ is ignorable, and the corresponding integral is 

— :=constant, 
oB 

or (r* + ^) B = constant 

The integral of energy is 

fj + r^ + jp^2 = constant 

Dividing the second of these int^prals by the square of the first, we have 



(T+rap + ;5^n^=«» ^^®^ ^ « * constant, 



or ^+con8tant=: [{(r8+ii«)(cr«+JcP-l)}"*rfr. 

Writing cr»=s«, this becomes 



^+constant=a |{4»(«+Jc^ («4-JcP-l)}~*ci«. 




136 The Soluble Problems of Rigid Dynamics [oh, vi 

If therefore fp denotes the Weierstraasian elliptic function with the roots 

«i=i(-i+?<^). «2=i(2-ic^), «s=i(-i-i<^), 

dr 
which satisfy the relation e{>e{>e^ '^ dk'^ sufficiently great initially, we have 

« ai jf^ (^ - ^q) - 6^ , where ^q is a constant of integration ; 

since $ is positive, we have if^(^ — ^o}>^i ^^^ ^^^ values of 6y and consequently the 
constant 6^ is real 

The solution of the problem is therefore contained in the equation 

(ii) One cylinder rcUing on another under gravity. 

Let it now be required to determine the motion of a perfectly rough heavy solid 
homogeneous cylinder of mass m and radius r, which rolls inside a hollow cylinder of mass 
i/'and radius 12, which in turn is free to turn about its axis (supposed horizontal).^ 

Let ^ denote the angle which the plane through the axes of the cylinders at time t 
makes with the downward vertical, and let $ be the angle through which the cylinder of 
mass M has turned since some fixed epoch. The angular velocities of the cylinders about 
their axes sxe easily seen to be ^ and {{R-r)^- R6)lr respectively; and the moments of 
inertia of the cylinders about their axes are MI& and ^mi^ respectively; so the kinetic 
energy T of the system is given by the equation 

while the potential energy is given by the equation 

F«» ^mg{R-r) cos ^. 

The coordinate 6 is clearly ignorable ; the integral corresponding to it is 

ar 

— ;= constant, 

or MB^ - ^ mR {(/2 - r) ^ - RB) » k^ where ir is a constant. 

The integral of enei^gy is 

jTh- Vwmh, ' where A is a constant, 

or iJtri2^Him{(/2-r)^-i;^}« + im(/2-r)«^«-7W^(/2-r)cos<^-A. 

Eliminating 6 between the two integrals, we obtain the equation 

This is the same as the equation of energy of a simple pendulum of length 

the solution can be effected by means of elliptic functions as in § 44. 

(iii) Rod moving in a free circular fiume. 

We shall next consider the motion of a rod, whose ends can slide freely on a smooth 
vertical circular ring, the ring being free to turn about its vertical diameter, which is fixed. 





66] The Solvble Problems of Rigid I>ynamics 137 

Let m be the mass of the rod and 2a its length ; let i/' be the mass of the ring and r 
its radius : let ^ be the inclination of the rod to the horizontal, and ^ the azimuth of the 
ring referred to some fixed vertical plane, at any time t 

The moment of inertia of the rod about an axis through the centre of the ring 
perpendicular to its plane is m{f^-'^a^\ and the moment of inertia of the rod about the 
vertical diameter of the ring is m{(r*-a*)sin*d+Ja*cos*^}. The kinetic energy of the 
system is therefore 

T- im (r* - Ja«) *+ii/r«0«+im^« (r« sin* ^ -a« sin« ^ + J a^ cos« 6), 

The potential energy is 

F--«i^(r»-a«)*cos^. 

The coordinate fft is evidently ignorable ; the corresponding integral is 

—r«= constant, 

or ^Mf^^-^-m^ (r«sin« ^- a« sin« ^+ Ja«co8« 0)^k, 

where i: is a constant Substituting the value of ^ found from this equation in the 
int^ral of energy 

we have ' 

Jm (r* - la*) rf«= A +»wr(r*- a')* COS ^- 4 i-iZ3-; — ,^ - ^a — « - ^in . i < rsc* 

* ^ ' ' ^^ ' * ^i/r*+ m (r* sm'^- a* sm*^+}a* COS* ^) 

In this equation the variables 6 and t are separable; a further integration will 
therefore give $ in terms of t^ and so furnish the solution of the problem. • 

(iv) ffoop and ring. 

We shall next discuss the motion of a system consisting of a uniform smooth circular 
hoop of radius a, which lies in a smooth horizontal plane, and is so constrained that it can 
only move by rolling on a fixed straight line in that plane, while a small ring whose mass 
is 1/X that of the hoop slides on it. The hoop is initially at rest, and the ring is projected 
from the point furthest from the fixed line with velocity v. 

Let denote the angle turned through by the hoop after a time t from the commence- 
ment of the motion, and suppose that the diameter of the hoop which passes through the 
ring has then turned through an angle ^. Taking the ring to be of unit mass, so that the 
mass of the hoop is X, the moment of inertia of the hoop about its centre is Xa*, and this 
centre moves with velocity a^ while the velocity of the ring is compoimded of components 
o^ and a^, whose directions are inclined to each other at an angle ^. The kinetic energy 
of the system is therefore 

ir=iXa20«+iXa«<^«+i (a«^HaN^+2a?i^^cos ^) 
«i (2X + 1) a«^2+ ia«>jr«+a«^iir cos ^, 
and the potential energy is zero. 

The coordinate ^ is evidently ignorable, and the corresponding integral is 

—7 ""Constant, 

or (2X + 1 ) a^ + a^ cos ^ t. the initial value of this expression 

=av. 



138 The Soluble Problems of Rigid Dynamics [ch. vi 

Integrating thia eqiutioD, we bare 

(SX-l'l)0+sm^ — =the initial value of thiaeipression 
-0. 

*=2xTi(S-'"""'*')- 

Tliis equation detarminM ^ in terms of ^. 

The equation of energy ia 

T=\ta initial valuev^v*, 

andsubsUtutii^for^ita value («/<>— cos ^.^)/(2X-M) in this equation, we have 

so *=-n= (*(2X+ainV)*'i*- 

Writing sin ^ ~ x, tliia becomes 

itV2X Jo 
la order to evaluate thia integral, we introduce an amiliaiy variable «, defined by the 

»= /''(a+x>)-^{l-j^-*dr. 
Write 3? = 2X/{, where £ ie a new variable ; the last integral becomes 

which is equivalent to 

«-«>(K)-!(l-2«. 
where the function g> (it) ia formed with the roots 

«, = ia+«), «,=i(l-2X), «j=-|(l+X); 
these roots are real and satisfy the inequality ej>e,>e„ so P(u) is real and greater than 
«i for real values of w. 

Now we have dt--^(ik+a?)*a~x^-*dx, 

■ nVax 

Integrating, we have 



where f (u) denotes the Weierstrassian Zeta-function. 

Thus finaUy the eoordinat* ^ and the time t are expreettd in term* of an auxiliary 
varicMe it by the equaliim* 



^'-i(l+4X)» + f(«) + J 



66, 67] The Soluble Problems of Rigid Dynamics 139 

67. Initial motions. 

We have already explained in § 32 the general principles used in finding 
the initial character of the motion of a system which starts from rest at 
a given time. The following examples will serve to illustrate the procedure 
for systems of rigid bodies. 

(i) A particle hangs by a string of length h from a point in the cvrcumference of a disc 
of twice its mass and of radius a. The disc can turn about its axis, which is horizontal^ and 
the diameter through the point of attachment of the string is initially horizontal. To find the 
initial path of the particle. 

Let $ denote the angle through which the disc has turned, and <(> the inclination of the 
string to the vertical, at time t from the beginning of the motion : let m be the mass of the 
I>article. The horizontal and (downward) vertical coordinates of the particle with respect 
to the centre of the disc are 

acos^+6sin<^ and asin^+^cos^, 
ao the square of the particle's velocity is 

d^+b^^-'2aham(6+il>)d^ 
and the kinetic energy of the system is 

T ^ma^+iimb^*- mob Bin {6 'k'<l>)i4>f 

while the potential energy is 

r«a - mg (a sin ^+6 cos <f>). 

The Lagrangian equations of motion are 

dt\di) d3 dB' 



d f^T\_dT^_dy 



r2a'^-a6 cos (^+<^) ^-^a cos ^-a5 sin (^+^) ^*>0, 
or J 

[ 6*^-06 cos (^+^) ^+^6 sin ^- oft sin (^+^) (9=0. 

Initially the quantities $, 6, ^, 6, are all zero : these equations therefore give initially 
O^gl^a and <^aO, so the expansion of 6 begins with a term gt^j^ and that of ^ with a 
term higher than the square of t. Assuming 

substituting in the above differential equcmons, and equating powers of t, we can evaluate 
the coeflacients A, 5, C, ... ; we thus find 

4a ^ 

Now if X and y are the coordinates of the particle referred to htK^2on|;al and (downward) 
vertical axes through its initial position, we have 

^-a(l-co8(9)-6sin<^-ia^-6*=l^^,apP«>ximaU;,y^ 

and y =a sin ^+ 6 (cos (^ - l)=a^='^ , approximately. 



140 The Soluble Problems of Rigid Dynamics [en. vi 

Eliminating t between these equations, we have 

y'=s30a6jp, 

and this is the required approximate equation of the path of the particle in the 
neighbourhood of its initial position. 

(ii) A ring of mass m can dids freely on a uniform rod of mass M and length 2a, which 
can turn about one end. Initially the rod is horizontal^ with the ring at a distance Tq from 
the fixed end. To find the initial curvature of the path of the ring in space. 

Let (r, B) denote the polar coordinates of the ring at time ty referred to the fixed end of 
the rod and a horizontal initial line, 3 being measured downwards from the initial line. 
For the kinetic and potential energies we have 

F=5 - mrg sin B - Mag sin $. » 

The Lagrangian equations of motion are 

^d /dT\ ZT dV 



i 






dt\drj dr" 8r ' 

\dt\^^) do "55' 

r-ri'-gsinB^O. 
or •( . .. . 

l^Ma^a + m$^'B+2mrrB — Mga cos ^ - mgr cos ^—0. 

Since r, B, and 6 are initially zero, we can assume expansions of the form 

substituting these expansions in the differential equations, and equating coefficients of 
powers of t, we find 

0,-0, 03=0, a4-^6j(5r+46jro), 

«""2(4J/o«+3mro«)* 

The coordinates of the particle, referred to horizontal and vertical axes at its initial 
position, are 

x=^r COB B-rQ and y«r sin tf, 

or approximately *=(«4-i^oV)^> y=**o^8^- 

The ciurvature of the path is given by the equation 

1 , 2j7 204 L 

p y* o^*''o ^0 

and on substituting the above values of 6, e-*^ ®4> ^® ^*v® 

1 iifa(4o~3ro ) 
7/'9ro«(i/b+ni/^)' 

This is the required initial ci -^^^^^^^ o^ ^^^ P**^ of the ring. 

Example. Two unifr-^^ ^"^ds AB, BC^ of masses m^ and wi,, and lengths o and h 
respectively, are free''^ hinged at B, and can turn round the point -4, which is fixed. 
Initially, AB is hr^^^i^^ and BC vertical Shew that, if (7 be released, the equation of 
the initial path ^^ *^® Poi°t of trisection of BC nearer to C can be put in the form 

y»«60 (l+2mjm{) abx. 

(Camb. Math. Tripos, Part 1, 1896.) 



67, 68] The Soluble Problems of Rigid Dynamics 141 

68. The motion of systems with three degrees of freedom. 

The possibility of solving by quadratures the motion of a system of rigid 
bodies which has three degrees of freedom depends generally (as in the case 
of systems with two degrees of freedom) either on the occurrence of ignorable 
coordinates, giving rise to integrals of momentum and angular momentum, or 
on a break-up of the kinetic potential into a sum of parts which depend on 
the coordinates separately. The following examples illustrate the procedure. 

(i) Motion of a rod in a given Jidd of force. 

Consider the motion of a uniform rod, of mass m and length 2a, which is free to move 
on a smooth table, when each element of the rod is attracted to a fixed line of the table 
with a force proportional to its mass and its distance from the line. 

Let (x, y) be the coordinates of the middle point of the rod, and $ its inclination to the 
fixed line. The kinetic energy is 

and the potential energy is 

F= -— I (y + r sin B)^ dr, where /x is a constant, 

or V^fitn (iy'+ia* sin* 6). 

The Lagrangian equations of motion are therefore 

y - - My, 

^(2^*)+/i Bin 2^-0. 

The first two equations give 

\x^ct+d, 

|y=/8iu(/i*^+€), 

where c, d, /, c, are constants of integration ; the centre of the rod therefore describes a 
sine curve in the plane. The equation for 6 is of the pendulum type, and can be 
integrated as in § 44. 

(ii) Motion of a rod and cylinder on a plane. 

We shall next discuss the motion of a system consisting of a smooth solid homogeneous 
circular cylinder, of mass J/ and radius c, which is moveable on a smooth horizontal plane, 
and a heavy straight rail of mass m and length 2a, placed with its length in contact with 
the cylinder, in a vertical plane perpendicular to the axis of the cylinder and passing 
through the centre of gravity of the cylinder, and with one extremity on the plane. 

Let 6 be the inclination of the rail to the vertical, and x the distance traversed on the 
plane by the line of contact of the cylinder and plane, at any time t. The coordinates of 
the centre of the rod referred to horizontal and vertical axes, the origin being the initial 
point of contact of the cylinder and plane, are easily seen to be 

4?-ccot( — - rj+asin^ and acos^. 

Let ^ be the angle through which the cylinder has turned at time t. The kinetic 
energy of the system is 

T^ima^+^U^iccoseG^(^''^.d+aGose.dy-hima*^ 



142 The Soluble Problems of Rigid Dynaamcs [CH. vi 

The potential energy is given by the equation 

Fasm^a cos tf. 

The coordinates x and <^ are evidently i^orable ; the corresponding integrals are 

^-rs constant 
ox 

(which may be interpreted as the integral of momentum of the system parallel to the axis 
of x) and 

—r«=s constant 

(which may be interpreted as the integral of angular momentum of the cylinder about its 
axis). These integrals can be written 

m j^-^ccosec^f J — -).rf+acoe^.rf[-+i/';»=constant, 

^Mc^^ ss constant. 

Substituting for x and ^ the values obtained from these equations in the integral of 

energy 

T+ F:= constant, • 

we have the equation 

rf* io'+<**8in*^+— -^-lacos^-^ccosec'f^-- ^U \=^d-2gacoa3, 

where c? is a constant This equation is again integrable, since the variables t and B are 
separable ; in its integrated form it gives the expression of B in terms of t : the two 
integrals found above then give x and ^ in terms of t 

69. Motion of a body abovt a fixed point under no forces. 

One of the most important problems in the dynamics of systems with 
three degrees of freedom is that of determining the motion of a rigid body, 
one of whose points is fixed, when no external forces ar6 supposed to act. 
This problem is realised (§ 64) in the motion of a rigid body relative to its 
centre of gravity, under the action of any forces whose resultant passes 
through the centre of gi-avity. 

In this system the angular momentum of the body about every line which 
passes through the fixed point and is fixed in space is constant (§ 40), and 
consequently the line through the fixed point for which this angular momen- 
tum has its greatest value is fixed in space. Let this line, which is called the 
invariable line, be taken as axis OZ, and let OX and OF be two other axes 
through the fixed point which are perpendicular to OZ and to each other. 
The angular momenta about the axes OX and OY are zero, for if this were 
not the case the resultant of the angular momenta about OX, OY, OZ, would 
give a line about which the angular momentum would be greater than the 
angular momentum about OZ, which is contrary to hypothesis. It follows 
(§ 39) that the angular momentum about any line through making an 
angle with OZ is d cos 0, where d denotes the angular momentum about OZ. 



68, 69] The Soluble Problems of Rigid Dynamics 148 

The position of the body at any time t is suflSciently specified by the 
knowledge of the positions at that time of its three principal axes of inertia 
at the fixed point: let these lines be taken as moving axes Oosyz; let (0, <^, ^jr) 
denote the three Eulerian angles which specify the position of the axes Oxyz 
with reference to the axes OXYZ, let {A, B, C) be the principal moments of 
inertia of the body at 0, supposed arranged in descending order of magnitude, 
and let (oi^, cus, a>,) be the three components of angular velocity of the system 
about the axes Ox, Oy, Oz, respectively, so that (§§ 10, 62) 

Aa>i =s — d sin ^ cos yjt, 

Ba}2 = dsin sin yjt, 

. Co)j = d cos Of 



or (§ 16) 



• • • • cL 

sin*^— ^ sin cos -^ = — -r sin ^ cos y^, 

< 6 cos -^ + <^ sin ^ sin -^ = ^ sin sin yjr, 

•^ + <^ cos ^ = To^ COS ^. 



These are really three integrals of the differential equations of motion of 
the system (only one arbitrary constant however occurs, namely d, our special 
set of axes being such as to make the other two constants of integration 
zero); we can therefore take these instead of the usual Lagrangian differ- 
ential equations of motion in order to determine 0, <l>, yfr. 

Solving for 6, ^, y^, we have 

^ (A--B)d . . , . , 
~ ^ — In — sm ^ cos -^ sm yjr, 

<^=:-jCOS»^ + ^sm»^, 
'd d 



•»^ = (p- -T cos''^— oSin^V^j cos^. 



The integr^f of energy (which is a consequence of these three equations) 
can be writt^^ down at once by use of § 63 ; it is 

where c is a constant: replacing a>i, a>3, q>8 ^y their values in terms of dand^, 
this equation can be written in either of the forms 



A-B 



. p sin' cos" -^ = — 



Bc-d* B-G 



Bd' 



+ 



BC 



cos"^, 



A-B 



Ac-d" A-C 



or 



^^ 8m'<?8in'V^ 3^-^iC 



COS* ft 



144 



The Solubte Problems of Rigid Dynamics [oh. vi 



Since il > B > (7, the quantity {cA - d«) or B (il - B) to^ + C(il - C) (»,« is 
positive, and (cC — d^) is negative: the quantity (Bc — d^) may be either 
positive or negative : for definiteness we shall suppose it to be positive. 

The first of the three differential equations can, by use of the last equa- 
tions, be written 



d 
dt 



(cos^) = -d J- 



Bc-d" B-C 



'>n 



*Uc-d« A-C 



Ad} 



AC 



cosfff 



}'• 



This equation shews that cos ^ is a Jacobian elliptic function of a linear 
function of t ; and the two preceding equations shew that sin cos '^ and 
sin ^ sin ^ are the other two Jacobian functions. 

We therefore write 

sin^cos-^sPcnu, sin^8in'^ = Qsnu, cos^ = Rdau, 

where P, Q, R, are constants and u is a linear function of t, say Xi + € ; the 
quantities P, Q, R, \ and the modulus k of the elliptic functions, are then to 
be chosen so as to make the above equations coincide with the equations* 

i* en* M = — k'* + dn" u, 

A* sn* w =a 1 — dn' u, 



dm 

^- dn u =5 — A;* sn w cn w, 
du 



The comparison gives 



P» = 



d^iA-C)' 



^_ B(d^^cC) j^_ C(cA^d^) 



d'(A^C) 



ifc«- 



(il - B) (d» - cC) 



V=: 



_(J3-C)(cil-6p) 



ABC 



Thus finally the values of the Eulerian angles and '^ at time t are given 
by the equations 

sin ^ cos ^ = P en (\^ + c), 

sin ^ sin -^ = Q sn {\t + c), 

cos^ = jBdn(\^ + €), 

where the constants P, Q, iJ, X, k, have thej^bove, values, and € is an arbitrary 
constant. The equation for k^ shews that k is real, and the equation 

shews that 1 - i" is positive, i.e. that k<l. The quantities P, Q, JR, \, are 
also evidently real from the above definitions. 

* WhitUker, A Course of Modem Analysu, §§ 190, 191. 



L 



69] 



The Soluble Problems of Bigid Dynamics 



145 



vhere 



When cP^cB, we have Ifl^ly and the elliptic functiona degenerate into hyperholic 
functions ; this is illustrated by the following examples. 

Example 1. A rigid body is moving about a fixed point under no forces: shew that if 
(ut the notcUion used above) d*=Bc, and if »^ is zero when t is zero^ a>i and o), being initially 
positive^ then the direction-cosines of the B-axis at time tj referred to the initial directions of 
the principal axes, are 

a tanh ;( — y sin /i sech X) cos/isech;^, 7^tanh;^-fasin/isech;^, 

_dt dt f (A'-B)(B-C) \^ ( A(B'C) \^ (C{A^n 

^"B' ^"BX AC J' ""tj?(^-C)j ' '^'"\B{A-C)] ' 

(Camb. Math. Tripos, Part I, 1899.) 

To obtain this result, we observe that when Bc^d\ the differential equation for the 
coordinate 6 becomes 

the int^;ral of which is 

coB^Biysechx) 

where y and x ^^^ ^^ quantities above defined. The equation 



}'• 



then gives 

and the equation 

gives 



A-B . ,^ . , I Ac'd^ A-C J. 
-;j^sm«^sm«V^=^^--j^co8«d 

sin 6 sin ^sstanhxi 
^= -J cos* ^+ » sin* ^ 



sin (<^ - fi) = - y sin ^. 

These equations shew that the direction-cosines of the J^-azis referred to the axes 
OXYZ, which (§ 10) are 

— cos^cos^sin^-sini^cos^, -sin<^cos^sin^+cos<^cos^, sin^sin^, 

can be written 

- sin /i sech ;f , cos/isech;^, tanh^* 

But if a»io, cD^y ttjQ, denote the initial directions of the principal axes, since 

so that An-^^ad and Cn^'^ydf we see that the direction-cosines of a»]o, m^, o»^, referred to 
OXYZy are given by the scheme 

X 7 Z 



«io 



«ao 



co< 



ao 



y 





a 





1 





— o 





y 



and hence the direction-cosines of the J?-axis, referred to o»iO) ^so> ®90> ^^^ 

— ysinfisech^+atanhx) cosfisechx, asin/isechx+ytanh^* 

W. D. 



10 



146 



The Soluble Problems of Rigid Dynamics [ch. vi 



Example 2. When cP^cB, shew that the axis Oy describes, on a sphere with the 
fixed point as centre, a rhumb line with respect to the meridians passing through the 
invariable line. (Coll. Exam.) 

We now require the expression of the third Eulerian angle <f> in terms of 
the time : for this and many other purposes the above expressions for and 
y^ in terms of Jacobian elliptic functions can advantageously be replaced by 
expressions in terms of the Weierstrassian functions. It is known* that 



sn« {(61-63)*^} = 



e. — e. 



where the Jacobian functions are formed with the modulus A:=(6,- 6,)*(ci— 6^)"*, 
and the Weierstrassian function is formed with the roots e^ e^, c,. Let us 
therefore determine quantities 61, 61, 6s, from the equations 



these equations give 



^j — ^ — \ J 



gj — 6; 



ei-e. 



?«A;»; 



^ (B''C)(cA-d^)-{C-A)(cB-cP) 

61 « 



SABC 



(C - A)(cB -d?)-(A- B)(cG -cf) 
*•" . 9ABG 



«.= 



{A-B)(cC-d')-(B-C){cA-d') 

2ABC 



The preceding equations shew that (ei — «,) and (e. — «i) are positive; 

while the equation 

, , _(A-C)( Bc-d^) 
*'"*•" ABC 

shews that ei — e, is also positive : the three real quantities eu e,, ^, therefore 
satisfy the inequality ej > 69 > ««. 

With these values of e, , e,, e«, and choosing the origin of time so as to 
omit the constant additive to t, we have therefore 

/ • ./I ,. A(d^-cO)oit)-ey 

• •/! • • . B(d}-cC) Ci-e, 
sm.^sm«t=^^^5rc)Vw^.' 

cP(A-C) i»(0-ei 

* Whittaker, A. Count of Modem Analyiu, § 202. 




69] The Soluble Probiemg of Rigid Dynamica 147 

These equations can be exvJf^^^ j^ ^ more symmetrical form by intro- 
ducing 8 new constant. We ^^^ ^^^^ ^^^ definitions of e„ e^, e.) 

{A-B){A-C)d? ^ _ {A - B) {cA - d') 



{A-C)icA-iP)_ 

let i be a new constant such' 
then it is readily seen that ti 

sin»^ 



A'BG 



A*B 



that each of these expressions is equal to fp{l) ; 
le above equations take the form 



rcos''^ = 



(?(o-e.}{Ko-«.r 



e. 



cos' 



(if (0-«.Hp (<)-«.}• 

These are the finai e ^^ \pre88ion8 for the Eulerian angles and -^ in terms 
of t and the constants ^i, 'I ^^^ ^^^ ^ j^. fon^^g fr^j^ ^h^ j^g^ equation that I is 
the value of t corresponcy' -j^^^g ^^ ^j^^ ^^^.^ ^^j^^ ^f ^. |^^j. ^j^j^ cannot be 
regarded as a physical int^ terpretation, for never attains this value in the 
actual motion, and I is imaj \ginary. 

The third Eulerian angJ veA ^ ^^^ ^^^ ^^ ^^^^ ^ r^.^^ differential equation 
for ^ is '^y- 



i^] 



JUCi 



^ = — 008**^ + -^ sin'*^. 



jlI 



A 



B 



But from the *'^^L^Ae equations, we have 



cos' 






8in'^ = - y<^>"^' 



and therefore 



s d (il-^)d{y (0-6.1 
''"A AB[f{t)-^{l)] 



But we have . (^ -B)'d' ^ (pffl-e,) {yffl-e.} 
But we nave ^,^, |>(0-«i 

and we can therefore write the equation for ^ in the form 



% 



^ = T + 5 



P'(0 



^ 2 j,(0- «)(/)• 

Expressing the fraction on the right-hand side of this equation as a sum 
of f-functions*, we have 

^=2+|{r(«-o-r(^+o+2r(0}. 



* Whittaker, A Courte of Modem Analy$is, § 211. 



10—2 




148 Tlu SolubU Problems dif Rigid Dyruimics [ch. vi 

Integrating this equation, we have 

where ^o is a constant of integration. This eU^^®'"^^ ^ n j i. • j 
of t: the three Eulerian angles (e. i>, f) are ^Hhus now all determined as 

functions of the time. 

„ , - ^,. • jijf lii • lii . X * .^ ma-functioDB; and hence exprcaa 

Example 1. Obtain d"*', cos J^, sin J^, in terms of sigrr i"^ * 

the Klein's parameters (a, fi, y, d) of § 12 in terms of L *n^ 

^V -fixed, and moves -under the 

Example 2. A uniform circular disc has its centre 0\ iT^.^- ^ u ^ ^* i. 

• Irftlocities O about a diameter 

action of no external forces. The disc is given imtial angular / *..,/%». oi. At. x 

• J- VLU i^A • J u X -x • -J- J ith Of in space. Shew that 

coinciding with 0( in space, and n about its axis coinciding w/ ^ 

at anj subsequent time _ 

he : -1 
«•= cot-if ?5_tan{(0>+4n«)Va^*'H' 

where x is the angle between Of and the axis of the disc OzP — •'^ * ^ 

planes f Of and fOz. h " (^"- ^^^'^"^^ 

For let OZ denote as usual the invariable line, and con< ^ *>d«r the spherical triangle Zfi^ 
whose vertices are the intersections of the lines OZ, OC, JP^ 0^ respectively with a sphere of 

centre 0, In this spherical triangle we have Zz^By fii-^ ^ Moreover 
disc C-^B^2Ay so 



and 



|-(0*+4»«)*. 



The equations of motion for 6 and ^ therefore become 

rf-0, 0-cfM=(O«+4n«)*, 



so 



d«irf-cos-i 



2n 



(02+4n»)*' 

In the spherical triangle iTfs, we have therefore 

2n 



0=(O"+4n«)*f. 



ZC^Zz^coi,'^ J^ f^z=(0«+4n«)*f, -2(?«=«, f^-^, 



and hence 



and 



(0"+4n«) 
sin ^ B sin iTf sin ^f 2^ s= 



cot o» « cos 2f tan ^f Zz I 



which are the required equations. 



(0«+4n«) 

2n 

(0«+4n«)* 



-Tsin{(0«+4n2)*.if} 



tan{(0«+4»2)*.iO» 



70.' Poinsot's kinematical representation of the motion; ttie polhode and 
herpolhode. 

An elegant method of representing kinematically the motion of a body 
about a fixed point under no forces is the following, which is due to Poinsot^ 



f 




\ 



► 



69, 70] The Soluble Problems of RigM Dynamics 161 

The equation of the momental ellipsoid of the bo. yoot ei ; this equation 
referred to the moving axes Oxyz, is "^e time. 

Aa? + J5y« + C^« = 1. \t For this we 

Consider the tangent-plane to the ellipsoid which is perpencft®"^^^ whose 
invariable Kne. If p denotes the perpendicular on this tangent^^"^ fixed 
the origin, we have (since the direction-cosines of p are Aooi/d, Ba^/cl^^^^^' 



=s-^, which is constant. 

Since the perpendicular on the plane is constant in magnitude and 
direction, the plane is fixed in space : so the momental ellipsoid always 
touches a fixed plane. 

Moreover, if (x\ y', z') are the coordinates of the point of contact of the 
ellipsoid and the plane, we have on identifying the equations 

Axx' + ByiZ + Czz' ^1 and AcdiX + Ba^y + Ca^z ss pd 

the values a?'^ — = — v' = — =— ^« — = — 

pd a/c' ^ pd is/c* ^ pd hjc' 

and hence the radius vector to the point {x\ y\ z') is the instantaneous axis 
of rotation of the body. It follows that the body moves as if it were rigidly 
connected to its momental ellipsoid, and the latter body were to roll about the 
fixed point on a fiaed plane perpendicular to the invariable line, withoiU 
sliding ; the angular velocity being proportional to the radius to the point of 
contact, so that the component of angular velocity about the invariable line is 
constant 

ExampU 1. If a body which is moveable about a fixed point is initially at rest and 
then is acted on continually by a couple of constant magnitude and orientation, shew that 
Poinsot's construction still holds good, but that the component angular velocity about the 
invariable line is no longer constant but varies directly as the time. (Coll. Exam.) 

For in any interval of time dt the addition of angular momentum to the body is Ndt 
about the fixed axis OZ of the couple ; sp that the resultant angular momentum of the 
system at time t\& Nt about OZ, Now the components of angular momentiun about the 
principal axes of inertia Oxyt are ^coj, ^c*,, Cn^, wherp A, B,C are the principal moments 
of inertia and (a»i, a»2) ^) ^^^ ^he components of angular velocity : hence we have 

A^i = ~^i( sin ^ cos ^, B»fmiJVt sin $ sin yjt, Cm^^Nt cos By 

where B^ <t^ ^ are the Eulerian angles which fix the position of the axes Oxyt with 
reference to fixed axes OXYZ, But these equations differ from those which oocmr in the 
motion of a body under no forces only in the substitution of tdi for dt ; so the motion will 
be the same as in the problem of motion under no forces, except that the velocities are 
multiplied by t ; whence the result follows. 

ExiatnpU 2. In the motion of a body, one of whose points is fixed, under no forces, let 
a hjrpierboloid be rigidly connected with the body, so as to have the principal axes of 





'^he Solvhuf Prohletm of Rigid Dywxtmm 

Integratuig this u^^int ns axes, and to have the squares of its axes i 

Be, d'—Ce, where A, B, C are the moments of ine 

ie twice its kinetic energf, and d is the resulta 

i^that the motion of this hjperboloid cab be represented bj o 

1 . .I'ng on a circular cylinder, whose axis passBB through the fixed [ 

*^ ° « axis of resultant angular momentum. , (S 

of t: the jr 

fuDCtionr^'^'^s which in Poinsot's constructipn ib traced on the i 
id by the point of contact with the fixed plane is called th 
iquatioDS, referred to the principal moments of inertia, are c1 
equation of the ellipsoid together with the equation p = constant, 

Example 1. Shew that when A =B, the polhode is a circle. 

Example % Taking A^B^C, shew that there are two kinds of polhodei 
consisting of curves which surround the axis Ot of the momenta! ellipsoid, and 
to cB>d*>eC, while the other kind consists of curves which surround tb 
and cerreapond to eA>d*>eB; and that the limiting case between these tv 
polhodefi is a singular polhode which correeponds to cS— <f =0, and oonaists of t 
which pass through the extremities of the mean axis. 

The curve which is traced on the fixed plane by the point of con 
the moving ellipsoid is called the herpolkode. 

To find the equation of the herpolhode, let p, x be the polar cc 
of the point of contact, when the foot of the perpendicular from 
point on the fixed plane is taken as pole. If {of, y, z') denote the cc 
of the same point referred to the moving axes Oxyz, we have 
x'* + y' + «'' = square of radius from point of suspension to point ol 

Substituting for x', 'j/, z', theii- values as given by the equations 

(k' = l^l,/^/c " — d sin ^ cos -^lA v'c, 
y' = «,/Vc= dsin-^sin -^jBtJc, 
^ = "Wj/Vc = d cos 6lC>Jc, 
we hare 

p'™— j; + -T^sin'5co8'ilr+-=-- sin'^ sin'U- + =-cos'ft 
Replacing 6 and -^ by their values in terms oft, this becomes 
(cA--^)(<e-cO)l (B-C)(A-B)d- \ 



\ 



i 



/ 



70, 71] The Solvble Problems of Rigid Dynamics 



161 



where oi denotes the half-period corresponding to the root e^) this equation 
expresses the radius vector of the herpolhode in terms of the time. 

We have next to find the vectorial angle % ^^ terms of t For this we 
observe that *Jcp^xld is six times the volume of the tetrahedron whose 
vertices are the fixed point, the foot of the perpendicular from the fixed 
point on the fixed plane, and two consecutive positions of the point of contact, 
divided by the interval of time elapsed between these positions, and that this 
quantity can also be expressed in the form 



Acx'jd\ Bcy'/d\ Ccz'/d^ 



•I 



y> 



C til 

or -j-.xyz 



1. 
A, 

X /x, 



1, 

B. 



1 

C 



ifW, i'/^' 



\ 



All the quantities involved, except % are known functions of t: on 
substituting their values in terms of t, and reducing, we have 



"^ jB{j»(0-|>(f + a>)) 
which can be written in the form 



{p(0~ 



A-G 



% 



d i fp' (I + w) 



X"" « "^ o 




nation ca n be integrated in the same way as the equation for the 
, and gives 

onstant of integration. The current coordinates (p, x) of ^^^ 
ce thus expressed as functions of t 

150 ]fe&f#i. A particle moves in such a way that its angular momentum round the 
inear function of the square of the radius vector, while the square of its 
^</&Wriitfea quadratic function of the square of the radius vector, the coefficient of the 
F^atixalte^-jfjwer being negative ; shew that the path is the herpolhode of a Poinsot motion, 
% ti tie loi r however Ay B, C are not restricted to be positive. 

'^""'^ '^.triple 2. Discuss the cases in which the polhode consists of (a) two ellipses 
^^TOt^^ixig on the mean axis of the momental ellipsoid, (/3) two parallel circles, (y) two 
Pwtot^ ; shewing that in these cases the herpolhode becomes respectively a spiral curve 
jse equation can be expressed in terms of elementary functions), a circle, or a point. 




% 71. Motion of a top on a perfectiy rough plane; determination of the 
^^vlerian angle 0. 

A top is defined to be a material body which is symmetrical about an axis 
and terminates in a sharp point (called the apex or vertex) at one end of 
the axis. 

We shall now study the motion of a top when spinning with its apex 
placed on a perfectly rough plane, so that is practically a fixed point. The 
problem is essentially that of determining the motion of a solid of revolution 

er the influence of gravity, when a point on its axis is fixed in space. 



162 The Soluble Problems of Rigid Jjynamics [ch. vt 

Let {Af A, G) denote the moments of inertia of the top about rectangular 
axes Oxyz, fixed relative to the top and moving with it, the origin being the 
apex and the axis Oz being the axis of symmetry of the top ; let {0, <t>, ^) be 
the Eulerian angles defining the position of these axes with reference to fixed 
rectangular axes OX YZ, of which OZ is directed vertically upwards. 

The kinetic energy is (§ 63) 

where a>i, a>t, a>s denote the components relative to the moving axes of the 
angular velocity of the top, so that (§ 16) we have 

G>i = d sin -^ — ^ sin ^ cos sir, 
tot^d cos -^ + ^ sin ^ sin ^^, 
ft), = ^ + ^ cos ^ ; 

the kinetic energy is therefore 

T^iAd^ + Jil<^« 8in« + ^C(yjr + <f> cos 0)\ 

and the potential energy is V^ Mgh cos 0^ where M is the mass of the top 
and h is the distance of its centre of gravity from the apex 

The kinetic potential is therefore 

L = r- F= Ji4d« + ii4<^«sin»^ + i(7(^ + <^ cos ^)«- if^rA cos ft 

The coordinates ^ and '^ are evidently ignorable; the corresponding 
integrals are 

— r = constant, and — j =* constant, 

or A<j> sin« ^ + (7(^ + ^ cos ^) cos ^ = a, 

C('^ + ij>cos0) =6, 

where a and b are constants : these may be interpreted as integrals of angular 
momentum about the axes OZ and Oz, and so are obvious d priori from 
general dynamical principles. 

The modified kinetic potential (§ 38) is 

iJ = i — a^ — fr^ 

ijiit (a-6cos^)* b* mjt J, n 
g^iig'— ^ - . . _^^ — 57V — ifoAcosg. 

The term — &'/2(7 can be neglected, as it is merely a constant; the 
equation of motion is 






71] The Soluble Problems of Rigid Dynamics 168 

so the variation of is the same as in a dynamical system with one degree 
of freedom for which the kinetic energy is j^Ad^ and the potential energy is 

(a - 6 cos ^)« , ,, , ^ 
c% A ' .A + Mgh cos 0. 
2A sm« ^ 

The connexion between and t is therefore given by the integral of 
energy of this reduced system, namely 

where c is a constant. 

Writing cos d = ^, this equation becomes 

ul«ir« =s - (a - 6a?)« - 24 Mgh (a? - a;*) + 2il c (1 - «»). 

The right-^and side of this equation is a cubic polynomial in x\ now 
when d? a= — 1, the cubic is negative ; for some real values of 0, Le. for some 
values of x between — 1 and 1, the cubic must be positive, since the left-hand 
side of the equation is positive ; when 0;=: 1, the cubic is again negative ; and 
when d? SB + 00 , the cubic is positive. The cubic has therefore two real roots 
which lie between —1 and 1, and the remaining root is also real and is 
greater than unity. Let these roots be denoted by 

cos a, cos)8, cosh 7, 

where cos /9 > cos a, so that a > /9. 

The differential equation now becomes 

(Mghl2A)^ (2^ =s (4 (a? — cos a) (a; — cos /8) (x — cosh 7)}""* dx. 

If we write 

2A ,, ^ ' , 2i4 . 2Ac + b' 

we have therefore t + constant =« | {4 (« — c,) (s — «i) (^ — «»)}"* dt, 

where the constants ei, e^, 6$ are given by the equations 

Mgh , iAo + b' 
^^=_|-coshy— j2^. 

Mgh a 2ilc + 6* 
Mgh 2Ac + i* 

SO that 01, e^, ei are all real and satisfy the relations 

ei + Ci + 6,«0, ei>e^>et. 



^^_ _- ~—^ ' 

154 The SolvhU Problems of Rigid Dynamics [oh. vi 

The cotinexion between z and t is therefore 

where 6 is a constant of integration, and the function ^ is formed with the 
roots ^1, 62, 63; and hence we have 



x^ 



Mgh^^^'^^^'^ QAMgh' 



sec^Bl+sech 



Now in order that x may be real for real values of t, it is evident that x must 
lie between cos a and cos/9, Le. |f>(^ + e) must lie between e^ and e, for real 
values of t : and therefore the imaginary part of the constant € must be the 
half-period o), corresponding to the root ^. The real part of e depends on 
th& epoch from which the time is measured, and so can be taken to be zero 
by suitably choosing this epoch. We have therefore finally 

and this is the equation which expresses the Eulerian angle in terms of 
the time. 

Example 1. If the circumBtances of projection of the top are such that initiaUy 

e^m\ rf=0, i^ = 2{MghlZA)\ ^ = {ZA''C){MghlZAC^\ 
shew that the value of B at any time t is given by the equation 

(v/¥'). 

SO that the axis of the top continually approaches the vertical. 

For in this case we readily find for the constants a, b, c, the values 

a=6-(3%Ail)*, c^Mgh, 
so the differential equation to determine x is 

whence the result follows. 

Example 2. A solid of revolution can turn freely about a fixed point in its axis of 
symmetry, and is acted on by forces derived from a potentval-energy function fi cot' By where 
B is the angle between this axis and a fixed line; shew that the equations of motion can be 
integrated in terms of elementary functions. 

For proceeding as in the problem of the top on the perfectly rough plane, we find for 
the integral of energy of the reduced problem the equation 

\ aM (a-6co8^)* cos*^ . 
Writing 008^=47, this becomes 

The quadratic on the right-hand side is negative when x^l and x^ -l, but is positive 
for some values of x between - 1 and + 1, since the left-hand side is positive for some real 



71, 72] The Soluble Problems of Rigid Dynamics 165 

values of B : the quadratic has therefore two real roots between - 1 and + 1. Calling 
these cos a and co6/3, the equation is of the form 

X*i" = (cos o - 4?) ( j: - cos ^), 
the solution of which is 

^bcos a sin* (^/2X) + cos /9 cos* (</2X). 

72. DetermincUion of the remaining Eulerian angles, and of Klein's 
parameters; tlie spherical top. 

When the Eulerian angle has been obtained in terms of the time, as in 
the last article, it remains to determine the other Eulerian angles (f> and '^. 
For this purpose we use the two integrals corresponding to the ignorable 
coordinates : these, when solved for ^ and yp^, give 



(+- 



b (a — 6 cos 0) cos 
C A sin« 



If we regard the motion as specified by the constants of the body 
(M, A, C, h) and the constants of integration (a, b, c), it is evident from 
these equations and the equation for that C does not occur except in 
the constant term of the expression for ^; and therefore an auxiliary top 
whose moments of inertia are (il, A, A), can be projected in such a way that 
its axis of symmetry always occupies the same position as the axis of symmetry 
of the top considered, the only difference in the motion of the two tops being 
that the auxiliary top has throughout the motion a constant extra spin 
b(C^A)/AC about its axis of symmetry. A top such as this auxiliary top, 
whose moments of inertia are all equal, is called a spherical top. It follows 
therefore that the motion of any top can be simply expressed in terms of the 
motion of a spherical top, and that there is no real loss of generality in 
supposing any top under consideration to be spherical. 

If then we take C^A, the equations to determine (f> and yjt become 

, _ a — 6co8^__ a + b a — b 

*"■ ilsin*^ ''2A{Gos0-\-l)^2A(cos~0^Ty 

:_6 — acos^__ a + b a — b 

"^ A sin*^ " 2A (cos^ + l) "*" 2A (cos^- 1) ' 

Substituting for cos its value from the equation 

^ 2A ,^ . 2Ac + b* 

and writing 

'^^^~ 2A 124' ' 

a(U^ JfgA 2^10 + 6* 

'^^ ' 24 124* ' 



166 The Solvble Problems of Rigid Dynamics [oh. vi 

so that I and k are known imaginary constants (being in fact the values of 
^ + 6), corresponding to the values and ir of 0\ the differential equations 
become 



, Mgh (a + b) 
* = 4l^ 



4 



>(« + ft),) - jf> (fe) 
1 



, _ Mgh (a + b) 

^" 44' X«+a),)-jf)(fc) 



Mghia-b) 1 

Mghja-b) 1 

4.1» >(e + ft>,)-.if>(0* 



Now the connexion between the function fp and its derivate p' can be at 
once written down by substituting for x from the equation 



in the equation 






^»gy = -(a-6icy-2il3f^A(a?-««) + 2ilc(l-a^); 

if the argument of the jf)-function is k, it follows from the definition of k that 
the corresponding value of a; is — 1 ; and so the last equation gives 

A*. {2ilp' (k)IMghY = - (a + 6)S 

or p' (k) = iifflrA (a + 6)/2^«. 

Similarly we have 

p' (0 = iJf<7A (a - b)l2A\ 

and therefore the equations for <f> and yft can be written in the form 

p'W p'(0 



Now the function 



jf)(e+fi>,)-jf>(fc) 

is an elliptic function, whose poles in any period-parallelogram are congruent 
with t + m^^k and ^ + ai, » — A, the corresponding residues being 1 and — 1 ; 
and the function is zero when ^ + ft>3 = 0. Hence * we have 



jf> (e + ft),) - jf) (*) 

and therefore 



= f(« + ft),-*)-f(« + a), + *) + 2{:(A), 



f P (k)dt , a-(e + ft),-*:) , o,,,;x . , . . 

I .^ ^ / TTi ~ log — TT-: — - — Tx + 2? (A) ^ + constant. 

jf)(« + ft),)-f)(fc) *a-(^ + ft), + i) ^^^ 



* Whitiaker, A Coune of Modem AmOyrit, f 211. 



72] 



The Solvble Problems of Rigid Dynamics 



167 



The integrals of the equations for ff> and '^ can therefore be written in 
the form 

-j2i(*-^)^gt{C(*).C(0}< <^(^-Ha),-A:)cr(^-hft), + Q 

where ^o ^^^ '^o ^^ constants of integration. 

These equations lead to simple expressions for the Klein's parameters 
a» /8» 7, S (§ 12), which define the position of the moving axes Oxyz with 
reference to the fixed axes OXYZ: for by definition we have 

a = 008^5. «*•(♦+*), )8 = isini^..e*»(*-*), 
7 « f sin i ^ . 6**(*-« 8 = cos J ^ . «-**<♦+*). 

But we have 

2cos*i^«l+co8^ 

, 2A ,, ^ 2ilc + 6» 

2A 



=':^^«^(^+^->-*^<*>^ 



or 



^ 2-4 o" (^ + fii, + ir) o" (^ + ft)i - ir) 
""MgJi' <7« (ifc) cr» (t + ft),) ' 

Similarly we find 

and on combining these with the expressions for 6^*^ and e'^^ already found, 
we have 

\Mgh) ' a(k) • <r(« + «,) ' 

„_/-£\* e**^-*^ ait + a^ + l) ,f^l^■ 
^~\Atgh) ' '<r(l) ' <r(< + «,) ' 

\Mgh) ' G{k) ■ <r(< + «,) 

These equations express the parameters a, /8, 7, S as functions of the 
time. 



1 




sin' tf coa 2* " j^ (v/3/2 + cca *)*, 



158 The Soltible Problems of Rigid Di/namies [ch. vi 

Example 1. A ggroitat of maa U movei about a fixed point in iti axit of tymm^ry: 
tie momentt of inertia <dMnit the aria of figure and a perpendicular to il tkroagh the fixed 
point are C and A retpectioely, and the centre of gravity it at a distance h from the fixed 
point. The gyrottat it held to that it* axis motet an angle oot'^ 1/^/3 with the dovmvrard 
vertical, and it given an angular velocity ij AMgh -JS/C about ite axil. If the axit be now 
left free to move about the fixed poitit, tkew that it vpill detcribe the cone 

sin'tf sin 2^=(-co8fl- 1/^3)' (-coBfl+^/3)* 

2V2 
^/3 #3 ^'• 
tahere <(> ii the aiimutbal angle and 6 the inclination of the axit to the upward vertical. 

(Camb. Math. Tripos, Port I, 1894.) 
For in thiH problem we have initially 

cosfl=-l/V3, *=0, rf=0, 0=0, ^=-ifISgQ3IC, 
aod theee initial values give 

a---JMA^lilZ, b=i!Z-fM^h, e--Mghl^ 
Substituting in the general diSerential equation for 0, namelj 

ne have 

Ai»»\D*6=-Mgh(cois6+ll^){y/3+2cm6){-ooeg+^3), 
while the equation 



8in*tf 
Dividing this equation bj the square root of the preceding equation, we have 

= 3* (( - cos fl - l/v'3)' ( V3 + 2 cos «)-* ( - cos fl + V3)-* cosec tf iM, 

or 0=3* j(j;- 1/^)' (V3- 2«)-*(r+ V3)~*(l -a^')-' dx, where *= - 

Now if we write 

«-{:i- l/V3)»(x+v^)» (s/3/2 -x)-*, 
we have by differentiation 



- i (1 - a^) {* - 1 /V3)* (^+ V3)-» (V3/2 - X)-* 



and l + 5-u'-i 



t*a-^* 



8W3/i-j;)- 
We have therefore 



3* I du 



or 20-tan-'(3*2"*u), 

or tan2</.-3*2~*C-costf-l/V3)*{-cofltf+V3)*(V3/2+coatf)"*, 

which is equivalent to the result given above. 



f 



• T 



72, 73] The Soluble Problems of Rigid Dynamics 

Example 2. Shew ihat the logarithms of Klein's parameters, considered as functiv. 
of cos ^, are eUiptifc integrals of the third kind. '^« 

Example 3. Obtain the expreMions foimd above for Klein's parameters as functions of 
the time t by shewipg that they satisfy differential equations typified by 

where Y denotes a lioably-periodic function of ty these equations being of the Hermite- 
Lamd type which i^ soluble by doubly-periodic functions of the second kind. 

A simple type of motion of the top is that in which the axis of symmetry 
maintains a 46on8tant inclination to the vertical ; in this case, which is 
generally kn/own as the steady motion of the top, 6 and d are permanently 
zero; since %e have 

\ AAi (a — ftcostfy* ,^ , ^ 

it follows that 



^ d ((a - 6 cos tf)» , --. , ^ 



Perfonniiig the differentiation, and substituting for (a — 6 cos 0) its value 

= - 6^ + A^^ cos tf + Mgh. 

^^ equation gives the relation between the constants ^, 0, and h (which 
depenqg on the rate of spinning of the top on its axis) in steady motion. 

7S. Motion of a top on a perfectly smooth plane. 

^ e shall now consider the motion of a top which is spinning with its apex 
m ^M"''^^*^ ^lljjj ^ smooth horizontal plane. The reaction of the plane is now 
Y ertwij^ g0 the horizontal component of the velocity of the centre of gravity, 
^ > ^^the top is constant ; we can therefore without loss of generality suppose 
^ ^' this component is zero, so that the point G moves vertically in a fixed 
1 ^ which we shall take as axis of Z; two horizontal lines fixed in space and 

^endicular to each other will be taken as axes of X and Y, 

Let Gxyz be the principal axes of inertia of the top at 0, and {A, A, C) 
d moments of inertia about them, Oz being the axis of symmetry : and let 
> ^, '^) be the Eulerian angles defining their position with reference to the 
-ces of Z, F, Z. 

The height of G above the plane is A cos 0, where h denotes the distance 

*°'. G firom the apex of the top ; the part of the kinetic energy due to the 

>tion of G is therefore ^Mh* sin' . 6^, where M is the mass of the top ; and 

\ ' as in § 71, the total kinetic energy is 

/ 
L . r = im«sin«tf.tf» + iil^« + iil^»sin»tf + iC(i^ + <^costf)«, 

'"^and the potential energy is 

V = Mgh cos 0, 



I 



*y 




158 / ^^ Soluble Problems of Rigid Dgnmrms [ch. vi 

/■ Proceeding now exactly as in § 71, we have two integrals corresponding to 
A\ie ignorable coordinates ^ and '^, namely 

y fil<^sin«tf+C(^4-<^co8tf)co8tf-sa, 

/ I (7(^+^C0Btf)-ft, 

/ where a and b are constants ; and on performing the process of ignoration of 

coordinates we obtain for the modified kinetic potential the expression 

i (il 4- m« sin« 0) 6' - ^ V/ ^"^^^ - %* cofl ^p 
^ 2 A sm* ff ^ . 

so the variation of is the same as in the system with one degree of freedom 
for which the kinetic energy is 

^{A'^Mh^sin^0)6\ 
and the potential energy is 

(a - 6 cos ^)» , J, , ^ 
2A sm* S ^ 

The connexion between and ^ is given by the integral of energy of this 
latter system, namely 

i(A + Mh^siu' 5)^« = - (a - 6 cosy _ ^ . ^^^ ^ 
'^ ^ 2il sin* ^ ^ 

where c is a constant. Writing cos 0^x, this becomes 

il (^ + JtfA« - if AV) i^ == - (a - 6a:)« - 2il%A (a? - a^) + 240 (1 - «•)• 

The variables x and ^ are separated in this equation, so the solutic^ ^^^ 
be expressed as a quadrature ; but the evaluation of the integral in^^'*'?^ 
will require in general hyperelliptic functions, or automorphic functi**^ ^*f 
genus two. 

74. Kowalevskis top. 

The problem of the motion under gravity of a body one of whose point 
fixed is not in general soluble by quadratures : and the cases considered 
§ 69 (in which the fixed point is the centre of gravity of the body, so tl 
gravity does not influence the motion), and in § 71 (in which the fixed poi, 
and the centre of gravity lie on an axis of symmetry of the body) were fc 
long the only ones known to be integrable. In 1889 however Mme. S. vo 
Kowalevski* shewed that the problem is also soluble when two of t\ 
principal moments of inertia at the fixed point are equal and double t) 
third, so that -4 = 5 = 2(7, and when further the centre of gravity is aituat*' 
in the plane of the equal moments of inertia. I 

Let the line through the fixed point and the centre of gravity be take, 
as the axis Ox, and let the centre of gravity be at a distance a from the fixecrv 

• Acta Math. xn. p. 177. ^ ! 






t 



r 



J 



k 



78, 74] The Soluble Problems of Rigid Dynamics 161 

point ; let (0, ^, i/r) be the Eulerian angles which define the position of the 
principal axes of inertia Oxyz with reference to fixed rectangular axes OXYZ^ 
of which the axis OZis vertical; let ((Uj, ci>„ a>,) be the compouents along the 
axes Oxyz of the angular velocity of the body, and let M be its mass. The 
kinetic and potential energies are given by the equations 

= (7 {^« -f <^» sin» tf + i (^ + ^ cos ^)»}, 
F= — Mga sin 5 cos i/r. 

The coordinate ^ is evidently ignorable, giving an integral 

— r s= constant, 

or 2^8in'0 + (^ + ^cos9)co8d«A;, 

where A; is a constant : and the integral of energy is 

I' -f F= constant, 

or ^« + ^»sin«tf + i(i^ + <^co8tf)»-^- 8intfcos^ = A. 

Mme. Eowalevski shewed that another algebraic integral exists, which can 
be found in the following way. 

The kinetic potential is 

i r= Cd « + (7<^« sin« tf + i C (^ + ^ cos tf )« 4- Mga sin cos ^, 

and the equations of motion are 

dtVad) 3^" ' 
d /dL\ _ 3^ ^ 



the first of these is 



de(a^)"°' 



Mga 



2^ = (^ cos tf - •^) <^ sin d + - ^ cos tf cos -^j 
and on eliminating y(r between the second and third, we obtain 

2^^(^sin5)«-(^co8tf-i^)^-f^co8tfsin^. 

• * 
Adding the first of these equations mukipliedlby t to the second, we have 



2 ^ (^ sin +%d) = t (^ cos ^ - ■^) (^ sin ^ t^) + 1 . ^ cos Be'**, 

W. D. 11 



74, 76] Tfie Soluble Problems of Rigid Dynamics * 163 

«,V=(2a>,a, + to) {(«3„^+ ^y . «3V} 

Shew by use of Kowalevski's integral (without using the integrals of energy or angular 
nioiiQeDtum) that the equations of motion can be written in the form 

where F is a function of .r and y only, so that the problem is transformed into that of the 
motion of a particle in a plane conservative field of force. (KolosofH) 

Liouville* has shewn that the only other general case in which the motion under 
gravity of a rigid body with one point fixed has a third algebraic integral is that in which 

1®. The momental ellipsoid of the point of suspension is an ellipsoid of revolution. 

29. The centre of gravity of the body 'is in the equatorial plane of the momental 
ellipsoid. 

3^ If {A J A J C) are the principal moments of inertia at the point of suspension, the 
ratio 2CIA is an integer : this integer can be arbitrarily chosen. 

Example, A heavy body rotates about a fixed point 0, the principal moments of 
inertia at which satisfy the relation A=^B^^C: and the centre of gravity of the body lies 
in the equatorial plane of the momental ellipsoid, at a distance h fi*om 0. Shew that if 
the constant of angular momentum about the vertical through vanishes, there exists an 
integral 

o>3 {»i + »J) •\-gho>i cos 6 B constant, 

where o»i, a>2, a>3 are the components of angular velocity about the principal axes Oxyz^ 
Ox being the line from to the centre of gravity ; and hence that the problem can be 
solved by quadratiu'es, leading to hyperelliptic integrals. (Tshapliguine.) 

75. ImpiUsive motion. 

As has been observed in § 36, the solution of problems in impulsive 
motion does not depend on the integration of differential equations, and can 
generally be effected by simple algebraic methods. The foUowitig examples 
illustrate various types of impulsive systems. 

Example 1. Two um/orm rods AB, BC, each of length 2a, are emooMy jointed at B 
and rest on a horizontal table with their directions at right angles. An impulse is applied to 
the middle point of A By and the rods start moving as a rigid body: determine the direction 
of the impulse that this may be the case^ and prove that the velocities ofA,C wiU be in the 
ratio V13 : 1. (Coll. Exam.) 

We can without loss of generality suppose the mass of each rod to be unity. Let {x^ y) 
be the component velocities of B referred to fixed axes Ox, Oy parallel to the undisturbed 
position BA, BC of the rods, and let ^, ^ be the angular velocities of BA and BC. The 
components of velocity of the middle point of AB are (i*, y4-«^), and the component of 
velocity of the middle point of BC are (.r - a^, y), so the kinetic energy of the system is 
given by the equation 

* Acta Math. xx. (1897), p. 239. 

.11—2 



164 The Soluble Problems of Rigid Dynamics [ch. vi 

Let the components parallel to the axes of the impulse be /, J, The components of 
the displacement of the point of application of the impulse in a small displacement of the 
system are (&p, dy +ad^) ; and hence the equations of § 36 become 

IT ^ IT J IT J IT ^ 
'bx d^ d6 d^ 

while the condition that the system moves as if rigid is ^^^ These equations give 

Hence /=</, which shews that the direction of the impulse makes an angle 45^ with BA ; 
and as the components of velocity of A are {x^ y+2a6), and the components of velocity of 

C are {i - 2a^, y\ we have for the velocities of A and of C the values V65y and slhy 

respectively, so the velocity of il is Vl3 x the velocity of C\ which is the required result 

Exam'pU 2. A framework in the form of a parallelogram U made by smoothly jointing 
the ends of two pairs of uniform bars of lengths 2a, 26, masses m, m', and radii of gyration 
k, if. The parallelogram is moving without any rotation of its sides, and with velocity F, in 
the direction of one of its diagonals ; it impinges on a smooth fixed ukUI icith which the sides 
make angles 6, <t> and the direction of the velodty V a right angle, the vertex which impinge* 
being brought to rest by the impact. Shew that the impulse on the wall is 

2 r{(m+m')-*+(«il;8+m'a«)-i a« cos* ^+(m6«+m'ir'«)-i 6« co8« 0}-i. 

(ColL Exam.) 

Let X and y be the coordinates of the centre of the parallelogram, x being measured at 
right angles to the wall and towards it The kinetic energy is 

The ;r-coordinate of the point of contact is :i7+a sin d-f 6 sin 0, so the displacement of the 
point of contact parallel to the axis of x corresponding to an arbitrary displacement 
{dx, by, b6j d0) is &p4-acosdd^-f 6cos<^d^. The equations of motion, denoting the 
impulse by /, are therefore 

f^T /dT\ 

dx-ywo — ' 

dT fdT\ - 

dT fdT\ ., . 

'2(m+m')(i- F)=-/, 

.2(w6«+m'ife^)^ =-/6co8^. 
Moreover since the final velocity of the point of contact is zero, we have 

X + <rcos ^.^+6cos0.^sO. 



or 



76] The Soluble Problems of Rigid Dynamics 165 

Eliminating i, 6^ ^ from these equations, we have 

1 a*coe*^ 6*cos*<^ 



■-'{^ 



which is the result stated. 

The next example relates to a case of sudden fixture ; if one point (or line) 
of a freely-moving rigid body is suddenly seized and compelled to move in a 
given manner, there will be an impulsive change in the motion of the body, 
which can be determined from the condition that the angular momentum of 
the body about any line through the point seized (or about the line seized) 
is unchanged by the seizure ; this follows from the fact that the impulse of 
seizure has no moment about the point (or line). 

Example 3. A uniform circular disc is spinning with an angular velocity O about a 
diameter when a point P on its rim is suddenly fixed. Prove that the subsequent velocity of 
the centre is equal to \ of the velocity of the point P immediately before the impact, 

(Coll. Exam.) 

Let m be the mass of the disc, and let a be the angle between the radius to P and the 
diameter about which the disc was originally spinning. The original velocity of P is 
Qc sin a, where c is the radius of the disc. The original angular momentum about P is 
about an axis through P parallel to the original axis of rotation, and of magnitude ^m4^Q; 
and this is unchanged by the fixing of P, so when P has been fixed, the angular momentum 
about the tangent at Z' is ^mc^Q sin a. But the moment of inertia of the disc about its 
tangent at P is ^mc^, and so the angular velocity about the tangent at P is ^O sin a. The 
velocity of the centre of the disc is therefore ^Ocsin a, which is ^ of the original velocity 
of P. 

Example 4. A lamina in the form of a parallelogram whose mass is m has a smooth 
pivot at each of the middle points of two parallel sides. It is struck at an angular point 
by a particle of mass m which adheres to it after the blow. Shew that the impulsive 
reaction at one of the pivots is zero. (ColL Exam.) 

Miscellaneous Examples. 

1. Prove that for a disc free to tium about a horizontal axis perpendicular to its plane 
the locus on the disc of the centres of suspension for which the simple equivalent 
pendulum has a given length L consists of two circles ; and that, if A and B are two 
points, one on each circle, and L' is the length of the simple equivalent pendulum when 
the centre of suspension is the middle point of AB, the radius of gyration k of the disc 
about its centre of inertia is given by the equation 

it»Z'«= (ii;« - c») (Z'« - iZ»+c»), 
where 2c is the length of AB. (ColL Exam.) 

2. A heavy rigid body can turn about a fixed horizontal axis. If one point in the 
body is given through which the horizontal axis has to pass, discuss the problem of 
choosing the direction of the axis in the body in such a way that the simple equivalent 
pendulum shall have a given length ; shewing that the axes which satisfy this condition are 
the generators of a quartic cone. (ColL Exam.) 

3. A sphere of radius b rolls without slipping down the cycloid 

a*sa(tf+8in^, y=a(l-ooetf). 



/; 



166 The Soluble Problems of Rigid Dynamics [ch. vi 

It starts from rest with its centre on the horizontal line ^=2a. Prove that the velocity V 
of its centre when at the lowest point is given by 

y^=^g (2a - 6). (CoU. Exam.) 

4. A uniform smooth cube of edge 2a and mass M rests symmetrically on two shelves 
each of breadth b and mass m and attached to walls at a distance 2c apart. Shew that, if 
one of the shelves gives way and begins to turn about the edge where it is attached to the 
wall, the initial Angular acceleration of the cube will l)e 

M g {c—af (c-6)4-^mgr6 (c~ g) (g- ft+g) 
""i/(c-a)2{^+(c-6)8} + /(c-6+a)» ' 

where Jfifl and / are respectively the moments of inertia of the cube about its centre and 
of the shelf about its edge. (Camb. Math. Tripos, Part I, 1899.) 

5. A homogeneous rod of mass Af and length 2a moves on a horizontal plane, one end 
being constrained to slide without friction in a fixed straight line. The rod is initially 
perpendicular to the line, and is struck at the. free end by a blow / parallel to the line. 
Shew that after time t the perpendicular distance y.of the middle point of the rod from 
the line is given by the equation 

•1 

(1 - i^)* (1 - ^)~* dx = 3Itj2lfa, (CoU. Exam.) 

y/a 

6. Four equal uniform rods, of length 2a, are smoothly jointed so as to form a 

rhombus ABCD, The joint A is fixed, whilst C is free to move on a smooth vertical rod 

through A, Initially C coincides with A and the whole system is rotating about the 

vertical with angular velocity m. Prove that, if in the subsequent motion %a is the least 

angle between the upper rods, 

a»^ cos a = 3^ sin' a. 

(Camb. Math. Tripos, Part I, 1900.) 

7. A disc of mass M rests on a smooth horizontal table, and a smooth circular groove 
of radius a is cut in it, passing through the centre of gravity of the disc. A particle of 
mass \M\H started in the groove from the centre of gravity of the disc. Investigate the 
motion. Prove that if o^ is the arc traversed by the particle and 6 the angle turned 
round by the disc, then 

(a*+/:2)« ^ 

Mh^ being the moment of inertia of the disc about a vertical line through its centre 
of gravity. (Coll. Exam.) 

8. A rigid body is moving freely under the action of gravity and rotating with angular 
velocity a> about an axis through its centre of gravity perpendicular to the plane of its 
motion. Shew that the axis of instantaneous rotation describes a parabolic cylinder of 

latus rectum (\/4a+\/2^/a»)', whose vertex is at a distance s/^gajta above that of the path 
of the centre of gravity of the body; where 4a is the latus rectum of the parabola 
described by the centre of gravity. (Coll. Exam.) 

9. A particle of mass m is placed in a smooth uniform tube which can rotate in a 
vertical plane about its middle point. The system starts from rest when the tube is 
horizontal. If ^ is the angle the tube makes with the vertical when its angular velocity is 
a maximum and equal to od, prove that 

4 (mr« -f J/*>) »♦ - Bmgrw^ cos ^ + mg^ sin" ^ = 0, 

where Mi^ is the moment of inertia of the tube about its centre and r the distance of the 
particle from the centre of the tube. (Coll. Exam.) 



CH. vi] The Solvble Problems of Rigid Dynamics 167 

10. Four uniform rods, smoothly jointed at their ends, form a parallelogram which 
can move smoothly on a horizontal surface, one of the angular points being fixed. 
Initially the configuration is rectctngular and the temework is set in motion in such a 
manner that the angular velocity of one pair of opposite sides is X2, that qf the other pair 
being zero. Shew that when the angle between the rods is a maximum or minimum, the 
angular velocity of the system is O. (Coll. Exam.) 

11. Two homogeneous rough spheres of equal radii a and of masses m, m' rest on a 
smooth horizontal plane with m' at the highest point of m. If the system is disturbed, 
shew that the inclination of their common normal to the vertical is given by the 
equation 

a^2(7wi-|-5m'sin2^) = 5^(»H-m') (1-cos^). (Coll. Exam.) 

12. A uniform rod AB ia of length 2a and is attached at one end to a light inexten- 
sible string of length c. The other end of this string is fixed at to a point in a smooth 
horizontal plane on which the rod moves. Initially OAB is a straight line and the rod is 
projected without rotation with velocity V in the direction perpendicular to its length. 
Prove that the cosine of the greatest subsequent angle between the rod and string is 
1 - a/6c (Coll. Exam.) 

13. To a fixed point are smoothly jointed two uniform rods of length 2a, and upon 
them slides, by means of a smooth ring at each end, a third rod similar in all respects. 
Initially the three rods are in a horizontal line with the ends of the third rod at the 
middle points of the other two and, on the application of an impulse, the rods begin to 
rotate with angular velocity fi in a horizontal plane. Shew that the third rod will slide 
right off the other two unless 

0« > 2ffla^S. (Coll. Exam.) 

14. A hollow thin cylinder of radius a and mass M is maintained at rest in a 
horizontal position on a rough plane whose inclination is a, and contains an insect of mass 
m at rest on the line of contact with the plane. The cylinder is released as the insect 
starts off with velocity V : if this relative velocity be maintained and the cylinder roll up 
hiU, ahew that it will come to instantaneous rest when the radius through the insect 
makes an angle 6 with the vertical given by 

F* {1 - cos {6 - a)] +a^ (cos a - cos ^)= (1 +M/m) ag (^- a) sin a. 

(Coll. Exam.) 

15. A uniform smooth plane tube can turn smoothly about a fixed axis of rotation 
lying in its plane and intersecting it : the moment of inertia of the tube about the axis 
is /. Initially the tube is rotating with angular velocity fi about the axis, and a particle 
of mass m is projected with velocity F within the tube from the point of intersection of 
the tube with the axis. The system then moves under no external forces. Prove that, 
when the particle is at a distance r from the axis, the square of its velocity relative to the 
tube is 

F» + V^ o8. (CoU. Exam.) 

16. A uniform straight rod of mass M is laid across two smooth horizontal pegs so 
that each of its ends projects beyond the corresponding peg. A second imiform rod of 
mafls m and length 21 is fastened to the first at some point between the pegs by a 
universal joint. This rod is initially held horizontal and in contact with the first rod ; and 
then let go, so as to oscillate in the vertical plane through the first rod. Prove that if & 



168 The Soluble Problems of Rigid Dynamics [CH. vi 

be the angle which the second rod makes with the vertical at any instant, and x the 
distance through which the first rod has moved from rest, 

(ir+ m) a: + »i? sin ^ = m/, 
and U - ^x^-coft* B\li*=^2gcoB$. (CJolL Exam.) 

17. A plane body is free to rotate in its plane about a fixed point, and a second plane 
body is free to slide along a smooth straight groove in the first body, its motion being in 
the same plane ; shew that the relation between the relative advance x along the groove 
and the angle of rotation $ (no external forces being supposed to act on the system) 
is of the form 

where F and Q are re8i>ectively linear and quadratic functions of a^, (ColL Exam.) 

18. A pendulum is formed of a straight rod and a hollow circular bob, and fitting 
inside the bob is a smooth vertical lamina in the shape of a segment of a circle, the 
distances of the centre {€) of the bob from the point of suspension (0) and from the 
centre of gravity (G) of the lamina being I and c respectively. Prove that if My m are the 
masses of the pendulum and lamina, k and kf their respective i*adii of gyration about 
and O, $ and the angles which OC and CG make with the vertical, then twice the 
work done by gravity on the system during its motion from rest is equal to 

(J/*«+«i^rf2+m(ifc^+c«)4>«+2fwc^cos(^-0)^<^. (ColL Exanu) 

19. A particle of mass m is attached to the end of a fine string which passes round 
the circumference of a wheel of mass M^ the other end of the string being attached to a 
point in that circumference, a length I of the string being straight initially, and the wheel 
(radius a and radius of gyration h) being free to move about a fixed vertical axis through 
its centre; the particle, which lies on a smooth horizontal plane, is projected at right 
angles to the string, so that the string begins to wrap round the wheel ; prove that, if the 
string eventually unwinds from the wheel, the shortest length of the straight portion is 

(/« - a2 - Mlc^jmf. (ColL Exam.) 

20. A carriage is placed on an inclined plane making an angle a with the horizon and 
rolls straight down without any slipping between the wheels and the plane. The floor of 
the carriage is parallel to the plane and a perfectly rough ball is placed freely on it. Shew 
that the acceleration of the carriage down the plane is 

14¥+4i/'+14??i 
14ir+4Jf' + 21m^^'''"' 

where M is the mass of the carriage excluding the wheels, m the sum of the masses of the 
wheels, which are imiform discs, and M' that of the balL The friction between the wheels 
and the axes is neglected. (ColL Exam.) 

21. A imiform rod of mass m^ and length 2a is capable of rotating freely about its 
fixed upper extremity and is initially inclined at an angle of Yr/6 to the vertical. A second 
rod, of mass m^ and length 2a, is smoothly attached to the lower end of the first and rests 
initially at an angle of 2ir/3 with it and in a horizontal position. Shew that if the centre 
of the lower rod commence to move in a direction making an angle fr/6 with the vertical, 
then 39i4»14m2. (ColL Exam.) 

22. A uniform circular disc is symmetrically suspended by two elastic strings of 
natural length c inclined at an angle a to the vertical, and attached to the highest point of 



OH. vi] The Soluble Problems of Rigid Dynamics 169 

tbe diaa If one of the strings is cut, prove that the initial curvature of the path of the 

centre of the disc is 

(c sin 4a — 6 sin 2a)/6 (6 — c), 

'where h is the equilibrium length of each string. (CoU. Exam.) 

23. Two rods AC, CB of equal length 2a are freely jointed at (7, the rod AC being 
freely moveable about a fixed point A, and the end B of the rod CB is attached to il by 
an inextensible string of length 4a,V3. The system being in equilibriiun, the string is 
cut; shew that the radius of curvature of the initial path of ^ at ^ is 

4 /41» 

isiVT-'*- 

(Camb. Math. Tripos, Part I, 1897.) 

24. A rod of length 2a is supported in a horizontal position by two light strings which 
paas over two smooth pegs in a horizontal line at a distance 2a apart and have at their 
other extremities weights each equal to one half that of the rod. One of the strings is 
cut ; prove that the initial curvature of the path of that end of the rod to which the cut 
string was attached is 27/25a. (Coll Exam.) 

25. A heavy plank, straight and very rough, is free to turn in a vertical plane about 

a horizontal axis from which the distance of its centre of gravity is c. A rough heavy 

sphere is placed on this plank at a distance 6 from the axis, on the side remote from the 

centre of gravity ; the plank being held horizontal. The system is now left free to move. 

Prove that the initial radius of curvature of the path of the centre of the sphere is 

216^/(5 -lid), where 6 = {mb'-Mc)l(mb+Ma)y m and M are the masses of the sphere and 

the plank, and Jfab is the moment of inertia of the plank about the axis. 

(ColL Exam.) 

26. A light stiff rod of length 2c carries two equal particles of mass m at distances k 

from the centre on each side of it ; to each end of the rod is tied an end of an inextensible 

string of length 2a on which is a ring of mass m'. Initially the string and rod are in one 

straight line on a smooth horizontal table with the string taut and the ring at the loop ; 

the ring is then projected at right angles to the rod, shew that the relative motion will be 

oscillatory *if 

c^/k^ > 1 +2»i/m'. (CJoU. Exam.) 

27. Three equal uniform rods, each of length c, are firmly joined to form an equilateral 
triangle ABC of weight W; a imiform bar of length 2b and weight W* is freely jointed to 
the triangle at C, This system rests in equilibrium in contact with the surface of a fixed 
smooth sphere of radius a, AB being horizontal and in contact with the sphere, and the 
bar being in the vertical plane through the centre of the triangle; the bar, and the centre 
of the triangle, are on opposite sides of the vertical line through C. Provejthat the 
inclination of the plane of the triangle to the horizon is the angle whose tangent is 

[atfi + 2cX«] -r [n/x (a« + i c») + X V - 2a&;] , 

where X«=a«+ ic»-J6c, /4«=12a«-c», and n^W/W. 

(Camb. Math. Tripos, Part 1, 1896.) 

28. A body, under the action of no forces, moves so that the resolved part of its 
angular velocity about one of the principal axes at the centre of gravity is constant ; shew 
that the angular velocity of the body must be constant, and find its resolved parts about 
the other two principal axes when the moments of inertia about these axes are equal. 

(Coll. Exam.) 

29. Shew that a herpolhode cannot have a point of inflexion. C (M. de Sparre.) 



1 70 The Soluble Problems of Rigid Dynamics [ch, vi 

30. In the motion under no forces of a body one of whose points is fixed, shew that 
the motion of every quadric homocyclic with the momental ellipsoid relative to the fixed 
point, and rigidly connected with the body, is the same as if it were made to roll 
without sliding on a fixed quadric of revolution, which has its centre at the fixed point, 
and whose axis is the invariable line. (Gebbia.) 

31. In the motion of a body under no forces round a fixed point, shew that the three 
diameters of the momental ellipsoid at the fixed point and the diameter of the ellipsoid 
reciprocal to the momental ellipsoid, determined respectively by the intersection of the 
invariable plane with the three principal planes and with the plane perpendicular to the 
instantaneous axis, describe areas proportional to the times, so that the accelerations of 
their extremities are directed to the centre. (SiaccL) 

32. When a body moveable about a fixed point is acted on by forces whose moment 
round the instantaneous axis is always zero, shew that the velocity of rotation is 
proportional to that radius vector of the momental ellipsoid which is in the direction of 
this axis. 

Shew that this theorem is still true if the body is moveable about a fixed point and 
also constrained to slide on a fixed surface. (Flye St Marie.) 

33. A plane lamina is initially moving with equal angular velocities fi about the 
principal axes of greatest and least moment of ineHia at its centre of mass, and has no 
angular velocity about the third principal axis; express the angular velocities about 
these axes as elliptic fimctions of the time, supposing no forces to act on the lamina. 

If d be the angle between the plane of the lamina and any fixed plane, shew that 

^.s<.(..-(D')'an<.,.{o.-(g)')^.. 

(Camb. Math. Tripos, Part 1, 1896.) 

34. A rigid body is kinetically symmetrical about an axis which passes through a 
fixed point above its centre of gravity and is set in motion in any manner ; shew that in 
the subsequent motion, except in ope case, the centre of gravity can never be vertically 
over the fixed point ; and find the greatest height it attains. (Coll. Exam.) 

35. In the motion of the top on the rough plane, shew that there exists an auxiliary 
set of axes O^riC whose motion with respect to the fixed axes OXYZ and also with respect 
to the moving axes Oxyz is a Poinsot motion ; the invariable planes being the horizontal 
plane in the former case, and the plane perpendicular to the axis of the body in the 
second case. (Jaoobi.) 

36. A uniform solid of revolution moves about a point, so that its motion may be 
represented by the uniform roUing of a cone of semivertical angle a fixed in the body 
upon an equal cone fixed in space, the axis of the former being the axis of revolution. 
Shew that the couple necessary to maintain the motion is of magnitude 

Jfl« tan a {(7+(C'-^) COB 2a}, 

where O is the resultant angular velocity and A and C the principal moments of inertia at 
the point, and that the couple lies in the plane of the axes of the cones. (Coll. Exam.) 

37. A vertical plane is made to rotate with imiform angular velocity about a vertical 
axis in itself, and a perfectly rough cone of revolution has its vertex fixed at a point of 



CH. vi] The Solvble Problems of Rigid Dynamics 171 

ft 

that axis. Shew that, if the line of contact make an angle with the vertical, and 
/3 and y be the extreme values of 6^ and a be the semi-vertical angle of the cone, 



Ka)'-** 



sin' a (cos 6 - cos ff) (cos y - cos 6) 
cos a cosj9 + cosy 



where h is the distanee of the centre of gravity of the cone from its vertex, and k its 
radius of gyration about a generator. (Camb. Math. Tripos, Part I, 1896.) 

38. A body can rotate freely about a fixed vertical axis for which its moment of 
inertia is 1 : the body carries a second body in the form of a disc which can rotate about a 
horizontal a3d8, fixed in the first body and intersecting the vertical axis. In the position 
of equilibrium the moments and product of inertia of the disc with regard to the vertical 
and horizontal axes respectively are A^ B, F, Prove that if the system start from rest 
with the plane of the disc inclined at an angle a to the vertical, the first body will oscillate 
through an angle 

2F ... (B^ sina] ,n ^^ ^ \ 

r tan * ^ < - . V . (Coll. Exam.) 

39. A gyrostat consists of a heavy symmetrical flywheel freely mounted in a heavy 
spherical case and is suspended from a fixed point by a string of length / fixed to a point 
in the case. The centres of gravity of the flywheel and case are coincident. Shew that, 
if the whole revolve in steady motion round the vertical with angular velocity X2, the 
string and the axis of the gyrostat inclined at angles a, /9 to the vertical, then 

Q' (^ sin a + a sin /3 + 6 cos 3) ^^ tan a, 
and 7X2 sin 0- AQ^ sin fi cos ^^Mg sec a {a sin O - a) + 6 cos O- a)}, 

where M is the mass of the gyrostat, a and h the coordinates of the point of attachment of 
the string with reference to axes coinciding with, and at right angles to, the axis of the 
flywheel, I the angular momentum of the flywheel about its axis and A the moment of 
inertia about a line perpendicular to its axis. (Camb. Math. Tripos, Part I, 1900.) 

40. A system consisting of any number of equal uniform rods loosely jointed and 
initially in the same straight line is struck at any point by a blow perpendicular to the 
rods. Shew that if k, i*, w be the initial velocities of the middle points of any three 
consecutive rods, it-f 4i;+tP«0. (ColL Exam.) 

41. Any number of uniform rods of masses A, B, C, ..., Z are smoothly jointed to 
each other in succession and laid in a straight line on a smooth table. If the end Z be 
free and the end A moved with velocity F in a direction perpendicular to the line of the 
rods, then the initial velocities of the joints {AB), (BC\ ... and the end J? are a, 6, ..., e 
where 

0=^(K+2a)+5(2a+6), 0=5(a + 26) + C(26+c), ..., 0= r(a?+2y)+-^(2y+«), 

and y+2«a»0. (ColL Exam.) 

42. Six equal imiform rods form a regular hexagon loosely jointed at the angular 
points: a blow is given at right angles to one of them at its middle point, shew that the 
opposite rod begins to move with ^ of the velocity of the rod struck. 

(Camb. Math. Tripos, 1882.) 

43. A body at rest, with one point fixed, is struck : shew that the initial axis of 
rotation of the body is the diametral line, with respect to the momental ellipsoid at 0, of 
the plane of the impulsive couple acting on the body. 



172 



The Solvble Problems of Rigid Dynamics [ch. vi 



44. The positive octant of the ellipsoid a^/a*+y^lb^'\'Z^/c^^l has the origin fixed. 
Shew that if an impulsive couple in the plane 






act upon the octant, it will begin to revolve about the axis of z. 



(ColL Exam.) 



45. An ellipsoid is rotating about its centre with angular velocity (a>i, a)^, M3) 
referred to its principal axes; the centre is free and a point (or, y, z) on the surface is 
suddenly brought to rest. Find the impulsive reaction at that point (ColL Exam.) 

46. Two equal rods AB, BC inclined at an angle a are smoothly jointed bX B\ A \a 
made to move parallel to the external bisector of the angle ABC: prove that the initial 
angular velocities of ABy BC are in the ratio 



2+3sin«^:2-158in«^. 



(ColL Exam.) 



47. A uniform cone is rotating with angular velocity a> about a generator when 
suddenly this generator is loosed and the diameter of the base which intersects the 
generator is fixed. Prove that the new angular velocity is 

(l+AV8P)»sina, 

where h \a the altitude, a the semi-vertical angle, and k the radius of gjnration about a 
diameter of the base. (Coll. Exam.) 

48. A rough disc can turn about an axis perpendicular to its plane, and a rough 
circular cone rests on the disc with its vertex just at the axis. If the disc be made to 
turn with angular velocity fi, shew that the cone takes an amount of kinetic energy 
equal to 

\Q^j{co&^ ajA +8in» ajC), (CoD. Exam.) 

49. One end of an inelastic string is attached to a fixed point and the other to a point 
in the surface of a body of mass M, The body is allowed to fall freely under gravity 
without rotation. Shew that just after the string becomes tight the loss of kinetic enei*gy 
due to the impact is 



*-/(^M'*j> 



where V is the resolved velocity of the body in the direction of the string just before 
in)pact, the string only touching the body at the point of attachment, (2, m, n, X, ^ y) are 
the coordinates of the string at the instant it becomes tight, and A, B, C are the principal 
moments of inertia of the body with respect to its principal axes at its centre of inertia. 

(ColL Exam.) 



I 



CHAPTER VII. 



THEORY OF VIBRATIONS. 



76. Vibrations about equilibrium. 

In Dynamics we frequently have to deal with systems for which there 
exists an equUibrium-configuration, i.e. a configuration in which the system 
can remain permanently at rest ; thus in the case of the spherical pendulum, 
the configurations in which the bob is vertically over or vertically under the 
point of support are of this character. If (q^ Qt, ..., qn) are the coordinates 
of a system and L its kinetic potential, and if (aj, cr,, ... , On) are the values of ^ss.7^f 
the coordinates in an equilibrium-configuration, the equations of motion y^/i«S7-f 

d (dL\ ai / 1 o X 

must be satisfied by the set of values 
?i = 0, ¥2 = 0, ..., ?n = 0, yi = 0, ja'^O, ..., g» = 0, yi«ai, q^-a^, ..., Jn^On. 

The values of the coordinates in the various possible equilibrium-con- 
figurations of a system are therefore obtained by solving for q^, 9t> •-•! 9n the 
equations 

g^^ = (r = l,2,...,7i), 

in which ji, $«, ..., ^n are to be replaced by zero. 

In many cases, if the system is initially placed near an equilibrium-con- 
figuration, its particles having very small initial velocities, the divergence 
from the equilibrium-configuration will never become very marked, the 
particles always remaining in the vicinity of their original positions and 
never acquiring large velocities. We shall now study motions of this type* ; 
they are called vibrations about an equilibrium-configuration. 

* More strictly speaking, we stady in this chapter the limiting form to which this type of 
motion approximates when the initial divergence from a state of rest in the equilibrium-configa- 
ration tends to zero ; the study of the motions which differ by a finite, thoagh not large, amount 
from a state of rest in the equilibrium-configuration is given later in Chapter XVI : the discussion 
of the present chapter may be regarded as a first approximation to that of Chapter XVI. 



/■ 



174 



Theory of Vibrations 



[cH. vn 



In the present work we are of course concerned only with the vibrations of systems 
which have a finite number of degrees of freedom ; the study of the vibrations of systems 
which bave an infinite number of degrees of freedom, which is here excluded, will be found 
in treatises on the Analytical Theory of Sound. 

We shall suppose that the system is defined by its kinetic energy T and 
its potential energy F, and that the position of the system is specified by the 
coordinates (g'l, q^, ..., gn) independently of the time, so that T does not 
involve t explicitly : we shall also suppose that no coordinates have been 
ignored ; the kinetic energy T is therefore a homogeneous quadratic function 
of ?i, ^a* •••! ?ni with coefficients involving Ji, 921 »••» ffn in any way. There 
is evidently no loss of generality in assuming that the equilibrium-con- 
figuration corresponds to zero values of the coordinates ^i, 99, ..., ^nl so that 
9i> 921 •••> ?ni ?i» ?2i •••> ?ni are very small throughout the motion considered. 

The coefficients of the squares and products of jj, y,, ..., g„ in T are 
functions of g'j, g,, ..., 9n> as however all the coordinates and velocities are 
small, we can in approximating to the motion retain only the terms of lowest 
order in 7, and so can replace all these coefficients by the constant vjilues 
which they assume when ji, q^, ..., jn are replaced by zero. The kinetic 
energy is therefore for our purposes a homogeneous quadratic function of 
9i> ?2> •••> ?n with constant coefficients. 

Moreover, if we expand the function V by Taylor's theorem in ascending 
powers of q^, jj, ...j^n the term independent of g'j, g,, ..., q^ can be omitted, 
since it exercises no influence on the equations of motion ; and there are no 

dV 

terms linear in g^, gj, ..., On, since if such terms existed the quantities ;: 

would not be zero in the equilibrium position, as they must be. The terms 
of lowest order in V are therefore the terms quadratic in Jug's, ..., ^n- 
Neglecting the higher terms of the expansion in comparison with these, 
we have therefore V expressed as a homogeneous quadratic form in the 
variables q^ q^t •••> qnt with constant coefficients. 

Thus the problem of vibratory motions about a configuration of equilibrium 
depends on the solution of Lagrangian equations of motion in which the kinetic 
and potential energies are homogeneous quadratic forms in the velocities and 
coordinates respectively, with constant coefficients. 



77. Normal cooid^inates. 



y" 



/ 



In order to solve the equations of motion of a vibr ating system, we write 
the expressions for the kinetic and potential energies .'m the form 

T-\ (a^iqi* + 02232* + . . . + annqn + 2aiagi j, + 2a„ j^r ^, + . . . + 2a„_i,n?n-i?n), 

» 

F= i (6,1 ?i* + b^q^^ + . . . 4- inn^n' 4- Sfcujj J, + 2baqi i78 + . . . + 26„_,,n3n-i?i») ; 



76, 77] 



Theory of Vihrdtions 



175 



of these T is (§ 26) a positive definite form ; and the determinant formed 
of the quantities Ors is i^ot zero (since if this condition is not satisfied, 
T will depend on less than n independent velocities). The equations of 
motion are 

dtW^'d^r (r = l,2,...,n); 

if a change of variables is made, such that the new variables (g/, g/, ..., q^) 
are linear functions of (g',, j,, ..., qn)i the new equations of motion will be 



dt \dq;) dq; 



(r=l, 2, ...,n), 



and these equations are clearly linear combii^ations of the original equations. 

Suppose then* that the original equations of motion are multiplied 
respectively by undetermined constants ?rii, m,, ..., Wn, and added together. 
The resulting equation will be of the form 



where 



Q=Aig'i + Ai,g2 + ••• + A„gn, 



provided the constants mi, Tn,, ..., ^m ^> '^» •••> ^m ^ satisfy the equations 

611^1 4- &]s^ + . . . + ftin^n — ^ (C^u Wi + OiaWlj + . . . + Om^ln) == XA^, 
ini Wli + tnam, + . . . + ^nn^n = ^ (^ni ^i + Ona Wlj + • • • + Onn W„) = XAn . 

These equations can coexist only if X is a root of the determinantal equation 



OuX — 6ji, dxsX — 61J, . . . , QinX — 6in 
CtuX — Oai, (XtQX — Ojni • • • > OsnX — Ojn 



= 0. 



a^iX — 6ni a„„X — 6 



fin 



Moreover, if Xi is any root of this equation, we can determine from the 
preceding equations a possible set of ratios for mi, m,, ..., vi^, Ai, Aj, .,., A„; 
these ratios may, in certain cases, be partly indeterminate, but in all cases at 
least one function Q can be obtained in this way, satisfying the equation 

Now let a linear change of variables be effected so that the quantity Q 
so determined is one of the new variables: there will be no ambiguity in 

* This method of proof is due to Jordan, Comptes ReiidtUt lzxiv. p. 1396. 



1 76 Theory of Vibrations [ch. \a 

denoting the new variables by g,, y,, ..., ^n; we shall take qi to be identical 
with Qy so that the above equations are satisfied by the values Ai = 1, As = 0, 
..,,An = 0- Since the form T is a positive definite form, the coeflScients 
a^, <hi» •••> (hin of the squares of q^, g,, ..., q^, will not be zero: so instead 
of qi, qs, '-•> qn we can again take new variables respectively equal to 

By this change of variables the terms in ^ij,, jij,, ..., gj^^i &re removed firom 
T: so we can assume that On, On, •.., ^m are zero. 

Now introducing the conditions Ai=l, ^2 = 0, A, = 0, ..., An = 0,.a2i = 0, 
• ••> OnisO, in the equations which determine mi, m,, ..., mn, hi. A,, ..., An, X; 
we obtain the values 

6n = XiOii, 6«i = 0, 6ji = 0, ... , 6„x = 0. 
It follows that the equation 

dt [dqj dqi 
has the form -^ + ^i9i = 0, 

while the equations -7- f ^-- j = — — (r = 2, 3, . . . , n) 

have the form 5/(s^)"'^a~" (r = 2, 3, ..., n), 

where r = T - Ja,,gA F' = F - i\aiiqi\ 

so that r' and F' do not involve ji and qi. 

This last system of equations may be regarded as the system of equations 
corresponding to a vibrational problem with (n — l)^c(egrees of freedom. 
Treating them in the same manner, we can isolaJ^Tanother coordinate q^ 
such that if 

(where X^ and a^ are certain constants), then if" and F" do not involve g, or 
ja, and the coordinates g,, q^, ..., qn are vHetermined by the equations of 
a vibrational problem with (n - 2) degree^ of freedom, in which the kinetic 
and potential energies are respectively Ty' and F". 

Proceeding in this way, we shall fiy jally have the variables chosen so that 
the kinetic and potential energies of the original system can be written in 
terms of the new variables in the form 

^ = i («ii ji" + Oaja' -»^\ . . + anngn'), 
V^i On?i'+ i8ag»» + .\^+ )8„ng«»), 
where a^, Oasi ..., Onm ^ui /^xa •••> ^nn are CODStaiffits. 




rr-] 



Theory of Vibrations 



177 



If finally we take as variables the quantities Vctn?!! V^Oayai •••, ^^n<lny 
instead of ji, ^s* •••> 9n» the kinetic and potential energies take the form 

where /^jb stands for fikk/^kk- 

In this reduction it is immaterial whether the determinantal equation has 
its roots all distinct or has groups of repeated roots. The final result can be 
expressed by the statement that if the kinetic and potential energies^ of a 
vibrating system are given in the form 

F= i (buqi^ + h^q^ + . . . + 6nn9n" + 26j23iga + . . . + 26n-i.n?n-i?n), 

it is always possible to find a linear transformation of the coordinates such 
that the kinetic and potential energies, when expressed in terms of the new 
coordinates, have the form 

where the quantities ^ij, fi^, ».., fin are constants. These new coordinates are 
called the normal coordinatej i or principal coordinates of the vibrating system. 

Now it is a well-known algebraical theorem that the roots of the determi- 
nantal equation 






a«ii\ — b. 



"ni 



ni 



a^nX — b 



^nn 



nn 



= 



are the values of X for which the expression 

(o^iX — 611) 5i^ + (a«X - 629) ?2" 4- . . . + {ann^ - bnn) qn + 2 (a^X - 612) 3i92 + . . . 

can be made to depend on less than n independent variables (which will be 
linear functions of ji, j,, ..., jn)- Since this is a property which persists 
through any linear change of variables, we see that the determinantal equation 
is invariantive, i.e. if 3/, g,', "•* qn are any n independent linear functions of 
9i» ?2» •••» ?«» ^^^ ^^ ^ ^^^ ^ when expressed in tenns of 5/, q^, ''^t qn take 
the form 

y = i (oiiqi' + a^'q^'' + ... + 2a,,'q,'q,' + ...), 

^= i (6n V + &« V + • • • + 26„V?2' +•••)> 

12 



W. D. 



1 



178 



Theory of Vibrations 



[cH. vn 



then the roots of the new determinantal equation ^a^'X^hn^^O are the 
same as the roots of the original determinantal equation || Or^X — &„ || =s 0. 

But when the kinetic and potential energies have been brought by the 
introduction of normal coordinates to the form 

the determinantal equation is 

■ 

X-/AX 0... 

\-/i, 0... 
X-fh 0... 











... X — /An 



= 0, 



80 its roots are /^i, /lc,, . . ., /in. It follows that the constants fh,fit, • • •! /hn which 
occur as the coefficients of the squares of the normal coordinates in the potential 
energy, are the n roots (distinct or repeated) of the determinantai equation 
\\ar$\ — b„\\^0, where Ou, Ou, ..., &u> &ui ..• are the coefficients in the original 
expressions for the kinetic and potential energies. 

It will be seen that the problem of reducing the kinetic and potential eneigies to their 
expressions in terms of normal coordinates is essentially the problem of simultaneoiisly 
reducing each of two given homogeneous quadratic expressions in n variables to a sum of 
squares of n new variables : the fact that jT is a function of the velocities while F is a 
function of the coordinates does not affect the question, since the formulae of transforma- 
tion for the velocities q^q^y •••* 17» ^ure the same as the formulae of transformation for the 
coordinates q^^ q^^ ..., ^n. 

It might be supposed from the foregoing that it is always possible to transform 
simultaneously each of two given homogeneous quadratic expressions in n variables to a 
sum of squares of n new variables ; but this is not the case ; for example, it is not possible 
to transform the two quadratic expressions 



to the forms 



€ufi-{-hxy-{-as^ and ca^-\rdxy-\-c3^ 

where £, 17, ( are linear functions of Xy y, z. 

The conditions which must be satisfied in order that two given quadratic expressions 

^U*i* + ^M*2*+ ••• +26i,JPiX2+..., 

may be simultaneously reducible to the form 

<»11 fc* + «22&' + . • • + OiMif »*» 
ftlfl*+i328f2»+...+A^fn^ 



77, 78] 



Theory of Vibrcttions 



179 



are, in fact, that the elementary divisors {Elementarikeiler) of the determinant ||a,^-5„|| 
shall he linear*. If however one of the two given forms is a definite form (as we saw was 
the case with the kinetic energy in the dynamical problem), the elementary divisors are 
always linear, and the simultaoeous reduction to sums of squares is therefore possible ; 
this explains the circumstance that the reduction can always be effected in the dynamical 
problem of vibrations. 

The universal possibility of the reduction to normal coordinates for dynamical systems 
was established by Weierstrass in 1858t; previous writers (following Lagrange) had 
supposed that in cases where the determinantal equation had repeated roots a set of 
normal coordinates would not exist, and that terms involving the time otherwise than in 
trigonometric and exponential functions would occur in the final solution of the equations 
of motion. 

78. Sylvester's theorem on the reality of ike roots of the determinantal 
equation. 

We have seen in the preceding article that by introducing new variables 
mrhich are linear functions of the original variables, it is always possible to 
reduce the kinetic and potential energies of a vibrating system to the form 

The question arises as to whether this transformation is realj ie. whether 

the coefficients m^y m^ mn, Ai, ^2, ..., An which, occur in the trans- 

fonoation are real or complex. Since these coefficients are given by linear 
equations whose coefficients, with the possible exception of the roots Xi, X,, . . ., X» 
of the determinantal equation, are certainly real, the question reduces to an 
investigation of the reality or otherwise of the roots of the equation 






«i2^-6u Oin^ — 6in 

OuX — 6n aff^-^bfn 



ttniX — bni flna^ — bm an«X — b 



nn 



= 0; 



it being known that the quantities a^ and b„ are all real, and that 

Mi' + ^aSt' + ... + annqn^ + ^Oi^iq^ + ... + 2a,i-.i, „ J^^i ?n 

is a positive definite form. 

Let A denote^ the determinant ||a^\ — 6,t||, and let A^ denote the 
determinant obtained from it by striking out the first row and first column ; 
let As denote the determinant obtained from A by striking out the first two 



Cf. Math'fl treatise on Elementartheiler (Leipzig, 1899) ; Bromwioh, Proc. Lond, Math, Soc. 



XXXI.— 



t Ci.^V^ierBtrass' ColUcUd Works, Vol. i. p. 283. 

X The following proof is due to Nanson, Mess, of Math, xxvi. (1896), p. 69. 




12—2 



182 Theory of Vihratidns [ch. vii 

It appears from these equations that' if all the normal coordinates except 
one, say q^, are initially zero, and if the constant X^ corresponding to the 
non-zero coordinate is positive, then the coordinates (g, , ja, . . . , jr-i , 9r+i» • • • > ?n) 
will be permanently zero, and the system will perform vibrations in which 
the coordinate g,. is alone affected. Moreover the configuration of the 

system will repeat itself after an interval of time 2'7r/Vx^. This is usually 
expressed by saying that each of the normal coordinates corresponds to an 
independent mode of vibration of the system, provided the corresponding 

constant \r is positive ; and the period of this vibration is 27r/Vv. 

Moreover, if the system be referred to any other set of coordinates which 
are not normal coordinates, these coordinates are linear functions of the 
normal coordinates; and the normal coordinates perform their vibrations 
quite independently of each other ; thus every conceivable vibration of the 
system may be regarded as the superposition of n independent normal 
vibrations This is generally known as BemovlKs principle of the super- 
position of vibrations. 

If the quantities (Xi, \q, ..., Xn) are not all positive, it appears from the 
above solution that those normal coordinates qr which correspond to the 
non-positive roots X^ will not oscillate about a zero value when the system is 
slightly disturbed from a state of rest in its equilibrium position, but will 
increase so as to invalidate the assumption made at the outset of the work, 
namely that the higher powers of the coordinates can be neglected. In 
this case therefore, there will not be a vibration at all, and the equilibrium 
configuration is said to be unstable. If however the initial disturbance is 
such that these normal coordinates which correspond to non-positive roots 
Xf. are not affected, the system will perform vibrations in which the rest 
of the normal coordinates oscillate about zero values. 

The normal modes of vibration, which correspond to those normal 
coordinates for which the corresponding root V is positive, are said to 
be stable. If the constants Xr ^e all positive, the equilibrium-configuration 
as a whole is said to be stable. The condition for stability of the equi- 
librium-configuration is therefore, by the theorem of the last article, that 
the potential energy of the vibrating system shall be a positive definite form. 

This result might have been expected from a consideration of the int^ral of energy ; 
for this integral is 

where T and V are the quadratic forms which represent the kinetic and potential energies, 
and where A is a constant. This constant h wiU be small if the initial divergence from the 
equilibrimn state is small. But T is a positive definite form ; and if V is also a positive 
definite form, we must have T and V each less than A, so T and V will remain small 
throughout the motion : the motion will therefore never differ greatly from the equilibrium- 
configuration, i.e. it will be stable. 



79, 80] Theory of Vibrations 183 

80. Examples of vibrations aiout equilibrium. 

We shall now discuss a number of illustrative eases of vibration about 
equilibrium. 

(i) To find the mbration-period of a cylinder of any cross-section which cam, roll on the 
outside of a perfectly rough fixed cylinder. 

Let B be the arc described on the fixed cylinder bj the point of contact^ s being 
measured from the equilibrium position ; let p and p' be the radii of curvature of the 
cross-sections of the fixed and moving cylinders respectively at the points which are in 
contact in the equilibrium position ; p and p' being supposed positive when the cylinders 
are convex to each other : let J/ be the mass of the moving cylinder, MJc^ its moment of 
inertia about its centre of gravity, and c the distance of the centre of gravity from the 
initial position of the point of contact in the moving cylinder. 

If a denotes the initial angle between the common normal to the cylinders and the 
vertical, then a-\-Blp is the angle between the common normal at time t and the vertical, 
a-k-s/p-^-sIp' is the angle made with the vertical by the line joining the centre of cmrvature 
of the moving cylinder with the original point of contact in the moving cylinder, and 
e/p-he/p' is the angle made with the vertical by the line joining the last-named point to 
the centre of gravity of the moving cylinder. The angular velocity of the moving cylinder 
is therefore 

so its kinetic energy is 

The potential enei^ is 
V^Mg X height of the centre of gravity of the moving cylinder above some fixed position 

=Mg [{p + p) cos la +- j -p' cos f a+ - + - j +ccos ( - + - U . 
Neglecting ^ this gives 

The Lagrangian equation of motion, 

dt\di) 8<" ds' 
gives ir(^Ho2)(l + ^)V%{^cosa-c(^^')},=0, 

and the vibrations are therefore given by the equation 

where A and c are constants of integration to be determined by the initial conditions, and 
X is given by the equation 

The vibration-period is 2tr/k. 

(ii) To find the periods of the normal modes of vibration about an equU^nium-configura' 
tion of a particle m^mng on a fixed smooth surface under gravity. 

The tangent-plane to the surface at the point occupied by the particle in the 
equilibrium-configuration is evidently horizontal : take as axes of x and y the tangents to 



i 

f 



184 Theory of Vibrattom [oh. v 

the Uoes of curvature of the surface at this point, and as axis of ; a line drawn vertical 
upwards : so that the equation to the surface is approximately 

where p, and p, denote the principal radii of curvature, measured positively upwarc 
Hie kinetic energy and potential energy are approximately 

T-im (i>+yi) (whera m is the mass). 



K£-£)- 



It is evident from these expraasions that x and g are the normal coordinates: t! 
equations of motion are 

x+^x'^O and y + ^y-0, 
Pi P» 

and the periods of the normal modes of vibration are therefore 

(iii) To find the nwrnat model of vibration of a rigid body, one of whose point* u /in 
and wkteh it vibrating ahout a poeition of ttabU equilibrium under the action ofang tt/iti 

of conservative foreet. 

Take as fixed axes of reference OXTZ the equilibrium positions of the priodpol axes 
inertia of the body at the fixed point ; the moving axes will be taken as usual to be th« 
principal axes of inertia. We shall suppose the position of the body at any insta 
defined by the symmetrical parameters (f, ?;, ^, ;() of § ; we shall r^ard j, ij, ^ aa t 
independent coonlinat«a of the system, ^ being defined in terms of them by the equation 

The components of angular velocity of the body about the moving axes are (§ 16) 

U=2('!€-fi+xf-«)- 
On account of the smaUnesa of the vibration, we regard |, f, C ^ small quantities 
the first order; x therefore diSera from unity by a small quantity of the second order, a 
so we have, corroctly to the first order of small quantities, 

o.,-2i, -,-2i, »,=Bf, 
and the kinetic energy of the body, which is given by the equation 

where A, B, C are the principal moments of inertia at the point of suspension, can 
written 

The potential energy is some function of the position of the body, and therftfore of 
" (£• f, ; let it be denoted by V ({, i,, ()• 



r 



^VWI 



im 



185 



iipiilibrimn position, there will be 
jfing powers of (f , jy, {) : the lowest 
pi of higher order, we can therefore 



t+2Aft, 



tes is therefore the same as that 



.^2^, 



jmomental ellipsoid in its equi- 

I 

itKon is 

iirr=i, 

pr" ; and determine the common 
\Z^ he the coordinates, referred 
iKtes referred to the fixed axes 
^, Z') and (X, T, Z) he 



re reduced to the form 



1 coordinates in the dynamical 



t in which ^ alone varies, the 



are, to the first order of small 
Ine &bout which the rotation 
Di&l mode of vibration of the 
ose equation is 



9/* Vibr^^^^^'^ 

K>xid to thP"^^^^^^ initially by 
uncled in asc cosjd^, 
glecting te 



187 



^C^+y^C+^on is approximately represented by 

VXr cos jD^, 



oirmal coordii 



s 



ayapt; 



Cft 



•^^ hds of a vibrating system. 

+ C z^ 
» . ^^J ' ^®^ ^^ ^^® periods of normal 

^g^'^tion of stable equilibrium 
^«, % f). j^^ gyg^^^ ^ diminished by the 
&xed axes, of 

-+-5r»+(7Z*=*cified in terms of its normal 
xaadric whose e^^^ potential energies have the 

vial potential en. -v j^ jx . 
•<iric8. Let(X', '^^'n;, 
^int whose cooi^d by the equation 
connecting (JT'. ^ 

'h^'+m^F'-^-n^^ *^® function/ in ascending 
'^a-T'+jTisF'+nj®*' terms of the expansion: we 
ns of the quadric*^ 

■ , . ,, num-confaguration is supposed 

tch gives the noi,-i, i. i. x x -o 

^ill be no constant term. By 

•^^+m,y+ni«, ^ve thus have 

of vibration, say 
y in the ratio 4 ^ , . j ^ v* 

' is evident that f , constrained system are there- 
stion-cosines of t\ 
consequently the 
oscillation about a line* • j . » a \1 /> 

Z : F: Z^l^ : l^ : i ^ J 

(r=l, 2, ...,n-l), 

> (r = l,2, ...,n-l), 



J the common conjugate diameters of the 



186 Theorp, 

Hence finaUy we have the result 
oscOlatioM about the common wnjvgoit ^ 

of equal potential energy. tOTOttOm [GH. VH 

(iv) fo find the wrmal coordiwU^^^ and aa axis of « a line drawn verticaUy 
of three degreei of fretdom for «*»«»xpprorimately 

T =i (4 , 

' 'fJi 

where a is small in comparison curvature, measured positively upwards. 

qo from rest wXh y avd z »»^f<fy ^proximately 

yas the original one was mx, |-y*) (where m is the mass), 

The form of the kinetic and pote^ 

which gives j,j W 

\yj, and y are the normal coordinates: the 

The variable ij is therefore a nor^. ^ff q^ 
kinetic and potential energies to suif p, 

^ ' are therefore 

and then we have in (^\ . 

T^rj •\'Y^'^(^q^-jf a rigid body^ one of whose points is Jixed, 

8 (c9^iiiMum under the action of any system 

' \\} fb C are therefoiilibrium positions of the principal axes of 

' ' ig axes will be taken as us\ial to be these 

Suppose that initially we hav^ position of the body at any instant 

it) of § 9 ; we shall regard iyfjtC^ ^be 
jfined in terms of them by the equation 

an*' suppose that it is so small [=1. 

D ^!ted. Then to this degree oi ^^^^ ^.j^^ moving axes are (§ 16) 

\ vibrations of the normal 0I4. f f _ ,j^), 

!we regard (, 17, C ^ small quantities of 
small quantity of the second order, and 
Entities, 

^t equation can be wii«s=2f, 

4>»iifccosW^*^«^^*^^'^ 

<^s^l;co8fHia at the point of suspension, can be 

The potential e^. ^sition of the body, and therefore of the 

parameters (f , 1;, f ) ; k ). 

I 

! 

r 




80] Theory of Vibr'i^rations jg- 

Since zero values of (f, ij, f) correapond to tht repreeenfaj iaitiaUy by 
no tenns linear in ({, ;, fl) when V is expanded in ii«c » ^ 
terms are therefore of tlA second onjer ; neglecting tt 
■write I O. 

where a. 6, ^ / ^. ^ 4- constant. ^ " "PP-^^^t^^ rep«eented by 

The problem of determining the normal coordii 
of reducing the two quadratic expreBsions ' * co«^ ; 

to the form , "tods of a Vibrating avBtem 

t«,a:*+fc^ +■;,*', '""^ on the periods of normal 

where (X, y, *) are linear ftmctiona of (|, ,, f). *°%"ration of stable equilibrium 

^ ,, , J . .u c J the system is diminished by the 

Now the equation, referred tn the fixed axes, of j ^ 

librium poeition, is 

, ^j:"+Sr.+(;/.-peciaed i« terms of it, „„„„,, 

consider in conneiian with this the quadric whose ^ and potential cner^ea have the 

which we shall call the "ellipsoid of equal potential en 

set of conjugate diUnetere of these quadrics. Let (JT', "^ *»'9n') > 

"■ fr" "^H fjf";'*"' '",.'' t""' :^°" °?;scd by the equation 
are (^, r, Z), and let the equations connecting {X ^ 

I fX=ljX'+m,r-+n~ ' 

i r=i^'+«i,r+«^d the fiinction/ in ascending 

U='j^'+»4J"+n3rat terms of the expansion : we 

By tliis tianalbrmation the equations of the quadrion 

and therefore the tronsfonoation which gives the noi^'^'">^-'»ifigu ration is supposed 
problem is *ill be no constant term By 

p-'i*+'"iy+'H*. we thus have 

-jij—^^+n^y + Bir, 

It follows that in a normal mode of vibration, say J 

quantitiee (£, >j, will be permanently in the ratio 

. |,,,M:,.,,> + -^ 

But from Ithe defioitionB of § 9 it is evident that f , constrain 
qtia&titiea, proportional to the direction-cosines of tl 
of the rigid 'body takes place, and consequently the 
rigid body cnnsistB of a small oscillation about a line 

i.e. about ttte line 

r-0, z-=o, (r 

which ia ona of the common conjugate diameters of the t (r 



of Vibrdtions 



[CH. vn 



186 

Hence finally ve 
oicillations about the 
of equal potential energ^^ 

that the normal vibrattOTis of the body are email 

(iv) fo find the narw<a ^^.^^^, of the momefUal dlipeoid and the eUipeoid 
of three degrees of freet^on 

es and the periods of normat\ vibration in the eystenh 
where a is small in ^^^^^^=^^+^24.^2), ( 

V as the wiginal one vhx^ *i ^ and q ; and to shew that vf such a system he let 

cLi.^y^X'r^^^^iAzero. the vibration in x wiU hdve temporarily ceased 
The form of the kine^** > , i^ f.^ . . 

^ here will then be a vibration of the same amplitude lA 

j (ColL Exam.) 
itial energies suggests the transfiprmation 

The variable 1? is itt&^retk^ ^ ^ ^a ^^^^ 
dnetic and potential ^^^^^^^^^.^^^^^^.^ ^. 

joal coordinate : to reduce the remainiiig terms in the 



which gives 



and then we have 



us of squares, we write 



.y^ 



^-p> 



V^phj*-*- 



?}^^*(^^^}^' 



The variables ,, «*^ f, «**j^,_4^), , , . (4g»-2f«)a» ) ^ 

Suppose that initiaUra^*)*' J V* +* \? + {^-p')* J ^• 

e the normal coordinates. 



and suppose that * is ^^f.^ ^^q^ ^^q^ 
iS^ Then to thi»^^^^ .^^^ ._^_ 

\ that its product with other small quaniitieH can be 

/ \ vibrations of the f approximation we have initially 

)ordinates tf and <ji are therefore given b}' the equations 
equation caj -• I tf ~rf 



..AepOw 
parameters {^. 



I 



0, 



1 + 



2a« 



(J2«^)2 



tten 



*>(i-p^)}]' 



a«« 



at 



' «"?(?^)+** "° ^ '^p¥^^) ■ 



80, 81] 



Theory of Vibrations 



187 



or 



or 



The motion can therefore be approximately represented initially by 

After an interval of time irp {q*—p^/a^f the motion is approximately represented by 

ff=^k COS pt, if>s=-^k coaptj 

x=0 , y= -kcoapt ; 

which establishes the result stated. 

81. Effect of a new constraint on the periods of a vibrating system. 

We shall now consider the effect produced on the periods of normal 
vi'bration of a dynamical s}r8tem about a configuration of stable equilibrium 
when the number of degrees of freedom of the system is diminished by the 
introduction of an additional constraint. 

Suppose that the original system is specified in terms of its normal 
coordinates (^i, q^, ..., qn\ so that the kinetic and potential energies have the 
form 

T^==i(V3i' + V?2*+ ... + V?n»); 

and let the additional constraint be expressed by the equation 

/(?i, ?a, ..., 9n) = 0. 

Since qi, q^, ..., qn sure small, we can expand the function/ in ascending 
powers of qi^q^, ...> 9n> ^^^ retain only the first terms of the expansion : we 
can thus express the constraint by the equatipn 

where Aiy,..,An are constants. As the equilibrium-configuration is supposed 
to be compatible with the constraint, there will be no constant term. By 
means of this equation we can eliminate qni we thus have 

^ = i W+ 9a'+ ... -i-?Vi + 2^,(4i?i+ ... + ^n^i ?n-i)j , 

The Lagrangian equations of motion of the constrained system are there- 
fore the (n — 1) equations 

(r = l, 2, ...,n-l), 
or gr + V3r + /A^r = (I'ssl, 2, ..., n — 1), 



188 Theory of Vibrations [ch. vn 

where 

1 \f? 

SO the equations of motion of the constrained system can be written in the 
form of the n equations 

?r + V?r + Mr = (r = 1, 2, ..., »), 

where fi is undetermined. 

Now consider a normal mode of vibration of the modified system, defined 
by equations 

9i — ttiCOSX^, ^3=: OtsCOSX^, ..., ^AsOnCOsXi, /i8=|/C0S\^. 

Substituting in the equations of motion, we have 

ar(V->^0 + »'^r«O (r=l, 2,..., n). 

Substituting the values of a^, Oa, ..., etn given by these equations in the 
equation 

we nave 

X a _ "Xa "^ "xTZTa"^ "• "*" "X « — \.2 ~ 

This equation in X" has (w — 1) roots, which from the form of the equa- 
tion are evidently interspaced between the quantities Xi', X,*, ..., Xn*: the 
quantities 27r/X corresponding to these roots are the periods of the normal 
modes of vibration of the constrained system, and it therefore follows that 
the {n—l) periods of normal vibration of the constrained system are spaced 
between the n periods of the original system. 

82. The stationary character of normal vibrations. 

We shall next consider the effect of adding constraints to a dynamical 
system to such an extent that only one degree of freedom is left to the 
system. Let (gj, jj, ..., j^) be the normal coordinates of the original 
system ; the constraints may, as in the last article, be represented by linear 
equations between these coordinates, and can therefore be expressed in the 
form 

where /ii, /ii, ..., A^ ^^6 constants and g is a new variable which may be 
taken as defining the configuration of the constrained system at time t 

Let the kinetic and potential energies of the original system be 



81-83] Theory of Vibrations 189 

so 27rl\i, 27r/\9, ..., 27r/Xn are its periods of normal vibration: the kinetic 
and potential energies of the constrained system are then 

F= i (Va*i' + \W + ... + \nW) q'> 
The period of a vibration of the constrained system is therefore 27r/\, 
where X is given by the equation 

If the constraints are varied, this expression has a stationary value when • 

(n — 1) of the quantities /ii, /Aj, ..., /in are zero: this stationary value is one 
of the quantities W \j*, . . . , X^* : and thus we have the theorem that when 
constraints are put on the system so as to reduce its number of degrees of 
freedom to unity, the period of the constrained system has a stationary value 
foi' those constraints tvhich make the vibratio7i to be a normal vibration of the 
unconstrained system, (fi^^^f^, S.^ 1 1^^H , Ai4. s^,yt./^y-^^'^,^.*^. ^^^4)f''^^-f< 

83. Vibrations about steady motion, 

A type of motion which presents many analogies with the equilibrium- 
configuration is that known as the steady motion of systems which poteess 
ignorable coordinates: this is defined to be a motion in which the non- 
ignorable coordinates of the system have constant values, while the velocities 
corresponding to the ignorable coordinates have also constant values. 

One ejLample of a steady motion is that of the top, discussed in § 72 ; as another 
example we may take the case of a particle which is free to move in a plane and is 
attracted by a fixed centre of force, the potential energy depending only on the distance 
from the centre of force; for such a particle, a circular orbit described with constant 
velocity is always a possible orbit, and this is a form of steady motion, since the radius 
vector is constant and the angular velocity d corresponding to the ignorable coordinate 6 
is also constant. 

In many cases, if a system is initially in a state of motion differing only 
slightly from a given form of steady motion, the divergence from this form of 
motion will never subsequently become very marked ; we shall now consider 
motions of this kind, which are called vibrations about steady motion. 

The steady motion is said to be stable* if the vibratory motion tends to 
a certain limiting form, namely the steady motion, when the initial disturb- 
ance from steady motion tends to zero. 

I^^ {Pu pa> --->Pk) be the ignorable and (jx, jj, ..., jn) be the non-ignorable 
coordinates of the system. Corresponding to the ignorable coordinates, there 
will be k integrals 

g|^ = i8r (r=l, 2, ...,*), 

* This definition is due to Klein and Sommerfeld. 



190 



Theory of Vibrations 



[oh. vn 



where ^i, ^21 •••! 13k ^^ constants. We shall suppose that these constants 
have the same value in the vibratory motion as in the undisturbed steady 
motion of which it is regarded as the disturbed form ; this of course only 
amounts to coordinating each vibratory motion to some particular steady 
motion. 

We suppose the system conservative, with constraints independent of the 
time ; let its kinetic energy be 

I* I* n k k k 

where the coefficients Qij, b^, c^, are functions ofqu 9s» •••! ?fi- 
The integrals corresponding to the ignorable coordinates are 

% i 

Let C^ be the minor of Cij in the determinant formed of the coefficients 
c^, divided by this determinant; then solving the last equations for the 
quantities pr, we have 

Substituting for^,pi, ...,pk,ui the above expression for T, and utilising 
the properties of minors of determinants, we have 

r« i 2 (a<^ - :S.Cububj.) q^q^ + i 2 Cfeftft. 

Now perform the process of ignoration of coordinates. Let JS be the 
modified kinetic potential, so 

R^T-V- 2 prPr 
«i2(ac^-2(7to6«6^)g<g,.+ 2 Cr.prbuqi' htCuPi^.-V. 

We can without loss of generality suppose that the values of ji, 9,, ..., ^n 
in the steady motion are all zero. If then the coefficients in R are expanded 
in ascending powers of ^i, ja, ..., 9n by Taylor's theorem, and all terms in the 
expression of jR thus obtained which are above the second degree in the 
variables 91, q^, ..., ^n> Ju 921 •••> 9n are neglected in comparison with the 
terms of the second degree, we obtain for jR an expression consisting of terms 
linear and quadratic in ^1, j,, ..., q^ qu q%f •••> 9n- Now the terms which 
are linear in q^, q^, ..., q^ and independent of g^, j,* -•-! 9n> disappear auto- 
matically from the equations of motion 



dt \dqr) dqr 



(r = l, 2, ...,n), 



and these terms can therefore be omitted. Moreover, since the equations are 
satisfied by permanent zero values of q^ q^, •••> 9n> it is evident that no terms 



83, 84] 



Theory of Vibrations 



191 







(r= 1, 2, ..., n), 



linear in qu <!%* •••> 9» a&d independent of ^i, g,, ..., ^n can be present in jR. 
It follows that the problem of vibrations abotd steady motion depends on the 
soluticn of Lagrangian equations of motion in which the kinetic potential is a 
homogeneous quadratic function of the velocities and coordinates, with constant 
coefficients. 

The difference between vibrations about equilibrium and vibrations about 
steady motion consists in the possible presence in the latter case of terms of 
the type qrq$ (i.e. products of a coordinate and a velocity) in the kinetic 
potential. These are called gyroscopic terms. The vibrations about steady 
motion of a system are in fact the same thing as the vibrations about 
equilibrium of the reduced or non-natural (§ 38) system to which the problem 
is brought by ignoration of coordinates. 

The equations of motion for the vibrating system are therefore 

dt KdqJ dqr 
where jR can be written in the form 

12 = i 2 a„ jr?« + i 2 ^rWrJ. + 2 7rf ?r J« (r, s^ 1, 2, ..., n), 

r.« r.» r,# 

and where Or, = Ugr, fin = fitr, 

but where jn is not in general equal to jgr- The equations of motion in the 
expanded form are 

auji - /8u?i + aia32 + (7n - 71,) Ja ~ /8u?2 + ««}, + (7a - 71,) g, - /8„ft 
C%Ji + (7ia-7n)?i-^ii?i + fl^92-^aB?i + aag, + (7n-7a)g,-/8ag,+ ^ 

etc. 

These are linear equations with constant coefficients, which are of the same 
general character as the corresponding equations in the case of vibrations about 
equilibrium ; they differ only in the presence of the gyroscopic terms, which 
involve the coefficients (7»r — 7r#)- The presence of these terms makes it 
impossible to transform the system to normal coordinates* ; but, as we shall 
next see, the main characteristic of vibrations about equilibrium is retained, 
namely that any vibration can be regarded as a superposition of n purely 
periodic vibrations, which we shall call (as before) the normal modes of 
vibration of the system. 

84. The integration of the equations. 

We shall now shew how the nature of the vibrations can be determined, 
by integration of the equations of motion. 

* That is to say, impossible to transform the system to normal coordinates by a point-traM- 
fcrmoHan : it is possible to effect the transformation to normal coordinates by a contact-trans- 
formation, and this is actually done in Chapter XVI. 



192 Theory of VibrcUions [oh. vn 

It will be convenient first to transform them into a system of equations 
each of the first order. Let 12 denote the modified kinetic potential of the 
system, so that in the vibratory problem 22 is a homogeneous quadratic 
function of fj, jj, ..., q^, ji, g„ ..., jn- Write 

dR _ / _ 1 » \ 

rr ^n+r V — A, i, ..., 11), 

so that gn+i> ?n+8> •••, ?2n are linear functions of ji, g,, ..., ?n and vice versa ; 
the equations of motion can be written 

^^ /to \ 

qn^r = ^ (r=l. 2, ...,n). 

• Now if S denote an increment of a function of the variables ji, ja, ...,?»> 
9n+i> •••! 9»ti due to small changes in these variables, we have 

n 
« 2 (qn+r^qr -^^ qn+r^r) 

= S ( 2 ?n+r3r ) + 2 (qn+r^r-qr^n-n)' 
\r-l / r=l 

n 

Let the quantity 2 qn^rqr — -B, 

when expressed as a function of qi, q^^ ..., q^* be denoted by H, so that H is 
a known homogeneous quadratic function of the variables qi, Q^2i •••» Jsn) the 
last equation can be written 

Sfr= 2 (grSjn+r - gn+rSjr), 
r=l 

and therefore* the equations of motion, which consisted originally of n equations 
each of the second order, can be replaced by a system of 2n equations, ea>ch of 
the first order, namely 

dH dH /TO \ 

dqn+r oqr 

the independent variables being qi, q^, ..., },»• 

We shall now shew that the function H, which has replaced 12 as the 
determining function of the equations, represents the sum of the kinetic and 
potential energies of the dynamical system considered. 

For 12 contains terms of degrees 2, 1, and in the velocities, and 

^ . dR 

r=l oqr 

* This transformation is reaUy a case of the HamUtonian transformation given later in 
Chapter X. 



84] Theory of Vibrations 198 

is equivalent to twice the terms of degree two together with the terms of 
degree one, by Euler's theorem ; it follows that H, being defined as 

will be equal to the terms of degree two in the velocities in 12, together with 
the terms of zero degree in R with their signs changed : on comparing the 
expressions for T and R given on page 190, it follows that 

so S 18 the total energy of the dynamical system, expressed in terms of the 
variables ji, g,, ..., q^- 

In the case of vibrations about an equilibrium-configuration, we have 
seen that the condition for stability is that the potential as well as the 
kinetic energy shall be a positive definite form ; we shall now make a similar 
assumption for the case of vibrations about steady motion, namely that the 
total energy H is a positive definite form in the variables qi, q2t •••> ffsni on 
this assumption we shall shew that the steady motion is stable, and in fact 
that the equations of motion 

dqr dH^ dqn^_^_dH 

dt dqr^' dt " dqr ^r-i, A...,n; 

can be integrated in the following way *. 

Consider the set of linear equations in the variables qi,qi, ..., 99n> 






(r = l, 2, ...,n); 



if we denote the determinant of the system by f(s), and the minor of the 
element in the \th row and fith column by 

/(j?)am, (\, /^ = 1, 2, . . . , 2n), 

the expression of ji, g,, ..., g^ i^ terms of yi, y^, ..., y^ is given by the 
equations 

3''=^"'^y^' (m = 1,2, ...,2n) 

and the degree of f(s) in « is 2n, while the degree of f(s)K,t is not greater 
than (2n-l). 

In order to solve the equations of motion, consider expressions for 
?if 5a> •••> J2» of the form 

^^ = I ^^"'''"^ ^ (/^ = 1' 2, ..., 2n), 

* The method of mtegration which follows is due to Weierstrass, Berlin, MonaUherichte, 1879. 

W. D. 13 



194 



Theory of Vibrations 



[CH. vn 



where the integration is taken round a large circle C which encloses all the 
roots of the equation /(«) = 0. These values of ji, fj* •••> 9»» will satisfy 
the equations of motion, provided the equations 



//"-"{■ 



e^it-t^ \sp^^^ + 



dH(pi,Pi, -..iPjJn) 



Jc [ Opn+r 



ds 



= 



= 



(r = l, 2, ..., n) 



are satisfied. If therefore pi, psi •••> J'm ^^^ pol}rDomials in 8 so chosen that 
the expressions in brackets under the integral sign vanish when 8 is equal to 
one of the roots of the equation /(«) = 0, these equations will be satisfied, 
since the integrands will then have no singularities within the contour C7*. 
It follows that j>i, P2, '*»,Pfm must be a set of solutions of the equations 



^'^ d^^ """J 



(r = l, 2, ...,7i), 



when « is a root of the equation /{s) = ; this condition is satisfied by the 
expressions 

p^ («) = aif{B\i. + (hf(s)^ + . . . + (hnf(s)w,ii (/A = 1, 2, ... , 2n\ 
where Oi, a,, ..., 0^ are arbitrary constants. 

The equations of motion are therefore satisfied by the values 

9^ = coefficient of 1/8 in the Laurent expansion "f* in positive and nega- 
tive powers of 8 of the expression 



{<hf(8)i^ + as/(«V + . . . -f- Oan f(s)^i^} 



7W 



(/i=:l,2, ...,2n). 



Now on inspection of the determinant /(«) we see that minors of the 
types 

/(«)n+M, M aiid /(«)m. n+M (/i = 1, 2, . . . , w), 

are of degree (2n — 1) in s, and the other minors are of degree (2n — 2) in « ; 
so the coefficient of l/« in the Laurent expansion of /(«)am//(*) is zero unless 
\ = n + /Aor/i = n + \; in the former case it is — 1, and in the latter case it 
is 1. Hence on taking f == fo, we see that the quantities 



^ii ^> •••> ^hn 



are respectively the values of 



at the time L, 



* Whittaker, A Course of Modem Analytit, § 86. 
t Ibid. § 43. 



84]- Theory of VibraUom 

If therefore we write 



195 



m> 



4> (Oah = coeflBcient of l/« in the Laurent expansion ot'^-j^e*^^"*^, 

and if ^1, ^„ ..., ^^n are the values of ji, g,, ..., q^ respectively corresponding 
to any definite value ^ of t, we have 

!Zm= 2 {^tH^0(O..M-(?.*(OtH-.4 (/* = 1, 2, ..., 2n). 

In order to evaluate the quantities 0(<)am> it is necessary to discuss the 
nature of the roots of the determinantal equation /(«) = ; let K + Z, where 
k and I are real and i denotes V— h he any root of this equation ; then 
th^ 2n equations 






(a = l, 2, ...,n) 



can be satisfied by values of qi^ q^, ..., q^n which are not all zero. Let a 
system of such values be 

where fi, fa, ..., f«»» %> ^«i •••> ^9»* are real quantities. Then if we write 
we have, on separating the last equations into their real and imaginary parts, 

H (fi, fa, .-. , fan). + Zf«+« - *?i;»+« = \ 

-ff(fi, fa, ...,fanW-^f« + *^« =0 
S(rfi,rft, ... , l/an). + 2i7«h« + Arfn4^ = 

H (i/i, 17a, ..., i7a»)fi+« — ^« — Arf« = > 

But since iT is homogeneous and of degree two in its arguments, we have 

2n 
SJ^Cfi, fa, •••, fan)= 2 fx5(fi, fj, ..., f»,)x, 

and using the first two of the preceding equations this gives 



-(« = !, 2, ..., n). 



Similarly 



^^^(fi, fa, ..., fan)=^*? 2 (f.i7n+«- i7«f*i+«)- 

I* 

2-H'(l7i, 1/2, ...,l7a») = ^ 2 (f«l7n+a— '»7«fn+tt) 



(A). 



Moreover on multiplying the first of the preceding equations by rj^ and the 
second by i/n^, adding, and summing for values of a from 1 to n, we have 

2m * n 

2 17\ir(fi, fj, ..., fa»)A=i 2 (f«17fH-«""''7«fn+a), 



and similarly 



A=l 



8m M 

2 f\-ff(%i^aj •••» ^a»)A=» — ^ 2 (fttl7»+« — i7afn+«). 



13—2 



196 Theory of Vibrations [c 

Sioce the left-haad sides of these equations are equal, we must havi 

But from equatlone (A) we see that, as ^ is a positive definite form, n 
k nor X (f,ijn+. — '7.fn+«) c*"' ^ zero; we must therefore have I zero 
80 the equation /(»)= has each of its roots o/tke/orm tk, where k w 
quarUity differerU from zero. 

We shall next shew that in the case in which the equation /(«) = 
j-tuple root «', each of the functions /(s)*^ is divisible hy (« — ay-'. 

For let Ci, Ct, ..., Cm he a set of definite real quantities; define q] 
ties 9i, 9„ ..., q„ by the equations 

''"-■'^"■•" '-'• -"• |(«.1,2 „)..., 

-sq, + H(qi,qt, ..., ff«.)n+.= 'V* J 
80 that we have 

'•-3,7W''" <^-'''' ' 

Let Bii be any root of the equation f(B) = 0, and let m be the sic 
positive integer for which all the functions 

(' '■•' /(,) 

are finite for the value ff,i of a. When a is taken sufficiently near a,i, w 
expand q^ in a series of the form 

(g^ + hj) (s - s,i)-" + (ff/ +A;i)(s- s.i)-"+' + ... . 
where g^, h^, g^, A/, .... denote real constants; and we can suppos 
quantities Ci. Cf, ..., Cm so chosen that the quantities^,, and k^ are not 
Substituting this value of q^ in equations (B), and equating the coefBi 
of (s — S|t)". we have 

BQh,fh. •■.,*«). + Si?n+- = 
H{K.K. ■■.,A»)n4--M. = o) 

and on equating the coefficients of (a — «ir)"~', we have 

„ , , , /\ I « fO when m > 1 



K«-l,2 ")( 



TTi ' r '\ . I 1 fOwhenm>l 



(«-l,2 n) 



84] Theory of Vibrations 197 

Now by Euler's theorem on homogeneous functions we have 
or by (C), 

n 

and similarly 

n 
I* 

from which it is evident that 2 (gah^^ — Kgn-^) is not zero. 
Moreover, the first two of equations (C) give 

2 hxH(gi,gi, ...,gin)K + Si 2 (A.A'n+a - A/An+a) = (E), 

and the last two of equations (C) give 

2s» n 

2 g/H(h^,hi, ....A^)a-«i 2 (5'./n+« - fl'.Vn+a) == (F). 



But firom the first two of equations (D), when m > 1, we have 

9n n n 

S hxH(gj\ gi, . . . , g^)^ - «i 2 (A.A'„+« - A.'A»+.) - 2 (goK^ - A.flrn+«) « 

(G), 

and from the last two of equations (D) we have 



2 gKff(fh\K\ .... Aw»')x+*i 2 igag^n+a" ga'gfH-) + 2 (jr. An+« — A.^,i^) = 

(H). 

Also since H is homogeneous of the second degree in its arguments, we 
have the identities 

2 h^'H(jgi,gif ...,S^an)A« 2 g\H(Wtfh\ -"»Kn')k (K) 

A=l A=l 

and 2 gKH{h^, Aj, .... A^)a= 2 KHiig^^gi, ....fi'tnOA (L). 

AbI A»1 



From equations (E), (H), (E) we have 

n n ^ 

2 (5^a A„+« — A.5rn+«) = «i 2 (A*A «+a - A.'A,i^) — tfi 2 (g^g'n+a-ga9n+a) 



n n n 

2 (5^a A„+« — A.5rn+«) = «i 2 (A*A «+a - A.'A,i^) — tfi 2 

asl aal asl 

and from equations (F), (G), (L) we have 



2 (S^aAn+a — AaJTn-K*) = - «l 2 (A.An+« — A«' An+«) + «i 2 (fl^afi^'ii+a — fl^/flTn-Ha). 
Lsl asl a-:l 

Comparing these equations, we have 



I* 

2 (5'«A„+«-A.5rii+«) = 0, 



198 Theory of Vibraiions [oh. vn 

which is contrary to what has already been proved. The assumption that 
m>l, which was used in obtaining equation (Q), must therefore be &lse; 
m must therefore be unity, and consequently when /(a) is divisible hy {s^ «!»)*» 
eocA of the fmctions f{s)KiL is divisible by (s — «ll)*~^ 

Now let «i, «8, . . ., «r be the moduli of the distinct roots of the equation/(«)=0, 
so that the functions f{s))^lf(s) are infinite only for « = ± s^i, ± «»*, ..., ± Sri ; 
then denoting the coefficient of (« — «pi)"^ in the Laurent expansion of 
f{^)i^lf{^) '^ powers of {s - «pi) by 

where (\, fi\ and (\, /i)p' are real, and observing that the only poles of the 
function /(9)a^//(5) are the points s^ ± Spi, and that these are simple poles, 
we have 

f(s) p«ii «-«pi *+v r 

and therefore <f> {t)xft, is the coefficient of 1/s in the Laurent expansion of 

^it-t^ £ [ (^> /^)p + ^' (X, m)/ ^ (X, m)p - 1 (X, A4)pn 

in powers of «. 

But the coefficient of 1/s in the Laurent expansion of 6'<*~*^/(« - *pi) is 
6'p<*"^*, and the coefficient of 1/s in the Laurent expansion of 6'<*"*^/(5 + tfpi) 
is r^p(*-*o)< ; we have therefore 

^(0Mi=2 2 {(\, /A)pCOS«p(^-^o)-(X, At)p'sin5p(^-^)}, 
P-i 

and so finally 

n r 

g^« 2 2 2 [j„+a {(a, M)pCos«p(^-^,)~(a, /a)p' sin «p (^ - ^)} 

-5«K^ + «»M)pCOS5p(e-<o)-(n + a,M)p's"i*p(^-^)}] (m=1,2, ..., 2n). 

This formula constitutes the general solution of the differential equations of 
motioTu Hence finally we see that when the total energy of a system vibraiing 
about a state of steady motion is a positive definite form, the vibratory miction 
can be expressed in terms of circular functions of t, and the steady motion is 
stable; the periods of the normai vibrations are 27r/«i, 27r/«a, ..., where ± isi, 
+ Wj, ... are the roots of the determinantal equation f(s) = 0, whose order in «* 
is equal to the number of non-ignorable coordinates of the system. 

The above investigation is valid whether the determinantal equation has 
repeated roots or not 

Between the coefficients (X, fi)p, (X, fi)p, there exist the relations 

(X, M)p--(fS X)p, (X, M)p''=(fb X)p', 
and 80 in particular 

(X, X)p-0. 



84, 85] 



Theory of Vibrations 



199 



These lelationB follow from equations which (in virtue of their definitions) are true for 
/(«X /(«)*„, namely 

/(*V-/(-V 

Bxample. If the number of degrees of freedom of the system, after ignoration of the 
ignorable coordinates, is even, shew that when the ignorable velocities are large (e.g. if 
the ignorable coordinates are the angles through which certain fly-wheels have rotated, 
this would imply that the fly-wheels are rotating very rapidly), half the periods of 
vibration are very long and the other half are very short, the one set being proportional 
to the ignorable velocities and the other set being inversely proportional to these velocities. 

86. Examples of vibrations about steady motion. 

A number of illustrative cases of vibration about a state of steady motion 
will now be considered. 

(i) A particle is descnbirig the circle r=ay z=bf in the cylindrical fidd of force 
in which the potential energy is V=(f>(r, z), where r*=s4;*+y*, it being given that dV/dz 
is zero when r^a^z^b. To find the conditions for stability of the motion. 

If we write 

x=rco8^, y=rsin^, 

we have for the kinetic and potential energies of the particle, whose^mass will be denoted 
by in, 

The integral corresponding to the ignorable -coordinate 6 is mr^air, where ir is a 
constant. The modified kinetic potential after ignoration of 6 is therefore 

R^T-'V-ld 

^im^+inu^-<t>{ryz)-^^. 

For the steady motion we must have 

the latter condition is satisfied by hypothesis, and the former gives Jl^=ma^d<l>/da. We 
have therefore 



R^imf*+inv?-^(r,z)-^^, 



Writing 



and Delecting terms above the second degree in p and (, we have 

As no terms linear m fi or C occur, this is essentially the same as a problem of 
vibrations about equilibrium, and the condition for stability is (§ 79) that 



200 Theory of Vibrations [oh. vn 

shall be a positive definite form, Le. that 

shall both be positive. These ore the required conditions for stability of the steady 
motion. 

CoroUary. If a particle of unit mass is describing a circular orbit of radius a in a 
plane about a centre of force at the centre of the circle, the potential enei^ being <ft (r) 
where r is the distance from the centre, the modified kinetic potential is 



i/»*-iP«(««+^«.), 



where r=a+py so the condition for stability is 

and the period of a vibration about the circular motion is 

(ii) To find the period of the vibrations about steady cvrcrdar motion of a particle 
moving under gravity on a surface of revolution whose axis is vertical. 

Let s=f{r) be the equation of the surface, where (e, r, 6) are cylindrical coordinates 
with the axis of the surface as axis of z. If the particle is projected along the horizontal 
tangent to the surface at any point with a suitable velocity, it will describe a horizontal 
circle on the surface with constant velocity. Let a be the radius of the circle ; we shall 
take the mass of the particle to be unity, as this involves no loss of generality. 

The kinetic potential is 

The integral corresponding to the ignoraUe coordinate $ is r^=ir, and the modified 
kinetic potential of the system after ignoration of ^ is therefore 

The problem is thus reduced to that of finding the vibrations about equilibrium of the 
system with one degree of freedom for which R is the kinetic potential The condition for 
equilibrium is 

and this gives 

^=i^ {1 +/"* (r)}-gf(r) - gcfif (a)/2r«. 

Writing r^a+p, where p is small, and expanding in powers of p, we have 

ii=i/>' {1 +/"(«)} - W {/" («)+^/' («)} • 

The equation of motion 

d fdR\ dR 
dt \dfi) " S^""" 



is therefore 



P {l+f'*(a))+ffp [f"(a)+lf' (a)}-0, 



86] 



Theory of Vibrations 



201 



and the condition for stability is 



/"(«)+i/'(«)>o, 



the period of a vibration being 



2n ( l+fHa) 1* 
s/9 t/"(«)+3/'(a)/aJ ' 



Examjle, If the surface is a paraboloid of revolution whose axis is vertical and 
vertex downwards, shew that the vibration-period is 



where I is the semi-latus rectum of the paraboloid. 



(iii) To determvM the vibrationa about steady motion of a top on a perfectly rough 
plane. 

Let A denote the moment of inertia of the top about a line through its apex perpen- 
dicular to its axis of symmetry, and let $ denote the angle made by the axis with the 
vertical, M the mass of the top, and h the distance of its centre of gravity from its apex : 
then we have seen (§ 71) that after ignoring the Eulerian angles and ^, the angle $ is 
determined by solving the dynamical system defined by the kinetic potential 



R=iA6^ - ^^. . o/ - Mgh cos 6, 



2A sin> B 
where a and b are constants dei)ending on the initial circumstances of the motion. 

Let a, n, be the values of $ and ^ respectively in the steady motion, so (§ 72) 

we have 

An^ cos a-^Mgh^bUj 

iln sin^ a=a— 6cos a. 

To discuss the vibratory motion of the top about this form of steady motion, we write 
6»a+s where ^ is a small quantity, and expand R in ascending powers of or, neglecting 
powers of x above the second and eliminating a and b by use of the last two equations ; 
we thus obtain for R the value • 

R = ^Ad^ - ^Aa^ {n« sin« a -H (n cos a - Mgh/A n)«}. 

The equation of motion for x is therefore 

X + {«* sin* a -h {n cos a - Mgh/An)*} x^O. 

As the coefficient of j; is positive, the state of steady motion is stable ; and the period of 
a vibration is 

2fr {»« - ^Mgh cos a/A -H IPg^h^lAH^) " K 

(iv) The sleeping top. 

If we consider that form of steady motion of the top in which a is zero, so that 
the axis of the top is permanently directed vertically upwards, the top rotating about this 
axis with a given angular velocity, the method of the preceding example must be modified, 
since now the form of steady motion in which a is a small constant is to be r^;arded as a 
vibration about the type of motion in which a is zero : so that we may now expect to have 
two independent periods of normal vibration, the analogues of which in the previous 
example are the period of the steady motion and the period of vibration about it. 



202 Theory of Vibratiom [oh. vn 

As in § 71, the kinetic and potential energies of the top are 

r=iJ^«+iu44>«8in«^+iC(4r+^coa^)», 

The integral corresponding to. the ignorable coordinate ^ is 

6»=C(^+^co8^), 

and hence after ignoration of ^ we obtain for the kinetic potential of the system the value 

R^\A6^+\Ai^ mi^ e-^h^cos A- Mgh co^e. 

In the two last terms we can replace cos 6 by (cos ^ — 1), since the terms — h^ and Mgh 
thus added disappear from the equations of motion. 

As ^ is not a small quantity throughout the motion, we take as coordinates in place of 
6 and ^ the quantities ( and 17, where 

(ssin^cos^, i^aasin^sin^. 

From these equations, neglecting terms above the second degree in {, 17, (, i}, we have 

^«+<^8in«^=f«+i78, 
^sin«^=^-i,f, 

l-cos^=i ««+,;»), 
and so we have 

The equations of motion are 

(A(+bij-Mgk(=0, 
or -{ . 

[Afj'^bS'-Mghrf^O. 

If 2ir/X is the period of a normal vibration, on substituting (ssJe^, f/^Ke^ in these 
differential equations- and eliminating J and K we obtain the equation 

-XU-'Mgk tb\ -0, 

-ih\ "X^A-Mgh 
or {\*A + %A)2 - 6 V = 0. 

The two roots of this quadratic in X* give the -values of X corresponding Jbo the two 
normal vibrations : we have therefore to determine the natiu^ of these roots. 

The solution of the quadratic is 

BO ±\^^{h±(h^''4AMgh)^}. 

The values of X are therefore real or not according as ft' is greater or less than AAMgK 
In the former case the steady spinning motion round the vertical is stable : in the latteif 
case, unstable. 

It must not be supposed, however, that in the imstable case the axis of the top 
neoessaiply departs very far from the vertical : all that is meant by the term *^ unstable " 
is that when b^<AAIfgh the distiurbed motion does not, as the disturbance is indefinitely 
diminished, tend U> a limiting form coincident with the undisturbed motion. 



86, 86] Thewy of Vibrations 203 

As a matter of fact, if h^-AAMgh, though negative, is very small, it is possible for the 
axis of the top in its " unstable " motion to remain permanently close to the vertical : but 
in this case the maximum divergence from the vertical cannot be made indefinitely small 
(for a given value of b) by making the initial disturbance indefinitely small*. 

86. VibraMons of systems involving moving constraints. 

If a dynamical system involves a constraint which varies with the time 
(e.g. if one of the particles of the system is moveable on a smooth wire or 
surface which is made to rotate uniformly about a given axis), the kinetic 
potential of the sjrstem is no longer necessarily composed of terms of degrees 
2 and in the velocities ; terms which are linear in the velocities may also 
occur. The equations which determine the vibrations of such a system will 
therefore in general include gjnroscopic terms, even when the vibration is 
about relative equilibrium : the solution can be effected by the methods above 
developed for the problem of vibrations about steady motion. The following 
example will illustrate this. 

Example, To find the periods of the normal vibrations of a heavy particle about its 
poeition of equilibrium at the lowest point of a swface which is rotating with constant 
angular velocity a> about a vertical axis through the point. 

Let {Xj y, z) be the coordinates of the particle, referred to axes which revolve with the 
surface, the axes of x and y being the tangents to the lines of curvature at the lowest 
point, and the axis of z being vertical Let the equation of the surface be 

z^- — }-§- +terms of higher order. 

The kinetic and potential energies of the particle are 

V=mgz. 
The kinetic potential of the vibration-problem is therefore 

The equations of motion are 

dt\dxj dx"^' *W/ ¥" 
X - 2«y +x (S. - S\ - 0, 

If the period of a normal vibration is 2ir/X, we have (substituting x^Ae*^, y=Be*^ m 
the differential equations, and eliminating A and B) 

-X8-©«+^/pi -2«iX «0, 

2«tX - X* — «* -k-glpi 

or (X»+«*-5^/pi) (X«+«2 -g/p^) - 4XV-0. 

The roots of this quadratic in X* determine the periods of the normal vibrations. 

* A discnsflioii of the stability of the sleeping top is given by Klein, BiM. Amer. Math. 8oc. in. 
(1897), pp. 129—132, 292, 



or 



'{-^i^°«{-o 



Theory of V^ationa [oa vn 



Miscellaneous Examples. 

iole moves on a curve which rotat^e unitbrmly about a fixed axis, the 
gy F(«) of the particls depeuding only on its position as defined bj the 
that the period of a vibration about a poeition of relative rest on the 

distance of the particle from the axis. 

line the vibrations of a solid horizontal circular cjlinder rolling inside a 
ital circular cjlinder whose axis is fixed, shewing that the length of the 
ent pendulum is (6-a)(3Jf+m)/{2Jf+m); where 6 is the rodius and Jf the 
iter cyhnder, and a is the radius and m the mass, of the inner cjlinder. 

(Coa Exam.) 
I hemif^herioal bowl of mass M and radius a is on a perfbctlj rough 
)e, and a particle of mass m is in contact with the inner surfiice of the bowl, 
ith. Shew that when the system performs small oscillatians, the motion of 
d the centre of gravity cf the bowl being in one plane, the periods of tha 
one are 2ir/VAt and Sir/VX], where X, and X, are the roots of the equation 

B«iV-(3-aX)(4jf-|<iX)if=0. {ColL Eiam.) 

g of length 4a is loaded at equal intervals with three weights m, JVand m 
id is suspended from two points A and B sjmmetricallj. Shew that if Jt 
vertical vibrKtions, the length of the simple equivalent pendulum is 

acosaco9<3ain(a-tf)coe(n-g} 



are the inclinationa of the parts of the string to the vertical. 

(ColL Exam.) 
Tm bar whose length is 2a is suspended by a abort string whose length is I ; 
i time of vibration is greater than if the bar were swinging about one 
le ratio 1 +9f/32a : 1 nearlj. (Coll. Exam.) 

jtic cjlinder with plaue ends at right angles to its axis rests upon two fixed 
dicular plauee which are each inclined at 4A° to the horizon. Shew that 
stable configurations and one unstable, and that in the former case the 
^uivalent pendulum is 

be lengths of the semi.aieB. (ColL Exam.) 

!i circular cylinder of radius a and mass m is loaded so that its centre of 
I distance A from the axis, snd is placed on a board of equal mass which 

smooth horisontal plane. If the nyatem is disturbed eligbtly when in & 
>le equihbriunt, shew that the length of the simple equivalent pendulum is 
A, where mi^ is the moment of inertia of the cylinder about a horizontal 
i centre of gravity. (ColL Exam.) 

1 of a uniform rod of length b and mass m is freely jointed to a point in a 

wall ; the other end is freely jointed to a point in the sur&ce of a uniform 



CH. vn] Theory of Vibraiions 206 

sphere of mass M and radius a which rests against the wall. Shew that the period of the 
vibrations about the position of equilibrium is 2ir/f, where 

/>*{8in^sin'(a-^)+ico8asin(Q-^)+|8in/3cos»^}— -j^ — (asinacos"Q+6sin^0O8»^), 

a and /3 being given by the equations 

a8ina+68in0 — a—0, 
(J«i+J0tan/3-J/tana=0. (ColL Exam.) 

9. A thin circular cylinder of mass M and radius 6 rests on a perfectly rough 
horizontal plane, and inside it is placed a perfectly rough sphere of mass m and radius a. 
If the system be disturbed in a plane perpendicular to the generators of the cylinder, find 
the equations of finite motion, and deduce two first integrals of them ; and if the motion 
be small, shew that the length of the simple equivalent pendulum is 

14ir(6-a)/(10ir+7m). 

(Camb. Math. Tripos, Part I, 1899.) 

10. A sphere of radius c is placed upon a horizontal perfectly rough wire in the 
form of an ellipse of axes 2a, 26. Prove that the time of a vibration under gravity about 
the position of stable equilibrium is that of a simple pendulum of length I given by 
l^dl = (a« - 6») (rf« + it«), where i^ « 2c«/6 and rf« = c» - ft^. (Coll. Exam.) 

11. A rhombus of four equal uniform rods of length a freely jointed together is laid 
on a smooth horizontal plane with one angle equal to 2a. The opposite comers are 
connected by similar elastic strings of natural lengths 2a cos o, 2a sin a. Prove that if 
one string be slightly extended and the rhombus left free, the periods during which 
the strings are extended in the subsequent motion are in the ratio 

(cos a)* : (sin a)*. (Coll. Exam.) 

12. A particle of mass m is attached by n equal elastic strings of natural length a to 
the fixed angular points of a regular polygon of n sides, the radius of whose circumscribing 
circle is c Shew that if the particle be slightly displaced from its equilibrium position in 
the plane of the polygon, it will execute harmonic vibrations in a straight hne, the length 
of the simple equivalent pendulum being 2mgac/iik(2c—a)f and that for vibrations 
perpendicular to the plane of the polygon, the corresponding length will be mgdcjnK (c-a), 
X being the modulus of each string. (Camb. Math. Tripos, Part I, 1900.) 

13. The energy-equation of a particle is 

/(^) i*— 2<^ (j?) + constant, 

and a is a value of x for which 0' {x) is zero. If <^W {x) is the first derivate of <^ {x) 
which does not vanish for 07=^0, shew that the period of a vibration about the position a is 



4 r(l/2p) i r(2;>)/(a)trU 
P-ir(l/2p+i) I 4p0(2p)(a) J ' 



AP-ir(i/2p 

where h is the value of {x-a) corresponding to the extreme displacement (Elliott) 

14. A cone has its centre of gravity at a distance c from its axis, there being in other 

respects the usual kinetic symmetry at the vertex. If the cone oscillates on a horizontal 

plane and the plane be perfectly rough, shew that the length of the simple equivalent 

pendulum is 

(cos a/J/c) {A sin* a + (7 cos* a), 

whereas if this plane be perfectly smooth, the length is 

(cos ajMc) (sin* aJA + cos* ajC). (Coll. Exam.) 



V 



Theory of VibrcUiona [ch. vn 

of equal uniform rods each of length 2a are freely joiDted at a common 
iged at equal anguLu- intervals like the riba of an umbrella. This cone 
a smooth fixed sphere of radius b, each rod being in contact with the 
I equilibrium. Shew that, if the sjatem be slightly disturbed so that 
vertical vibratioDS about the position of equilibrium, their period is 

alb. (Camb. Math. Tripoa, Part I, 1896.) 

octangular hoard is symmetrically suspended in a horizontal position 
,c strings attached to the comers of the board and to a fixed point 
centre. Shew that the period of the vertical vibrations is 



'(f-^)"'- 



ibrium distance of the board below the fixed point, a ie the length of 
<«»+«•)*, and X is the modulus. (Coll. Exam.) 

mina bangs in equilibrium in a horizontal position suspended by three 
e strings of unequal lengths. Shew that the normal vibrations ore 
it either of two vertical lines in a plane through the centroid, and 
ing parallel to this pluie. (Coll. Exam.) 

rod of length 2a is freely hinged at one end, at the other end a string 
led which is &stened at its further end to a point on the surfiMn of a 
9 of radius c. If the masses of the rod and sphere are equal, find the 
tern when slightly disturbed from the vertical, and shew that the 
iue the periods is 

(ColL Exam.) 
wire in the shape of an ellipse of aemi-aiea a, 6, rests upon a rough 
li its minor axis vertical and a particle of equal mass is suspended by 
^ I attached to the bigbeet point. If vibrations in a vertical plane 
I that their periods will be those of pendulums whose lengths are the 
f the equation 

{«(36-2a*/i;)+66"+i*}(«-0+'*6*'=Oi 
ius of gyration about the centre of gravity. (CoU. Exam.) 

[tensibte string has its ends tied to two fixed pegs in a horisont&I 
i apart is three-quarters of the length of the string. The string 
two small smooth rings which are fixed to the ends of a uniform 
length is half that of the string. The rod hangs in equilibrium 
ition and receives a small disturbance in the vertical plane of the 
initially its normal coordinates in terms of the time are L cos (pi -i- a) 
where ^ and — q* are the roots of the equation 

a^_^ |^_| g„o. (ColL Exam.) 

niform rod of length 2a, suspended from a fixed point by a string 
Xy disturbed &om its vertical position. Shew that the periods of the 
re 2ir/j>, and 2ir/p„ where p^ and p^ are the roots of the equation 
a*p'-<4a+36)sp'+3ff'-0. 



CH. vn] 



Theory of Vibrations 



207 



22. A circular disc, mass My is attached by a string from its centre C to a fixed 
point 0. A particle of mass m is fixed to the disc at a point P on the rim. Find the 
eqiiations of motion in a vertical plane in terms of the angles B and ^ which OC and CP 
make with the vertical, and prove that if the system vibrates about the position of 
equilibriun^ the periods in these coordinates are given by the equation 

(JZ-h w) (jp^a " g) {(ir+ 2«*) of - 2mg} - 2m^oap\ 

where a is the length of the string OC and c the radius of the disc. (ColL Exam.) 

23. A hemispherical bowl of radius 26 rests on a smooth table with the plane of its 
rim horizontal ; within it and in equilibrium lies a perfectly rough sphere of radius 6, and 
mass one-quarter of that of the bowL A slight displacement in a vertical plane con- 
taining the centres of the sphere and the bowl is given ; prove that the periods of the 
consequent vibrations are 2ir/pi *"^d 2ir//?,, where p^ and p^ are the roots of the 
equation 

l&66»^-260&r^+75^«=0. (Coll. Exam.) 

24. A uniform circular disc of mass m and radius a is held in equilibrium on a 
smooth horizontal plane by three equal elastic strings of modulus X, natural length Iq and 
stretched length l. The strings are attached to the disc at the extremities of three radii 
equally inclined to one another and their other ends are attached to points of the plane 
lying on the radii produced. Shew that the periods of vibration of the disc are 

2ir {ilI{^ - g}* and 2h- Oia/4 (a -h (/ - 1^}\ 
where n=2mll^lZK. (Camb. Math. Tripos, Part I, 1898.) 

25. A particle is describing a circle under the influence of a force to the centre 
varying as the nth power of the distance. Shew that this state of motion is unstable if n 
be less than —3. 

Shew that, if the force vary as «~«/r", the motion is stable or imstable according as 
the radius of the circle is less or greater than a. (Coll. ExanL) 

26. A particle moves in free space' under the action of a centre of force which varies 
as the inverse square of the distance rjid a field of constant force : shew that a circle 
described uniformly is a possible s+dte of steady motion, but this will be stable only 
provided the circle as view^j^r«-.m the centre of force appears to lie on a right circulcu: 
cone whose semi-vertical an^^is greater than cos-* J. (Coll. Exam.) 

27. A particle describes ai circle uniformly under the influence of two centres of force 
which attract inversely as the) square of the distance. Prove that the motion is stable if 
3 cos ^ cos ^< 1, where 6 an A ^ are the angles which a radius of the circle subtends 
at the centres of force. I (Camb. Math. Tripos, Part I, 1889.) 

2S. A heavy particle is pnwected horizontally on the interior of a smooth cone with 
its axis vertical and apex dowiy wards ; the initial distance from the apex is c and the 
semi- vertical angle of the cone isl «• Find the condition that a horizontal circle should be 
described; and shew that the \time of a vibration about this steady motion is that 
of a simple pendulum of length \ic aec a. (Coll. Exam.) 

29. A circular disc has a th V^ rod pushed through its centre perpendicular to its 
plane, the length of the rod being i \qual to the radius of the disc ; prove that the system 
cannot spin with the rod vertical \ unless the velocity of a point on the circumference 
of the disc is greater than the \ Velocity acquired by a body after falling from rest 
vertically through ten times the ra ^^ ^^ ^® ^^ (^^ Exam.) 



*! 



\ 



r 



208 Theory of VibrcUions 

30. Prove that for a aymmetricol top spinning upright with e 
velocitj for stability, the two types of motion, differing slightly from t 
in the upright positinD, which are determined by simple harmonic ftmc 
are the limits of steady motions with the axis slightly inchned to the 
the period of the vibrations is the limitiog value of that which com 
motion in an inclined position when the inclination is indefinitely dimini 



31. One end of a uniform rod of length 2a whose radios of gy 
end is i is compelled to describe a horizontal circle of radius c with 
velocity a. Prove that when the motion is eteatly the rod lies in t 
through the centre of the circle and makes an angle a with the vertical g 

•'(i' + accoseca) = aj?seca. 

Shew that the periods of the normal vibrations are 2ir/X|, 2fr/Xj, wh 
roots of 

(it»X' sin o - «'«) (HV sin a - c^c - n't* sin' a) = 46.«**X> sin' a . 
(Camb. Math. Tripos, 

32. Investigate the motion of a conical pendulum when disturbed 
steady motion by a small vertical harmonic oscillation of the point of si 
steady motion be rendered unstable by such a disturbance 1 

33. The middle point of one side of a uniform rectangle is fixed oni 
it to the middle point of the opposite side is constrained to describe 
of semi-angle a with uniform angular velocity. The rectangle bein) 
find the positions of steady motion and prove that the time of a vib 
position of stable steady motion is equal to the period of revolution dividi 

34. A solid of revolution, symmetrical about a plane through its i 
perpendicular to its axis, is suspended from a fixed point by a string of 
attached to one end of the axis of the solid, this axis being of lengtl 
of the solid is }/, and its principal moments of inertia at its centr 
(J, A, C). If the solid is slightly disturbed from the st^te o' steady mo 
string and axis are vortical, and the body is spinning on i .axis with a 
shew that the periods of the normal vibrations are Sir/jjj ^d iirjp^, whc 
the roots of the equation 1 

35. A symmetrical top spins with- its axis vertical, the tip of tl 
a fixed socket. A second top, also spinning, is placed on Jthe summit ol 
of the peg resting in a small socket. ' Shew that the orr^igement is st 

has all its roots real ; Q, O' being the spins of the unfiper and lower 
M, M' their massee, G, C their moments of inertia abd^ut the axis of t 
perpendiculars through the pegs, c, if the distances of [the centroids iro 
the distance between the pegs. (Car:^b. Math. Tripos 

36. A homogeneous body spins on a smooth hori^lontal plane in eta 
with angular velocity a about the vertical through tt^ le point of contact 
gravity. The body is symmetrical about each of t<9jro perpeodicuiar p 



\^ ■ ■ 

OH. vn] Theory of Vibrations 

vertioaL Tb< priDcipal radii of curratuTe at the vertex on which it raa 
moments of iiertia about the priacipal axes through the centre of gravit 
lioea of curnture) are respectivel; A and B, and that about the ve 
height of th( centre of gravitj above the vertex is a— Oj+pi'^acfp) 
waight of thi bodj. 

Shew that the following conditions muat be satisfied: 

(i) l!iat+A-C)(\a,+S-C)>0, 

(ii) \-a,A+a^B) <AB+{A-C) {B~ C), 

(til) Tb value of X must not lie between the two values 

if the two radiala in the expression are both reaL 

(Camb. Math. Tripos, 



CHAPTER VIII. 

NON-HOLONOMIC SYSTEMS. DI8S1PATIYE 8VSTB: 

87. Lagrangtfa equations mtk undetermined multtpli'irs. 

We Qow proceed to the consideratioQ of non-holonom j'c < I . i 
Id these systems, as was seen in § 25, the number of inrlon. , -, 
(?i, 3a, ••■> 3b) required in order to specify the configuratitnj -n- 1 
any time is greater than the number of degrees of freedom u 
owing to the fact that the system is subject to constraints ^ 
supposed to do no work, and which are expressed by a nui 
iutegrable* kinematicai relations of the form 

A^dqi + Aadq,+ ... + A^dq„ + Ttdt = (k = 
where Aa, Aa, •■■, Anm. ?it ^1. •■•. ^m, are given functions of ^ 
The most fomiliar example of such a system is that of a b 
constrained to roll without sliding on a given fixed surface : 
that no sliding takes place is expressed by two relations of it 
above. 

The number of kinematicai relations being m, the systi 
(n — m) degrees of freedom ; it is not possible to apply Lagranj 
directly to such a system, but an extension of the Lagrangian i 
now be given which will enable ub to. discuss the motion of i 
systems in a way aDalogous to that previously developed I 
systems. 

Consider then a non-bolonomic system, whose configuration i 

ia completely specified by n coordinates j,, 5, q„; let the i 

be T, and let the kinematicai conditions due to the non-holonon 
be expressed by the relations 

Attdqt + A^dqt + ...+ A^dgn +Tiidt = (fc = 1 

Now it is open to us either simply to regard the system 
these kinematicai conditions, or in place of these to regard f 
acted on by certain additional external forces, namely the foro 
to be exerted by the constraints in order to compel the system 

* If these relatioQB were Iutegrable, it would ba poBsible to eipiesB eome o 

(7i, g 9„) in tenm of the otben, and the n coordinates would therefore doi 

whiob ia oontrar; to oat asBomption. 



nie Systems. Disnpatim Sysiema 211 

ve shall for the present take the latter point of 

Bystem by these additional forces in an arhitrary 
, Sjn) (which is now not restricted to satisfy the 
nd let 

system by the original external forces in this dia- 
bstitution of additional forces for the kinematical 
lystem holonomic, we can apply the Lagrangian 
fore 

^,(|)-|=«-«'' <'— "> 

I of the system. 

Q„' are unknown : but they are ^uch that, in any 
ith the instantaneouB constraints, they do no work. 

•y 

the ratios dj, : t^i : ... : dq^ which satisfy the 
jii + A^dqt + . .. + 4«*d?« = ; 

X,^„ + X,^ri+ .-. +X«^™ {r = 1, 2, ... , n), 

X), ..., \m, ftre independent of r. We thus have 
[uations 

+ \,An + KA„+...+-K„Ar„ (r = l,2,...,n), 
4it9i+--.+^ni^n + 2*1 = (* = 1, 2, ...,m), 

to determine the (n + m) unknown quantities 
B,. The problem is thus reduced to the solution 

on referred to axes riwving in any manner. 
the preceding article depends essentially on the 
lomic system to a holonomic system by iabroducing 
i-bolonomic constraints. In practice, this is often 
by forming separately the equations of motion of 
! system. It is moreover frequently advantageous 
U~2 



212 Non-hcH&twmic Sy sterns. Dimpative j 

to use axes of reference which are not fixed either i 
and we shall now find the equations of motion of/ 
axes which have their origin at the centre of gv' 
turning about it in any manner*. < 

Let 6 be the centre of gravity of the body, and i 
axes. Let (u, v, w) be the components of velocity of e. 
resolved parallel to these axes, and let (^i, 0„ 0,) be . 
angular velocity of the system of axes Gan/z resolved alonj 
selves ; further let (cdi, m,, ck.) be tbe components of angul 
body, resolved along the same axes. Then (§64) the motion 
as that of a particle of mass M, equal to that of the b 
forces equal to the external forces which act on the bo 
forces of constraint, except the molecular reactions betwec 
particles of the body) ; let {X, Y, Z) he the components pi 
Qxjfz of these external forces. 

Tbe component of velocity of parallel to 02: is u, 
(§ 17) the component of its acceleratiou in this direction ia 
have therefore the equation 

Jf(ii-ftf. + w5,)-X, 
which can be written 

* dv '&» 

where T denotes the kinetic energy of the body, expr* 
(u, V, w, b),, or,, b),) ; and similar equations can be obtained 
parallel to the axes Oy and Oz. 

Consider next the motion of the body relative to 0, wh 
pendent of the motion of 6 ; from §§ 62, 63, we see that the ai 
of the body about the axis Ox is dT/dwi, so that the n 
angular momentum about an axis fixed in space and inst 
ciding with Ox is 



d/dT\ 
dt\5u)~ 



d/dT\_ 
dt \9(i»i/ 






If L, M, N denote the moments of the external force 
Oxyz, we have therefore (§ 40) 

d /dT\ ^ dT . dT 



d/dT\ 

dt\dw,}~ 



'9w, '3(w» ' 



and two similar equations. 

* In the applications of this method, the axes are asaall; ohoaea sal 
that the momeDta and pioduots of inertia of the body with respect to them 
condition ia not essential. 



^on-Aolonomic Systems. JHssipaUve Systems 

ally the motion of the body is determined by the six equatio 






f- 


iT 

as. 


i- 


dT 


1.^- 


ST 



d/dT\ 
dt [dmj ~ 

dfdT\_ 

dt \dtitf/ 



'3w, 



ato. 



dldT\_g^dT^gdT_j. 
dt \d(Ui/ 5q), d(U| 



observed that tbese are really Lagraogian e<]uationB of 
quasi-coordinates, and could have been derived by use 



If the origin of the moving aiee is not fixed in the bodj, let («, 
Deuts uf vetocit; of the origin of coordinatea, reeolvcd parallel 
position of the axes; let (d,, ^, d,) be the oomponenta of angular 
of aiea, resolved along tfaemselveB i let (v,, vg, v^) be the compo 
it point of the body which is instantaneooBlj situated at the < 
id let (u„ a>j, 0,) be the components of angular velocity of the b 
B moving axes. Shew that the equations of motion can be wi 



/ 



. "• . . >T_ 



aWJ— ■5F,+"■^^,- 



-+». 









-'.Z-'. 






, L, M, N) are the components and 
I moving axes. 



of the external ton 



iKation to special non-kolonomic problems. 
now consider some examples illustrative of the theory < 



Sph«re rolling on a fixed tpkere. 
quired to determine the motion of a perfectly rough sphere of radii 
■oils on a fixed sphere of radius b, the only external force being groi 
I be the polar coordinates of the point of contact, referred to the ( 
), the polar axis bung vertical We take moving axes OABC, wh 



214 Non-holonomic Systems. Dimpative Syatei 

the centre of the moving sphere, CfC ie the prolongation of the line joinii 
the apherea, OA is horizontal and perpendicular to OC, and OB is ] 
OA and OC, in the direction of $ increasing. 

With theee oxee we have, in the notation of the last article, 

6,= -i, tf,= -^8in 6, 4}-^ooa 

u--(a+b)^mn3, v~{a+b)6, m=0, 

, 2«V. 



r-im {«»+«"+«>■+ ^* (»,'+•,»+-,')} . 



and if P, F' denote the components of the force at the point of con 
OA and OB reepectirelj, we have 

X—F, T-mgma6+F', 

L=F'a, il—F^ Jf=0. 

The equations of motion of the last article become therefore 

m(i-pflj-f --|am(i,-d,«, + fl^, 
m(<J+irf»)-mffainfl=/"- |am(i,-fl,»,+tf^,), 

•1, — fl,r»,+ fli»j=0. 

Uoreover, the components parallel to the axes OA, OB of the velocit 
contact ore it—cut^ and v+aai, and consequeatly t he kinematical equatioi 
the condition o f no sliding at the point of contact are 

u-majoO, P+oui — O. 
Eliminating F, F', oi, v,, we have 

!"-''S,-?afli«j=0, 
w+w^-^a^jBi-fffsinS-O, 
i.,-0. 

The last aquation gives <&,=», where n is a constant ; wbik eubstituUr 
in the first two equations their values in terms of 0, 4, ^, we have 

1 Ca+6)^(^ainfl)+(a+6)^ooafl-?oB^- 

!(«+&) iJ-(a+6}^coe (9 8intf+|an^sin*-f^sin#= 

The former of these equations con be integrated at once after multip 
b; sintf, and gives 

(o + 6)^ain*« + fancoBfl = *, 

where i ie tt constant. Moreover, multipljdng the secocd equation throi 
the first equation bj ^sind, and adding, we obtain an equation which 
integrated, giving 

where A ie a constant ; this is really the equation of energy of the system. 



>mic Systems. Disnpatim Systems 216 

lieae two integr&I equations, we have 

i-|aiicoa^'-Vff(a+6)aiiiSflcoBfl + A(o + 6)>8m»tf; 
equation becomea 

i+6)»(l-*»)-{*-|anj;)'-V<?(o+6):r(l-^). 

X on the rigbt-band side of this equation is positive when 
I, positive for some real valuea of g, i.e. for Bome tsIum of * 
;ativ6 when x~~l; it has therefore one root greater than 

1 and - 1 ; we shall denote these roots by 
cosh 7^ coaj9, cobo, 
^en have 



..) - yifr^w-r „l (^ _ coe ^) (^ _ 0^ „jj-l ^_ 





■.-j"{4C.-«,)(«-<,)(«-*^}-*&, 
d with the roots 

301? (a+6) r 
7A(a+t)'+»a'n' l 

all real, and satisfy the relations 

lues of ( and (since £ is real) lies between cosa and ooe^; 
e, and e,; hence the imaginary part of the constant! in the 
I the half-period correeponding to the root e^, which we ehall 
' 4 may be token to be zero by suitably choosing the origin 

7A(g + 6)'+faV 

kriable 6 in terms of the time: the other coordinate <t> of the 
I then obtained by integrating the equation 



*-feS 



(a + 6)8in*0' 

id by a procedure similar to that uaed-(§ 78) to obtain the 
he position of a top epiunii^ on a perfectly rough plane. 



lie Systems. Dissipatim S 

ere, OC is the prolongation of tbe line 
a1 and perpeodicvilar to OC, and 
y{ 6 iocreasing. 

in the aotatioo of tbe last article, 
nponents of the force at the point > 



-r-^. ^^—±,.^g^ g^^^ 




nts of the force acting on the sphere « 
ictiTely, we have 

^, r^mgamO+F; 

F'a, M=-Fa, ^■=o, 

acome 

-*,«i+tfi«>,=0 

1 of the sphere Jf, we take moving ai 
a at J let (Q,, o,, a^ denote the . 
d along these uee. Then for the apht 
r-i*".fi'(0.*+Q,'+0,'), 



■i6JrC(l,-tf,0,+(j^)_/- 

V>M(iij-efi^+afi^=p-' 

Ai-fl,Q,+tf,Q, =0 

ig at the point of contact are 
w-a»,«6o„ o+ao.,--6Q, 

of equations we multiply equations (; 

sing (7), we have 

<»«) + Wij + tt^i + wtf, = 0, 

a«,+60»=an, whe 



ylonomie Syatenus. Disnpative Systems 216 

reen theae two integral equations, wo have 

i» Ci-|a»c«tf)>-Vff(o+6)Mn'«co8tf+A(o+6)»Bin»tfj 

z, thie equation becomes 

lial ID a^ on the right-hand aide of this equation is positire when 
n «-.!, positive for some real valaes of d, i.e. for some values of * 
id n<«ative when «- - 1 ; it has therefore one root greater than 
itween 1 and - 1 ; wo ahall denote theae roots by 

cosh y, cos j9, cos a, 
id wc then have 

5^gi^C»C"^a/^t«l)(*-««iS)(*-oos„)t-t<£^, 
6(if+.>.)yBip^ _ 

(A) and (B), from which and ^ are to be 4.g]i^.i^i„i . t, 

same character as the equations found for theT^^ • . of B 

a example : the former equations being in fact derivai,. .om the 

ing Jf very largo compared with m. The integration therefore 
1 the former case. 

niform sphere rolls on a perfectly rough horitoutal plane, under 
it passes through its centre. Shew that the motion of its centre 
a particle acted on by the same forces reduced in the ratio : 7. 

the equations of motion of a perfectly rough sphere rolling under 
right circular cylinder, the axis of which is inclinod to the vertical at 
r that, if the sphere be such that lfl-^\a\ a being its radius and k 

about any diameter, and if it he placed at rest with the axial plane 
laking an angle & with the vertical axial plane^ the velocity of 

the axis, when this angle ia 6, ia 

I* {ain ifl cosh - » (coa i# sec i5) + COB JiJ cos - ' (Bin Jfl cosec JS)}, 
liuB of the cylinder. (Camb. Math. Tripos, Part I, 1895.) 

1 of non-holonomic systems. 

consider the small vibratory motions of a non-holonomic 
»ear that bo far as vibrations about equilibrium are coo- 
ance between holonomic and non-holoDomic systems is 



e vibrations about equilibrium of a non-holonomic system 
It coordinates and (n— m) degrees of freedom, in which 
independent of the time. Let T be the kinetic and V the 
J that for the vibrational problem T will be supposed to be 
uiratic function of (f,,^i, ...,^„), and F to be a homogeneous 



218 Non-holonomic Systems. DissipeUive Systems 

qaadrstic function of {q,, 3,, .... q^), the coefficienba in both a 
coDstaats. There are m equations of the type 

A^q^+ AAqi+ ... + A^q^^O (i=l, 2, 
which express the non-holonomic constraints ; and the equations 
are (§ 87) 

From these equations it is evident that X,, X,, ..., X^ are in gei 
quantities of the order of the coordinates ; and therefore for the 1 
problem only the constant parts oi Au.Au, ■•■, ^nm need be conside 
vibrational motion is therefore the same as if the coefficients A^, A 
were constants independent of the coordinates; but in this case the 
Ajiq, + Attqt+ — + -4^9„=: (ft= 1, 2, 
can be integrated ; in fact, they give 

the constants of integration being zero since the values 

?,-0, 9, = 0, .... 9„=0, 
represent a possible position of the system. 

It follows that the vibratory motion of the given non-bolonomic 
the same as that of the hotonomic system for which the equatioi 
straint are expressible in the integrated form 

A,tq, + AAqj+... + j4^2„ = (A- 1, 2, 
we can therefore determine the vibrations by using these equations 
Date m of the coordinates (q„ q,, ..., }„) from 2" and V; we shall 
a holonomic system with (n - m) degrees of freedom, the kinetic anc 
energies being expressed in terms of (n — m) coordinates and 1 
sponding velocities : the vibrations of this system can be determin 
usual method described in the last chapter. 

As an ez&mple, we shall consider the following problem*. 

A heavy homogeneoui hemitpkert li rating in eqidltbrium on, a perfectly rcmg 
plane mth iii apherical mrfcice doienaardt., A teetmd heavy homogenanu hi 
rating in the tame way on (As perfectly rough plane face of thefirit, the poin 
being in the centre of tAe fiiee. The equHibrinm being slightly ditturbed, it 
to find the vibrationt of the syttem. 

T&ke aa axes of reference 
(1) A rectangular set of aiea Z^xyt fixed in the upper hemisphere, the 
its centre of gravity Z,. 

* Due to M«4ame Kerkhoveu-WythoS, ^timw ArchUfvoor WitktatAe, Deel it. 



Systems. JHssiptUi 

e Z| iqC, filed in the lower hi 

LOS Slmn fixed in spaca, the 
itact of the lower hemiaphere 

bj aupposing that id the «qi 
id therefore coiocident, wbi 
ad Rm being therefore also 
loordinates of a point refer 
latioDB 

' -'Y+Yx'+rty+y^ 

i-c+Ci£+<VJ+<^f- 
naformation-formulae oomple 
werer the system has only si 
hese coefficieote or their difii 

h<i,»=.l, a,S, + a^+<i^,=0 
hOj'-I, ai6,+aA+a^, = C 
uiter of the axes ; the remaii: 
[ DOW find. 

B lower and upper hemisphe 
avity from their plane faces, 
■/ of the upper hemisphere wii 

1 be at rest relative to the lo* 

+ d,J?,+^,+a^><0, 

i+ia,*,+3.y,+ft*,=o, 
;+^,^+y^,+^-o. 

vcB y+;,y,BO, which is the 
condition of coDtact of thf 

■ rolling of the upper od the 



-holonomic Syetems. JHssipative Systems [oh. vm 

ModitioD of contact of the lower hemisphere and the horizontaJ plane is 
IB of rolling are 

now obtained the IS equations connecting the 24 coefficients : taking 
, c,, as the 6 independent coordinates of the BjBtem, and solving for 
iciente in t«rn)s of theee, we find with the necessarj approximation 

.,.1-i (.,•+>,■), ja,.l-J(V+t,'). 

•n -e,(J!,-«, |i -*,(«,-!,), 



.«,+i,-i,|i-t(V+/S,')l. p -«,-',{i-J(«.'+».')). 
■-A, «■--«., 

ly.-l-l(yi'+S,>). U-l-ife'+V)- 

enei^ of the Bjstem is 

small quantities of the second order, 

s,Jf,-A«.i«-l*.«AS,+ftS,-Jf.«,+«,-(ft«,Jf,-A«.J« 

-I2!,B,,,„+^X,B,„'. 

xprees the coordinates ^, m, « of any particle of the upper or low«r 
inns of its coordinates relative to the axes Z^c and Z^(r|( reepectivelf, 
Q ^Zm(/) + ni*+n') foreachhemiaphere, neglecting terms above the second 
i^uantities, and ren^embering that the principal momenta of inertia of 
maas M and radius R at its centre of gravity are ^MR*, f^MR*, 
for the Icinetic energj' of the ajsteu the value T, where 

Jfi+-«.CliA.*+i-«A+^'))+2^iyi-tfA(S«i+ie^)+18yi'-M.*- 

I of motion evidently separate into three distinct sets, consisting of 

ons for the coordinates a, and a^ : theae coordinates give rise to Do terms 

i correspond to vibrations in the stricter sense ; in fact, the equilibrium 

if either of the hemispheres is turned through any angle about its axis of 

can therefore neglect these equations. 

.ions involving the coordinates b^ and |9,, 

ions for the coordinates c, and y, ; these are exactly the same as the 

and 0,, so we only need consider the latter. 



c Systems. Dissipative Syi 

, are, tn aetento, 

+j7(ffl.-J'".-S^M^j)6j-i 

lantol equation for X, where 2n-/V^ ii 

+is a 

I in X ; it is easily found that its roots i 

ability of the aqu^ibrium. 

boloiiomic ayatems about a state 
cussed by use of the equations 

II be illustrated by the followin; 

ation hat an equatorial plant of tymii 
le tolid being vertical. Thi» motion heii 

vitj of the solid, and let (C, A) be its 
le through Q perpendicular to tbe aii 
7z is the axis of the solid, Gy is perpei 
mtoct (so Qy is horizontal), and Qx is 
imponents of the force acting on thi 
e Oxt, F' being parallel to Oy, and R 
[u„ u,, u,) denote as usual the con 
i body respectively, and let (u, v, w) tx 
lie moving axes. Further, let p be the 
tt the equator, a the radius of ita eqi 
ertical, and tbe angle between Qy 

'-^sin^ 6i'=iat=e, 63=^coaS, 

fore give, if P is the point of contact, I 
i ON the perpendicular from on the 
, + «^) =F<xa6-{R-Mg)am0, 

\+uei, =f; 

i^+ve^) = (R- Mff)caB$+Fwa0, 
^j+C«sfl, = -/".(? J, 

-F-.PK. 
SP are measured poeitivel; parallel to 
ital projection of thb direction respecti' 



ymic Si/atems. DissipeUive 

liding at P &re 

jucosd+wsin«-<7iV.i.,»0, 

^ of the bodj and pUne is 

<ee-tiian6=j^{-aSooa$+PKaia 

line the motion in the geaeral case, w 
iposed to be small. When this lat 

*=f+X. •"j-B+w, v-^-an+n, 

Lnd F, F; u, to, »,, W], fl,, fl,, tf, are 
ihaTO jVi'aO-a)x- The eqmilioiiB 

■ir(K+anflJ R+Mg, 

lf(vi-ane,)-F, 

iy+Cne, =0, 
A^-Cn^i ^-Fa-ifg(j>-a)x, 

A =F'a, 

-OS, -0, 

faw -0, 

H,^^,--^, «,= (?,=;(, flj = 0. 

and replacing 4,, ^g, 6^, »„ ug b; tl 



1 of theae equations we see that d> ant 


le other three equations give, o 


aelimii 


Aij, 


Cx- 


i^ + A)x->r{C+Ma?)n^ + Mg(p 


-'■ix- 






ila*)x^{ilgA 0>-a) + (^'((7+jr«»)}, 


e period of a vibration is 




„ t Al,i+Mat 


_l' 



tem« ; Jrictional fvrcea, 

the consideration of systems for ' 

^al energy is not valid, the energ 



nic Systems. Disstpative Syatems 

•me other form (e.g, heat) which is not recog 
it cortaideT frictional systems. 
ich are uot perfectly smooth are in contact 
the point of contact may be reaolved into a 

normal to their surfaces at the point, whi 
, and a component in the common tangent-[ 
nal force. The frictional force is determine 
bas been established experimentally: The I 
r, provided ike frictional force required fo 
not exceed fi times the normal pressure, viki 
ImiHng coejfident of friction," which depends 
'.he surfaces in contact are composed. If cm 
orce required to prevent sliding is greater th 

there will be sliding at the point of contact 
Uo play wiU be ft times the normal pressure. 

3S illustrate the motion of systems invo 



ftide on a rough /ieed platu eurva. 

uticle which ia constrained to move on a rough fixei 
a plane cmve, under forces which depend solely 
and g{i) denote the components of force per unit 
on of the tangent and normal to the tube, where » 
me fixed point of the tube, measured along the arc : 
s moving; and let A be the normal reaction per unit 



deration of the particle along the tangent and norm 
elocity of the particle and p the radius of curvature 



de 



fW-fR, 
■S-W+it 






+ ^v*= 



lut depending on the initial circumstances of the mot 
» equation is a known function of «, sa; =F{t). 



miic Systems. Dissipative & 

adtin 

ts the solution of the problem. 

IT koop of moM M itandi on rough groi 
id of Ike korizOTUal diameter. To And w 

oiling motioD, oasumed poaaible, and so i 
M thia motion is, or is not, greater than 
imes the correeponding normal pressure 
op from the commencement of the moti 
re of gravity of the system, referred to i 
ts own initial position, bo that 

al energies are 

f ?■= JfaM»+ma^'(l -sin d), 

(F=. -myosin A 
m of motion is therefore 

[Jf+m(l-flin5)}] + ma^"co8fl=f>¥"coa 

this equation gives 

2(rf(Jf+m)=ny, 



_^2_ ii^^^a6= 



al force and R the normal pressure, we hi 

•-(M+m)s, R=(if+m)i-y+g), 

F X m(Jf+m) 

R ~y-¥0 2J^+4Jftn+nt*' 
roll or slide according aa the coefficient < 

iM^+AMm+m^' 
le moves under gravity on a rough cj 
horizontal : if ^ be the Inclination of th 
le equation of the cycloid can be written 

«=4aBin^ 
at of friction, shew that the motion is giv 

— ..,*..).".(^/5^,). 



lonomic Systems. IHseipative Systems 226 

cea depending on the velocity. 

of dissipative ayetem is illustrated by the motion of 
, as the resistance of the air depends on the velocity of 
eoeral rule can be formulated for the solution of prob- 
of this kind : a case of great practical interest, however, 
f a projectile under the influence of gravity and of a 
some power of the projectile's velocity, can be integrated 
ler. 

e the velocity of the projectile. A*" the resistance per 
nation of the path to the horizontal, and p the radius of 
lih. The components of acceleration of the projectile 
1 normal to its path are vdvjda and »*//} ; and hence the 



da ^ ' 

equation by the second, we obtain 

1 dv tan 6 k 

p"+' d6 V" gco&d' 



[^ 



1 <^ / 1 OV "^ /I 

- ~; -J7i (n log sec 6) = sec f. 



sec" + Constant = /see"*' ddB. 

es V in terms of 0. To obtain (, the equation ^ = pg cos 6 



gt = — Itisec^dd, 

function of 0, this equation gives t aa& function of 6. 
jinates {x, y) of the particle can now be found from the 

a!= \vco&0dt, y= ivsmSdt 

iroblem is thus reduced to quadratures. 

LTtiole in a resisting medium, when the reeiatauce depends on 
Bolved in maoj other casee in addition to that discussed above. 



3 Sygtems. DissipcUive Syti> 

I denotes the ratio of tiie reaietaDce to 
I effected in the four cases 

i=o+61ogi>, 

i = ai^+R + bv-', 

i=a(logi')>+Alogo+6. 

»ntii and R ie another constant dependii 

integrable caaes, of which the following u. 

'Jl+a(u-l)'"*''jl + 6(u + l)'"'"^' 
constants : this equation defines t> in 
; finite when e is rationaL 

■■Xa &lla vertically from re«t at the ori, 
y aa the vetocitj. Shew that the d: 



cle falls verticallj from rest at the ori) 
uare of the velocitj : shew that the dist 

- logC08h(Vs'F'Oi 

a per unit mass. 

on-function. 

\ci to external resisting forces whi 

!s of their points of application, it 

;ioa of the system in general coordi 

energies and of a single new functi 

> the' system by the action of the 

e m of the system, whose coordinati 

. (8a;, %, hz) be 

rai&c + kyifhy + k,zBe, 

B of X, y, z only. The equations of 

fore be 

imS = — kxA + X, 
my = -kyy+T, 

ments of the total force (external s 
>rce of ri;sistance. 

libn et du mouviment dafiitidtt, Paris, 174^ 
, cxun. (1901), p. IITS. 



holonomic Systems. Dissipative Systems 

action F be defined by the equations 

lation ifl extended over all the particles of the system 
called the dissipation-fuTiction, represents half the ra 
being lost to the system by the action of the resisting fo 
..., fn) be coordinates specifying the configuration ol 

the equations of motion of the particle m by dxjdqr, di 
ily, and summing for all the particles of the system, we 1 



e have 
, /■.. 3« .. 8,v ,.. 8«\ d/dT\ ST 

inetic energy ; and 

■S91+ ... + QnS9ii denotes the work done by the ext 
I the resistances) in an arbitrary infinitesimal diaplacec 

" 3?/ 
at the equations of motion of the system in terms of il 
..., 5„) can be urritten in the form 

dt KdqJ dq, d^r ^ 

he resisting forces depend on the relative (as opposed to the abi 
linta of applicatioD, so that the forces acting on two partidea (Xi, 
B the components 

-i.(i,-»J, -*,(j,-yj, -i.(i,-« 
-*.(«i-i,), -*,(jj-y,), -i.ft-l,) 
that the equatione in general coordinatea can be formed wit 

notion. llg^UjS,^,llt1H,l'f-llt-i, 



r-Qr (r=l,2,.. 



lomic Systems, Dissipative & 

dissipative syatema. 
lem is specified by its kinetic eaergj 
dissipation fuactioa, methods si 
ipplied in order to determine the i 
em abont an equilibriura-configura 
shall consider a system with two d 
hat for the vibrational problem the 
,11 be taken as homogeneous quadrat 
tential energy as a homogeneous qi 
^efficients in these fuDCtions being i 
variables which would be normal c 
nction, we can write these three fun 

■ /■ = i (aj,' + 2Ag, j, + bq*). 

be supposed positive, so that the eq 
I dissipative forces, 
lotion are 

i fe^V — +-- +^--0 

dt \dqj dqr Sjr Sqr 
9i + (K^i + Aji + X, ji — 0, 

nd a particular solution of these eqt 
q, = AeP'. J, = fie'^, 

values in the diGferential equations * 
A(_fP + ap + \) + Bhp = 0, 
Akp + B(p' + bp + \,) = 0, 

;hat p must be a root of the equatio 

' + op + \,-) { p" + 6p + X,) - A-*p' = 0. 

:he dissipative forces to be comparat 

ties a, h, b can be neglected ; on th 

ion are readily found to be 

Pi = i 'J\ - ^o, Pi = i Vxi - 46. 
the root px we have, ft'om the secoD 

A X,-X," 



{; 



94] Non-holonomic Systems. Dissipative Systems 229 

A particular solution of the differential equations is therefore given by 

?! = (>'i - Xa) er^ (cos *J\t + i sin V^O* 
gr, a h V\^^-*«* (t cos Vx^^ - sin ^/\t), 

and a second particular solution is obtained by changing i to — t in these 
expressions. It follows that two independent real particular solutions of the 
differential equations are 

J?i = (Xi - X,) e-^ cos \/\t (q^ = (Xi - Xs) c-*** sin »J\,t, 

Ija = - A Vxitf-*** sin Vv Ua = A Vx^e-^ cos »J\t, 

and therefore the most general real solution involving e^* is 

Ai = (Xi - Xa) ^e-^ sin {s/\t + e), 



ja = A VXi^e-*** sin TVXit + ^ + € j , 



where A and e are real arbitrary constants. This represents one of the normal 
modes of vibration of the system. Adding to this the corresponding solution 
in e^«*, we have finally the general solution of the vibrational problem, namely 

jj = (X, - X,) Ae-^ sin (s/\^t + e) + A ^T^Be-V* sin {^t + 1 + 7 V 

ft = A Vx^^«-4«< sin U/\t + 1 + e) + (X, - X,) Be-^ sin {slx^t + 7), 

where A, B, e, y are four constants which must be determined from the 
initial circumstances of the motion. 

Now we suppose the dissipative forces such that energy is being con- 
tinually lost to the system, so that J^ is a positive definite form, and therefore 
a and b are positive. The last equations therefore shew that the vibration 
gradually dies away, on account of the presence of the factors er^ and er^ : 
the periods of the normal vibrations are (neglecting squares of a, h, b) the 
same as if the dissipative forces were absent ; and in a normal vibration, the 
amplitude of oscillation of one of the coordinates is small compared with the 
amplitude of oscillation of the other coordinate, while the phases of the 
vibration in the two coordinates at any instant differ by a quarter-period. 

A similar analysis leads to corresponding results for systems with more 
than two degrees of freedom ; supposing that the dissipative forces' are small 
and that the dissipation function and potential energy are positive definite 
forms, we find that the periods of the normal vibrations are (neglecting 
squares of the coefficientsr in the dissipation function) unaltered by the 
presence of the dissipative forces, but that the vibration gradually dies away : 
and if (ft, ft, ..., qn) are the normal coordinates of the system when the 
dissipative forces are absent, there is a normal vibration of the system when 
the dissipative forces are present, in which the amplitude of the vibrations in 



L 



280 Non-holonomie Systemg. IHasipaiive Systems 

9ii 9it •••! 9n is small compared with the amplitude of the vibrati 
and the phase of the vibrations in q^, q,, ..., q„ differs by a quar 
from the phase of the vibration in j,. 

Exampk. DiBcuBa the vibrations of a ajetem which is acted on by periot 
forces which have the same period as one of the normal modes of free vibra 
system ; shewing the importance of dissipotive forces (even where smsil) in thi 

96. Impact. 

Another mode in which energy may be lost* to a dynamical ayt 
the collision of bodies which belong to the system; a collision 
results in a decrease of dynamical energy. 

The analytical discussion of collisions is based on the foUowii 
mental law ; When two bodies collide, Vie values of the relative veloi 
surfaces in contact {e^irnated normally to the surfaces) at instants in 
be/ore ctnd immediately after the impact bear a definite ratio to a 
this ratio depends only on the material of which the bodies are compt, 

This ratio will in general be denoted by — e. When e is zero, I 
are said to be inelastic. 

The general problem of impact reduces therefore to a problem in 
motion in which the unknown impulsive force at the point of cont 
bodies ia to be determined by the condition that the change i 
normal velocity of the bodies satisfies the above law. 

96. Loss of kinetic energy in impact. 

We shall now find the loss of kinetic energy when two perfect 
bodies impinge on each other. 

Let m typify the mass of a particle of either body, and let («j, i 

(u, V, w) denote its components of velocity before and after the in 

let {V, V, W) be the components of the total impulsive force (ex 

molecular) on this particle. The equations of impulsive motion (§ 

m (m - Uo) = IT', m{v — v„) = V, m(w — w,) -■ W. 

Multiplying these equations by (« + eu,), {v + eu»), (to + e«),) re; 

adding, and summing for all the particles of both bodies, we have 

2m {(« - u,) (m + eu,) + (u - «,) (v + evt) +{w~ w,) {w + eio,)] 

^t{U(u + eu,)+V{v + ev,)+W{w 
Now so far as molecular impulses are concerned, we have 

X(Uu+Vv+Wiv) = 0, and 2 ( (^«, + Vb. + Ww,) - 0, 
since the impulsive forces which correspond to each other in virtue 
of Action and Reaction will give contributions to these sums which 
destroy each other. 

' Le. lost lo the Byitem ooDudered as a dynamical ■yatein : the energy is not am 
appears in some othet manifestatiDD, e.g. heat. 



iolonondc Systems. DissipeUive Systems 231 

e part of (u + eug) due to the normal component of velocity 
e for each of the particles in contact at the point where the 
3 (in virtue of the law of impact) it follows that the impul- 
a the bodies does not contribute to the sum Sf7(u + eu,), 
I not contribute to the auras ^V(v + eVt) and 'ZW (vj + ew,). 

jfore 

[7(u + CM,) + V(v + efl„) + W(w+m,)} = 0, 



,) (« + ffu,) + (p - u,) (w + «ff,) +{w- w.) (w + ew,)] = 0. 
)- 2m («,* + !>„' + to,') 



can be expressed by the statement that the kinetic energy 
! tfl (1 — e)/(l + e) times the kinetic energy of that motion 
to be compounded with the motion at the instant be/ore the 

produce the motion at the instant after the impact. 

1 of impact 

change of motion consequent on the collision of two free 
«ce can be most simply determined by the following con- 

if each body before or after impact is specified by six 
3 three components of velocity of its centre of gravity and 
jnts of angular velocity of the body about axes through ita 
The total number of equations required to determine the 
of motion is therefore twelve. Of these, six are immediately 
onditioii that the angular momentum of each body about 
•he point of contact is unchanged (since the impulsive forces 

another equation is obtained from the condition that the 
system in the direction normal to the surfaces in contact 
ce the normal impulsive forces on the two bodies at the 
,re equal and opposite), and another by the experimental 
: the bodies are perfectly smooth, the remaining four equa- 
led from the condition that the linear momentum of each 
ction tangential to the surfaces in contact is unchanged 

tangential impulse if the bodies are smooth): if on the 
lies are perfectly or imperfectly rough, the condition that 
am of the system in any direction tangential to the surfaces 
toged gives two equations ; if the bodies are perfectly rough, 
the relative velocity of the bodies in any tangential direc- 



232 Non-holonomic Systems. DissipaUve 

tion after the impact is zero gives the other two: wh 

imperfectly rough, the coefficient of friction between th 

being fi., the remaining two equations are given by the cc 

(a) the relative velocity in any tangential directii 

impact, provided the tangential component of the impu 

does not exceed fi times the normal component of the im 

(j3) if the last condition is not satisfied, there is s 

equal to fi times the normal impulse between the bodies. 

In all cases, therefore, the required twelve equations ■ 

If the motion takes place in a plane, or if one of tb( 

procedure is still valid after making some obvious modifi< 

The following examples illustrate these principles: 

Example 1. An inelaitie tpkere of mat* m falls with velocity 

indattic indined plane of mau M and angle a, vhich retts on a . 

Sh«w that the vertical velocity of the centre of the tphere immediatdy 

5 (Jf+ni) Fain*.! 

7J&"+2jn+5i»8in»n' 

Let P be the velocity of the plane after impact, w the velocity i 

and relative to the plane, u the angukr velocity of the sphere, and 

The equation of horizontal momentum gives 

m^^tcoaa-lTj-^MU. 
The kinematical condition at the point of contact is aiii=u. 
The condition that the angular momentum of the sphere abo 
shall be the same before and after impact is 

■ »iFo9ina = iiBa««+ma{«-rcoaa). 
These three equations give, on eliminating u and U, 
B(y+ TO) rein*o 
** *"" " ° 7 Jf + 2m + 5m sin* o * 
which is the remit stated. 

Example !. A sphere of radius a rotating with angular ve 
inclined at on angle a to the vertical and moving, in the vertical pi 
with vdocity V in a direction moHng an angle a viitli the horizon, 
h»ritontal plane. If the plane be tangentially inelastic, find the 
plane containing the neic direction of motion mates with the old. 

Take rectangular axea Oxyi, where is the point of contact, ( 
the initial plane of motion ; and let wj and aij be the compone 
about Ox and Oy respectiyely after the impact, and M the mass of 

Equating the initial and final angular momenta about Ox, we h 
JfaFcosa=iJ/iiV- 

Equating the iniUal and final angular momenta about Oy, we Y 
%Ma*Qtaaa-\Ma*t,^. 



e Systems. Dissipative Si/stems 

OD of the new plane of motion to the pUne yC 
of the plane) a^oi, and this ia therefore equal to 



la p. tone. 

fh eiretdar due of mau M and radivi c imptnggi 
apahle of turning freely ahout a pivot at itt ae 
•am the centre of the rod, and the direction of moti 
8 VTtth the rod before and after colli$ion, lAea that 
ma*) ton (9 = 3 (ww' - 3X1*) ton a. (Coll. I 

■citj of the disc, and let « denote its final velocit 

le point of contact, we have 

vco8|3+c£l=c0. 
gular velocity of the rod, and bj I the nonnal 
equation of motion of the rod is 

Jb^ima*a. 
Moa of the disc in the direction normal to the ro< 

Jf(i.BiD;8+Faina)-/, 
e relation 

vsin^+bm-ersina. 
. angular memento of the disc about the point of 

Voma^vooafi-icQ. 
these equations, we have 
tMI^ + ma*) - 3 ton o {mea* - 3Jft*), 

tn motion without rotation in it* oicti plaae, impit 
ele in the plane. The velocity of the centre of 
'<m maiing an angle a with the edge, and the t 
tpuUive change of motion. 

iponents of velocit; of the centre of the hoop t 
ar to the edge, and let a be the angular velocit; 
P- 
to about the point of contact before and after the 

-ifa'u+Jfau-irracoea. 

I equation 

(+aw ia zero after the impact, provided the f 
: not exceed fi times the normal impulse : but 
tional impulse is /i times the normal impulse. 



234 Nonrkolonomic Systems. DmipaHve Systems [ 

Let F be the frictional and R the Donnol impulse : then we have 

M{u-Viima)=-F, i/(p+Fain<i)=fl, Jfe»»=-af. 
We have therefore R=M{\+t) Vsino, 

and if u+ow is eero, we shall have 

F~\XVet»a. 
The quantity u+aia will therefore be aero after the impact, provided 
^>cota/2(l + e); 
and if fi doee not satisfy this inequality, we shall have 
F-iiM[\+t) Fsino. 
Thus finally, if fi^cota/2(l+fl), the motion is determined by the equations 
UH FcoHo+oo, v— ^raino, t(+a«=0, 
while if /!< cota/2(l+e), the motion is determined by the equations 

a=Fco8a+!H», »=ePsino, oo>= -f*(l+«) Fsino. 



Miscellaneous Examples. 

1. A perfectly rough sphere of radius a is mode to rotate about a vertical 

which ia tiied, with a constant angular velocity n. A uniform sphere of n 

placed on it at a point distant aa troTa the highest point : investigate tl 

and determine in any position the angular velocity of the sphere. Shew tliat t 

will leave the rotating sphere when the point of contact is at an angular distai 

the vertei, where 

coatf-^°coaal ^ °'"''^*° 
17 "^119 (a+6)j ■ 

(Camb. Math. Tripos, Part I 

2. A rough sphere of radius a rolls under gravity on the surface of a eone of; 
which is compelled to turn about ita vertical aiia with uniform angular T 
its vertex being uppermost ; if a be the aemi-vertical angle of the cone, r sic 
distance of the centre of the sphere from the axis of the cone, ^ be the an( 
through, relatively to the cone, by the vertical plane containing the centre of tl 
and Bj be the rate of rotation of the sphere about the common normal, prove tha 






where A, B, C ate determinate constants. (Camb. Math. Tripos, Part I, 

3. A homogeneous solid of revolution of mass M with a plane circulai 
radius c rolls without slipping with its edge in contact with a rough horison 
Shew that B, w, d are determined by the equations 

Jftc^ cocoa" fl)-Jfc2QcoH«(9={(7+J^c«) COS fl^, 

{^ (C+ Jfc>) - J/'a'tfi} ^^ (O cos* fl) + C (C+ Jfc") » COB fl - Jfe CO coe»fl=( 

{J+jre*)^+JO*coe'tf-2Jfae»Oco8tf+(C+i^<!«)«*-|-2*y(awntf+oooetf)-( 



] Nonrholonomic Syatemg. DissipoHve Syxtems 235 

I tbe iDclioation of the fucie of the body to the horizon, Q the angulftr velocitj of 
al place containing its oils, « the angular velocity of the bodj about ita axis, 
iment of inertia of the body about a diameter of its base, C the moment of 
the body about its axis and a the distance of the centre of gravity from the 
(Camb. Uath. Tripoa Part I, 189S.) 

wheel with 4n spokes arranged eymmetricallj rolls with its axis horizontal on a 
rough horiEontal plane. If the wheel and spokes be made of a fine heavy wire, 
t the condition for etability is 

3 2Mjr 

is the radius of the wheel and V its velocity. (ColL Ezam.) 

body rolls under gravity on a 6ied horizontal plane. If this plane be taken as 
ys, shew that 

2m{(y-y^)i-Cx-z^)y}-Constant, 
y, i) are tbe coordinates of a particle m aod {xj, j/j, tj) of the pomt of contact, 
immation is extended over all the particles of the body. (Neumann.) 

le portion of a horizontal plane is perfectly smooth and the other portion 
ly rough. A uniform heavy ellipsoid of semi-aiee (a, b, o) has its b-axis vertical 
» with velocity v in the direction of its o-azis along the smooth portion 
tne towards the rough. Shew that, if 

aid will return to the smooth portion, i being the radius of gyration about 
, and that the motion will then consist of ao oscillation about a steady state of 

I special case a=2b, shew that after the return of the ellipsoid to the smooth 
he 6-axia can never make an angle with the vertical which is greater than 

(ColL Exam.) 

ahell in the form of a prolate spheroid whose centre of gravity is at its centre 
, symmetrical gyrostat, which rotates with angular velocity a about its axis and 
itre and axis coincide with those of the spheroid. Shew that in the steady 
the spheroid on a perfectly rough horizontal plane, when its centre describes a 
adius e with angular velocity Q, the inclination a of the axis to the I'ertical is 

r6c(aooto+6)-.d6coeo+(7(o8ino + e)}0'+(7'6«i.O-J(if6(o-6cotQ)=0, 

s the mass of the shell and gyrostat, A the moment of inertia of the shell and 
together about a line through their centre perpendicular to their axis, C, C 
the shell and gyrostat respectively about the axis, a the distance measured 
I the axis of the point of contact of the shell and plane ^m the centre and 6 its 
Tom the axis. (Camb. Math. Tripos, Part I, 1899.) 

uniform perfectly rough sphere of radius a starting from rest rolls down under 
itween two non-interaecting straight rods at right angles to each other whose 
Manoe apart is 2c and which are equally inclined at an angle a to the verticaL 
are the original distances of the points of contact from the points where the 



236 Nrni-Ticlonomic Systems. Diasipative Systems \ 

shortest distance ioteroectB the rods and p, p' their distances at s subBequent ' 
the Telocity is V, shew that 



'^f-^i^SE^;:^)-"!'{o.-'.v-P.')~..4(,^-A,--p-+p.-v 

(Camb. Math. Tripos, Part 
9. A particle moves under gravity on a rough helix whose axis is vertica 
the radius and y the angle of the helix, shew that the velocity v and arc < 
can be expressed in terms of a parameter $ by the equations 



-cosy "- ' i 



■*"jfl{;.cosy + fl(;icoay + 2ainy)|' 
ZcoeyV 0) 



10. A particle is projected horizontally with velocity u so aa to slide c 
inclined plane. Investigate the motion. 

Prove that if 

S > 2/1 cot a > 1, 
the partiole approaches asymptotically a line of greatest slope at distance 

*'■ , ''',"'•■ , 

where u is the coefficient of friction, and a is the iQclinatioQ of the plane. 

(CoU, 

11. A rough cycloidal tube has its axis vertical and vertex uppermoet. ] 
radius of the generating circle and a particle be projected from the vertex wi 
•Jiagam a, shew that it will reach the cusp with velocity equal to 

[4aff cos' a{l - 2 Bin <.*-<*—)'" •}]*, 
where a is the angle of friction. (Coll 

IS. A heavy rod of length Sa is moving in a vertical plane so that on< 
contact with a rough vertical wall and the other end moves along the ground s 
be equally rough ; and the coefficient of friction for each of the rough surfac 
Shew that the incUnation of the rod to the vertical at any time is given by 

il{i'+<i>coo2.)-o^sin2.-.ay8in{fl-2.). fCdl 

13. A thin spherical shell rests upon a horizontal plane and contains 
of finite mass which is initially at its lowest point. The coefBcient of friction l 
particle and the shell is given, that between the shell and the plane being 
infinite. Motion in two dimensions is set up by applying to the shell an imj 
gives it an angular velocity O, Obtain an equation for the angle through whii 
has rolled when the particle begins to slip. (CoU 

14. A circular disc of radius a is placed in a vertical plane touching a uni 
(p) board which can turn freely about a horizontal axis in the upper surface o: 
through its centre of gravity, the point of contact of the disc being at a distt 
tills axis. A string, parallel to the surface of the board, is attached to tt 
the disc furthest from the board and to an arm perpendicular to the board i 
and rigidly connected to the board. The centre of gravity of the board ac 



lolonomic Systems. DissipaUve Systems 237 

tern starts from rest in that poeition in which the centre of the disc 
plane through the aiia. Shew th»t slipping will take place between 
d, when the board makee an angle with the vertical given by 

i^A+lpa^ + ^ah ' 
lit of inertia of the board about tha aiis divided hy the moss of the 

{ColL Eiam.) 
ejected with velocity 7 dowu a plane of inclination a, the coefficient 
■ tan a). It has initially such a backward spin Q that after a time t^ 
lill and continues to do so for a time f,, after which it once more 
■^ if the motion take place in a vertical plane at right angles to the 

({,+'))9Bioa-aQ- V. (ColL Exam.) 

lius a is fixed on a smooth horizontal table ; a second ring is placed 
:he first and in contact with it, and ia projected with velocity V, 
in a direction parallel to the tangent at the point of contact Find 
before slipping ceases between the rings if the coefficient of friction 
id prove that the point of contact will in this time describe an arc of 

n that will ensue if at the moment slipping ceases the fixed ring be 
to move, and prove that during the time that the inner ring rolls 
one the centre of the latter will be displaced a distance 

s?i. '-»>(-■+"'• 

ntisaes of the inner and outer rings and b is the radius of the inner 

(Camb. Math. Tripos, Part I, 1900.) 
ioal motion of a heavy particle descending in a medium whose 
the square of the velocity, shew that the quantity 



istacce, and a and ^ are the distances described in ' 

me, depends only on r and ia independent of the initial velocity. 

(Coll. Eiam.) 
heavy particle, let fall from rest in a medium in which the resistance 
of the velocity, will acquire a velocity {7tanh {gtjV), and describe a 
'V)jg in a time t, where U denotes the terminal velocity in the 

r the complete trajectory of a projectile in such a medium, the angle 
totes is given by 

t"/P'=sinh-'cotfl+cot«cosectf, 
ty when the projectile meves horizontally. (ColL Exam.) 

;he horizontal and vertical coordinates {x, y) of a particle moving 
nedium of which the resist^ce is R satisfy the equation 

ttH i>*coe><^ ' 
and ^ the inclination of the tangent to the horizontal. 

(ColL Exam.) 



238 Non-holonomic Systems. DisaiptUive Systems 

so. A particle is moTing, under gniTit;, in a medium in which the resist! 
the valocitj. Shew that the equation of the tn^ectot? referred to the verti< 
and a line parallel to the direction of motion when the velocity woa inl 
written in the form 

y=tlog(*/o). (C. 

21. Prove that in the motion of a projectile through a resisting medium 
a retardation leifl, where k ia veiy small and the particle is projected hori 
velocity V, the approiiroat« equation of the path is (neglecting i*) 






the axis of :r being in the direction ofprojectionoodthe ftiia of y vertically d 

(O 

22. A particle moves in a straight line under do forces in a medium wh' 
is (b* — t^log*)/*, where v is the velocity and # the diBtaoce from a given poic 
Shew that the connexion between « and t is given by an equation of the form 

(•■o+ic«'+*log*, 
where a and e are constants. 

23. A particle is moving in a resisting medium under a central attraotio: 
if £ be the retardation due to the resistance of the medium, and v the velocil 
description of areas by the radius vector to the fixed centre of force variea as 

.-!-,'. (c 

S4. Prove that in a resisting medium, a particle can describe a parab< 
action of a force to the focua which varies as the distance, prwided the 
a point, where the velocity ia r, be h[v{y~e^)^; where v^ is the vel 
vertex. Determine h. (C 

25. A particle tnovea in a resisting medium under a force P tending to < 
If £ be the resistance, shew that 

r being the radius vector and p the perpendicular on the tangent 

If u=l/r, P=fM\ and A»ih^, and we n^lect i? and higher powei 
the di^rential equation to the path is 

k being a certain constant. (C 

26. A particle is moving under a central force ^ (r) repelling it &om tfa 
resisting medium which impoaes a retarding force equal to k times the vel 
that the orbit is given by the equations 

r^=A«-« r+i#-AV-»e-«*'=^(r), 
where A is a constant quantity. (C 

27. A particle is moving in a circle under a force of attraction to an : 
varying as the distance ; the resistance of the medium is equal to its densi 
by the square of the velocity. Shew that the density at any point is propoi 
tangent of the angle between the linea joining it to the centre of force ai 
of the circle. (0 



i-Adlonomic Systems. Diaaipaiive Systema 239 

length a is rotating about one eitremity, which is fixed, under the 
le except the resiatauce of the atmosphere. Suppoaii^ the retarding 
anoe on a small elemeut of length <£v to be Adx. (velocity)', shew that 
■J at the time ( ia given by 

aoment of inertia about the fiied extremity, and O is a constant 

(ColL Eiam.) 
1 oval disc of mass M, turning on a smooth horizontal table with 
but without any traoslalional velocity, strilEee a smooth horizontal rod 
B middle point Prove that the angular velocity ia diminished in 

tfficient of elasticity, x the distance of the centre of gravity &om the 
it of impact and i the radius of gyratiMi about a vertical axis through 
ity. (Coll. Exam.) 

each of length a and mass m, are jointed together at their upper ends 
Falls symmetrically, with its plane vertical, on to a smooth inelastic 
re impact the joint has a velocity V and each rod has an angular 
; to increase its inclination a to the horizon. Shew that the impulse 
and the plane is 

ni(ia+c'sin»a)(r+oOcoeo)/{i" + <!'+a(a-2c)cos'o}, 
ance of the centre of gravity of each rod from the joint and nbf is the 
of each rod about its centre of gravity. (Coll. Exam.) 

lal uniform rods AB, BC, CD, each of length 2a, and hinged at B and C, 
bt line and moving with a given velocity in a horizontal plane at 
lir lengths. The ends A and D meet simultaneously two fiied inelastic 
I A and D tc rest Determine when they will form an equilateral 
r that \ of the original momentum ia destroyed by the impacts. 

(Coll. Exam.) 
uniform cube is free to turn about a horizontal axis paaaing through 
opposite faces and ia at rest with two faces horizontal ; an equal and 
opped with velocity u and without rotation so aa to strike the former 
b1 to the fixed axis and at a distance c from the vertical plane containing 
uigulor velocity imparted to the lower cube is 

c>+i'+o»(l-sin2o)' 
iclination to the horizon of the lower face of the falling cube, Sa is 
ige, k the radius of gyration and e the coefficient of restitution. 
action of the upper cube immediately after the impact 

(Coll. Exam.) 

[y elastic circular disc of mass Jf and radius e impinges without rotation 
IS n» and length 2o which is free to turn about a pivot at its centre, the 
ling at a distance b from the pivot Prove that if the component of the 
tre of the disc normal to the rod be halved by the impact, Jft*=nia', the 
cient to prevent sliding. {Coll. Exam.) 



240 Non-holoTwmic Systems. Dissipative Sy^ei 

34. A perfectly rough sphere of radius a ie projected horizontAllj it 
teota a point at a height A above a horizontal plane. The sphere hi 
an angular velocity Q about its horizontal diameter perpendicular to ' 
motion. Shew that before it cesses to bound on the plane it passes o 
distance 



2*/2 



4r^ 



(5F+2aQ), 



where e is the coefficient of elaaticitj, and the distance is reckoned from 
aoctact. 

Compare the final with the initial kinetic energy. 

35. A homogeneous elaatic sphere (coefficient of elasticity e) is [i 
a perfectly rough vertical wall so that its centre moves in a vertical plan 
to the walL If the initial componenta of the velocity of its centre ai 
its initial angular velocity (Q) is about an axis perpendicular to the vej 
the subsequent motion after impinging on the wall, and shew that if it 
to its original position the coordinates of the point of impact referred to t 

T' T+lOe + Te* +■*• 

S« {(7«+6)r + gaO}{r(7 + 5B)-2a«0} 
jr" (7+108+7*")* ' 



where a is the radius of the spher 



CHAPTER IX. 

THE PRINCIPLES OF HAMILTON AND GAUSS. 

98. The trajectories of a dynamical system. 

The chief object of investigation in Dynamics is the gradual change in 
time of the coordinates (ji, ^a, •••. ?n) which specify the configuration of a 
dynamical system. When the system has three (or less than three) degrees 
of freedom, there is often a gain in clearness when we avail ourselves of a 
geometrical representation of the problem : if a point be taken whose rect- 
angular coordinates referred to fixed axes are the coordinates (gi, q^, g,) of 
the given dynamical system, the path of this point in space can be regarded 
as illustrating the successive states of the system. In the same way when 
w > 3 we can still regard the motion of the system as represented by the path 
of a point whose coordinates are (gi, 5a> •••» ?n) in space of n dimensions; this 
path is called the trajectory of the system, and its introduction makes it 
natural to use geometrical terms such as "intersection," "adjacent," etc., 
when speaking of the relations of diflferent states or types of motion in the 
system. 

99. Hamilton s principle^ for conservative holonomic systems. 

Consider any conservative holonomic dynamical system whose configur- 
ation at any instant is specified by n independent coordinates (ji, jj, ..., qn\ 
and let L be the kinetic potential which characterises its motion. Let a 
given arc AB in space of n dimensions represent part of a trajectory of the 
system, and let GD be part of an adjacent arc which is not necessarily a 
trajectory : it would however of course be possible to make CD a trajectory 
by subjecting the system to additional constraints. Let t be the time at 
which the representative point (^i, 9a, ..., ?n) occupies any position P on AB : 
we shall suppose each point on CD correlated to some value of the time, so 
that there will be a point Q on CD (or on the arc of which CD is a portion) 
which riorresponds to the same value ^ as P does. As the arc GD is 
describeld, the correlated value of t will be supposed to vary continuously 
in the isame sense. A moving point which describes the arc CD will 
therefora pass through positions corresponding to a continuous sequence of 
values oA 5i, 52» •••» 9n, *, and consequently to each point on CD there will 
corresponcl a set of values of ji, g,, . . . , g^. 

w. D.I 16 



242 The Principles of Hamilton and Gauss ' [c 

We shall denote by B the variation by which we pass from a point o 
to that point of CD which is correlated to the same value of the timi 
shall denote by ft,> *i, Ai +^U, ti + At, the values of t which correapond 1 
terminal points A, B, C, D respectively, and by Lg the value of the fui 
X at any point R of either arc. 

If now we form the difference of the values of the integral 

jliquqt, —, 9». ffi. ?). ■■■. ?n. t)dt, 
taken along the area AB and CD respectively, we have 
I Ldf-j Ldt = Ls^t, - L^AU + i ' ^Ldt 

J CD J AB J I, 

by Lagrange's equatio 

But if (^q,)B denote the increment of qr in passing from B to D, we 

and similarly if (A?r)^ denote the increment of q, in passing from A 
we have 

and consequently 

Suppose now that G coincides with A, and D coincides with B, and 
the times correlated to C and D are t, and ti respectively, so thiit 
Af], ..., Aqn, At, are zero at A and B: then the last equation become:) 



f Ldt-j 

J CD J A. 



Ldt = 0, 



which shews that the integral iLdt has a stationary value/or any pcirt 

actual trajectory AB, as compared with neighbouring paths CD wkuck 
tiie same terminal points as the actual trajectory and Jvr which theitin 
the same terminal values. This result is called Hamilton's principlet 



99, 100] The Principles of Hamilton and Gaus8 243 

If the kinetic potential L does not contain the time explicitly, we can 
evidently replace the condition that the time is to have the same terminal 
values by the condition that the total time of description is to be the same 

for AB as for CD, since 2 ^^ ^-r — -£, which represents the total energy of 

the system, is in this case constant. 

100. The principle of Least Action for conservative holonomic systems. 

Suppose now that the dynamical systenr.' considered is such that the 
kinetic potential does not involve the time explicitly, so that the integral of 
energy 

exists. Taking as before AB to be part of a trajectory and CD to be part of 
any adjacent arc, to the successive points of which values of the time are so 
correlated as to satisfy an equation of the form 

where AA is a small constant, we have 

= f (h + M)dt- I hdt+ ( Ldt- f Ldt 

J CD JAB J CD JAB 



= [2 pAg, + ^AA 



']B 
A 



B 
A 



If therefore we suppose that C coincides with A and D coincides with B, 
and that Ah is zero, we shall have 

. dL\ ,^ f /5 . dL 



Lii^'WHSi^-i)^- 



w^j^j^sbews that the integral If l^qr^jdt ha^ a stationary value for any 



jpart of an actual trajectory, cw compared with neighbouring paths between the 
r^ same termini for which the time is correlated to the coordinates in such a way 
as to satisfy the same eqvxition of energy. This is called the principle of 
Lecust Action, the integral 



being called the Action, 



/Ci'-D* 



16—2 



244 The Priiiciples of Hamilton and Gaitss 

la natural problems, for whicb L is the differeDce of a kinet 
bomogeQeous of the second degree in the velocities, and a pote 
V, independent of the velocities, we have (| 41) 

and the stationary integral can therefore in this case be written 

Example 1. Shew that the pruiniple of Leaat Action can be extended 
which the integral of energy does cpt exist, in the following fonn. Let 

Z jr ^ - -£ he denoted by h ; then t le integral 



/(. 



!,''i*'S)* 



haa a stationary value for any part of an actual trajectory, as compared wi 
between the same tenuiiial points for which A has the same terminal values 
Example 2. If a dynamical system which posaesaea an integral of energ 
a system of lower otder as in § 42, show that the principle of Leaat i 
original system is identical with Hamilton's principle for the reduced sj 

101. EsdCTmon of Eamiltona principle to non-conservativi 
systems. 

We shall now extend Hamilton's principle to holonoml 
systems in which the forces are no longer supposed to be conser^ 

Let T denote the kinetic energy of such a system, and 

denote the work done on the system by the external forces in 
displacement (S^,, hq^, ..., &q^; the equations of motion of the 
therefore 

dAdqJ-h-qr^' ^""^^ 

Let a denote a part of a trajectory of the system, and let j3 Ix 
arc having the same terminals, the times correlated to the pa 
terminals being the same as the values U ^^^ i\ of the time at t 
in the trajectory a ; then if S denotes the variation by which wf 
position on a to the contemporaneous position on ^, we have 

iI'(«^+J,«'«'')'"-/M,(i**'+a|>'+«'^')' 






'"'-''' H\i, 



'i:iii, 



m< 



The Principles of Hamiiton and Gauss 



/:(■ 



leorem of § 99, which is really a particular case of it) known as 
•incijAe. 

'enaion of Hamilton's principle and the principle of Least Action 

mtc systems. 

now shew that Hamilton's principle, when suitably formulated, 

or dynamical systems which are not bolonomic. 

a non-holonomic conservative system, in which the variations 

)rdinate8 (j^, f, q„) are connected by m non-integrable 

equations 

Atidg,+A^dqt+...+A^dqn+Tkdi~0 (i = l,2, ...,m) 

1 A„m, 2*1, ..., Tm, are given functions of q,, }„ ..., q^: so 

lotes the kinetic potential, the motion is determined (| 87) by 
ns 

b the above kinematical equations; the unknown quantities 

be part of a trajectory of the system, and let CD be a path 
AB by displacements consistent with the instantaneous kine- 
tions, i.e. the above kinematical equations with the -terms Ttdt 
I path CJ) will not in general be itself a path whose continuous 
ould satisfy the kinematical conditions, so CD ia really a kine- 
possible path. 

irallf be aaked why we do not take CD to be a kinematicaUy possible path : 
rhich is, that in that case the diaplaoetnentB from AB to CD would not be 
;oiiaist«nt with the kinematical equations : for in non-faolonomic STstems, 
. possible configurations are given, the displacement from one to the other 
ral a possible displacement ; there are iafinitel; more possible adjacent 
there are possible displacements from the given position. 

g as in the proof of Hamilton's principle given in § 99, S denoting 
tplacement from a point of AB to the contemporaneous point on 



Ldt = LBAt,-L^£i.t,+ r' £ (^^ &q,+^ Sq^dt 

CD Jab It, r=l \Oqr O^r I 



246 The Principles of Hamilton and Gattas [t 

Since the diaplftcementa obey the relations 

it follows that the terms of the type \,Ar,Sqr in the integral annu 
other, so we have 

From this point the proof proceeds as in § 99. We thus obtain the 
that EamilUm's principle applies to every dyitamiical system, whether hoi 
or not. In every case the varied path considered is to be derived fn 
aatual orbit by displacements which do not violate the kinematical eq\ 
representing the constraiitts ; but it is only for holonomdc systems th 
varied jnotion is a possible motion ; so that if we compare the actual 
with adjacent motions wAtcA obey the kinemaiical. equationa of com 
Hamilton's principle is true only for holonomic systems. 

The same remarks obviously apply to the principle of Least Actio 
to Hamilton's principle as applied to non-conservative systems. 

103. Are the stationary integrals actual minima f Kinetic fod. 

So far we have only shewn that the integrals which occur in Ham 
principle and the principle of Least Action are staiionary for the traje 
as compared with adjacent paths. The question now arises, whethe 
are actually maxima or miniTna. 

We shall select for consideration the principle of Least Action, a 
convenience of exposition shall suppose the number of degrees of ft 
in the dynamical system to be two, the motion being defined by a I 
energy 

I' = i«ii(5i. S»)9.' + OiJ?i. 3i)9i?i + ia«(9i.S.)gs'. 
and a potential energy 

The discussion can be esteoded without difficulty to Hamilton's pri 
and to systems with any number of degrees of freedom. The princ 
Least Action, as applied to the above system, is (§ 100) that the integr 

noiiji' + 2a,ij, j, + at,q^) dt 

has a stationary value for an actual trajectory as compared with other 
between the same termini for which dt is connected with the different 
the coordinates by the same equation of energy 
T+V=h. 
This latter equation gives 

Oiiji* + 2a,i jiji + Oa^i' = 2 (A - ^),. 
or dt = [2 (A - -f )}-* {Oy^dq^ + 2o„d5,djj + a^dq,')*. 



102, 103] The Principles of Hamilton and Gauss 247 

so the stationary integral can be taken to be 



J(A - ^/r)» (ttn + 2015?; + a«5/»)* d5a, 



where qi stands for dq^jdqiy this integral is to be taken between terminals, 
at each of which the values of qi and q^ are given. 

Writing this equation 

we shall discuss the discrimination of its maxima and minima (which was 
first effected by Jacobi) by a method suggested by Culverwell*. 

Consider any number of paths adjaceut to the actual trajectory. These 
paths will be supposed to have the same terminals, and to be continuous, 
but their directions may have abrupt changes at any finite number of 
points. For such a path let {q^, 99+S59) be a point corresponding to. a 
point (gi, q^ on the actual trajectory; we shall frequently write a<^ for Sg^, 
where a is a small constant the order of which determines the order of 
magnitude of the quantities we are dealing with, and ^ is zero at the terminal 
points. 

Let the expansion of the function 

in ascending powers of a be 

/(9i, 92, ?;) + a ( f/o<^ + f^if ) + i aH f^ooc^' + 2 I7o,<^f + t^^^^^^ 
let ZI denote the terms involving a in the first degree in 

and let S*/ denote the terms in a\ 

When the range of integration is small, and its terminals are fixed, the 
value of <f> at any point is large compared with the value of <f>. For since (f> 
is zero at the terminals, we have 






where P and R denote the terminals. If therefore /S be the numerically 
greatest value of <f>' between P and R, it follows that <t> can never exceed 
(?i(J2) ''iiiPi)^* and consequently by taking the range suflBciently small the 
ratio of (f> to ^' can be diminished indefinitely. 

• Proc, Lond, Math, Soc, xxiii. (1892), p. 241. 



248 The Principles of HamUton and Gauss 

Thus if till range is very small, the most importaDt term 

^ I U,i<l>'*dqi ; and as the sigQ of this is always the same as that of U, 

of dqi is takeD to be positive), we see that for small raDges, / is a 
or minimum according as Uu is negative or positive. Now 

"" - ^' ■ '^ - '*''' <°" + ^""i- + °-«-''>"' (»»«" - '■»>■ 

and this is positive, since the kinetic energy is a positive definite 
therefore 0,10^—011' is positive. We thus have the re^lt that 
ranges the Action is a minimum for the actual trajectory, y 

Now consider any point A on an actual trajectory, and let anot 
trajectory be drawn through A making a very small angle with the 
this intersects the first trajectory again, say at a point B, then th< 
position of the point B when the angle between the trajectories ( 
indefinitely is called the kinetic focas of A on the first trajectoi 
point conjugate to A. 

We shall now shew that for finite ranges the Action is a 
provided the final point is not beyond the kinetic focus of the initie 

For let P and Q be the terminals ; we have seen that if Q is 
to P, the quantity B'l is always positive and of order a* comparec 
value of / for the limits P and Q. It is therefore evident that as 1 
Q further from P, the quantity S'l cannot become capable of a 
value until after Q has passed through the point for which S*/ c 
for a suitably chosen value of aij>. 

Suppose then that PBQ is an arc of an actual trajectory. Q bein 
point for which it is possible to draw a varied curve PHQ for which I 
we shall shew that the varied curve PHQ must itself be a trajectoi 
it is not a trajectory between two of its own points A and C (sup] 
each other), let a trajectory ADC be drawn between these points, 
integral taken along ADC is less than that taken along AHC, so tl 
taken along PADCQ is less than that along PHQ, which by hy] 
equal to that along PBQ. Hence S'/ along PADCQ is negative, 1 
fore Q cannot be the first point for which, as we proceed from P, thi 
ceases to be positive ; which is contrary to what has been proved, 
that PAHCQ is a trajectory, and Q is the kinetic focus of P. 
Advm is a true minimum, provided that in passing along the tra^ 
final point is reached be/ore the kinetic focus of the initial point. 

Lastly we shall consider the case in which the kinetic focus of 
point is reached before we arrive at the final point. Suppose, with tb 
just used, that the initial and final points are P and R ; and let tw 
and F be taken, the former on the curve PHQ and the latter on tl 
these points being taken so close together that the trajectory Hi 



Hi 



l03, 104] The Principles of Hamilton and Gatiss 249 

vhem gives a true minimum. Since the integral taken along EGF is less 
i;han that along EQF, it follows that the integral taken along PEQFR is less 
tljian that along PEQR ; but the latter is equal to that along PBQR, since 
>th integrals are equal from P to Q ; and therefore the integral along PBQR 
is[ not a minimum ; hut it is not a maximum, since the integral taken along 
ai;iy small part of it is a minimum. Hence when the kinetic focus of the initial 
point is reached before we arrive at the final point, the Action is neither a 
mcucimum nor a minimum, 

A simple example illustrative of the results obtained in this article is furnished by the 
motion of a particle under no forces on a smooth sphere. The trajectories are great- 
circles on the sphere, and the Action taken along any path (whether a trajectory or not) 
is proportional to the length of the path. The kinetic focus of any point A is the 
diametrically opposite point A' on the sphere, since any two great-circles through A 
intersect again at A\ The theorems of this article amount therefore in this case to the 
statement that an arc of a great-circle joining any two points A and B on the sphere is 
the shortest distance from A to B when (and only when) the point A' diametrically 
opposite to A does not lie on the arc, i.e. when the arc in question is less than half 
a great-circla 

104. Representation of the motion of dynamical systems by mean^ of 
geodesies. 

The principle of Least Action leads to an interesting transformation of 
the motion of natural djrnamical systems with two degrees of freedom. 

Let the kinetic energy of such a system be 

i {(hi (ffi, 9a) ?i' + 2aia {qi , q^) Ji^a + «« (^i, q^) g,'}, 

and let its potential energy be -^ (jj, q^). By § 100, the orbits corresponding 
to that family of solutions for which the tot il energy is h are given by the 
condition that 



j (oiigi* + 2a„g,g2 + a^ij') dt 



is stationary for any part of an actual orbit, as compared with any other arc 
between the same terminals for which dt is connected with the differentials 
of the coordinates by the relation 

i (oiigi* + Soi^i^a + a^gaO + f(qu ga) = A. 
The integral 

l(A - yjt)^ (oiidqi^ + 201,^31^^2 + a^dq^)^ 

is therefore stationary. But this integral expresses the principle of Least 
Action for the motion of a particle under no forces on any surface whose 
linear element is given by the equation 

d^a = (A - i/r) (Oud^i^ + ^(hidqidqt + a^dq^\ 



250 The Frinciplea of Hamilton and Gauss [ch. i 

and is therefore the deficiDg coDdition of the geodesies on this sarfac' 
Consequently the equaiiona of the orbits in the given dynamical system are t? 
saine as the equations of the geodesies on this swrfaoe. 

Hxample 1. Shew tbat the paraboUc orbits of a free heavy projectile con ganp n 
to the geodeaicB on a certain surface of revolution. 

Example 2. Shew that the orbits described under a, central attractive force 0'(r) in a 
pLtue correspond to geodesies on a surface of revolution, the equation of whose meridia<a- 
curve is 2'=f(p), where 

and where r and p are connected by the relation p*—t^{~ii>{r)+k). 
Sfu* I kiuit. 106. The least-curvature principle of Gauss and Hertz. 

/^t^T"' u- ^® shall DOW discuss a principle which, like Hamilton's pvinciple, can be 
'■ ' ' used to define the orbits of a dynamical system, but which does not involve 

the sign of integration. 

In any dynamical system (whether holonomic or non-holonomic> let 
(f^r, Vrt ^r) be the coordinates of a typical particle jji, at time f, and 
{X,, Yr, Zr) the components of the external force which acts on the particle. 
Consider the function 

where the summation is extended over all the particles of the system, and 
where (i,, y^, ^r) refer to any kinematically possible path for which the 
coordinates and velocities at the instant considered are the same as in some 
actual trajectory. This function substantially represents what was called by 
Oauss the constraint and by Hertz (who however considered primarily the 
case in which the external forces are zero) the curvature* of the kinematically 
possible path considered. In what follows Hertz's terminology will be used. 

We shall shew that of all paths consistent with the constraints (which are 
supposed to do no work), the actual trajectoi-y is that which has the least curvature. 

In the simple case of a single particle moving on a smooth sui-face under no external 
forces, this result clearly reduces to the statement that the curvature in space (in the 
ordinary- sense of the term) of the orhit is the least which is conaistent with the condition 
that the particle is to remaia on the surface. 

To establish this result, let tiie equations which express the constraints 
(using X, to typify any one of the three coordinates of any particle) be 

lxi,dx^ = Q (k=l,2,...,m), 

where the coefficients Xh- are given functions of the coordinates, DiflFer- 
entiating these relations, we have 

S3:t,i,+ SS^±,^, = (fc = l, 2 m). 

* Strictly epeaking, the sqeaie root or thia faootian, and not Ihe fanotion itself, waa oalled 
the cnrvature by Heitz. 



ie Principles of Hamilton and Gausa 261 

. typical compoDeat of acceleration in the path considered 
sed to be kinematically possible, but is not necessarily the 
'), and let «„ be the corresponding component of acceleration 
ajectory. Subtracting the preceding equation, considered as 
actual trajectory, from the same equation, considered as 
Linematically possible path, we have (since the velocities are 
two paths) 

1xtr{xr-x„)-(> (&=1, 2, ...,m). 

n shews that a small displacement of the system, in which 
t Zxr of the coordinate ov is proportional to Qcf — icn), is con- 
equations of constraint, i.e. is a possible displacement, 
ents of the forces exercised by the constraints are typified by 
d in any possible displacement the forces of constraint do no 
i therefore 

t{mrXn - Xr) (Sr - X„) - 0. 

ch can be written in the form 

\ Wir' r \ ^r/ r 

the use of y's and ^'s) 

;)'-(-S)^(-£)] '■ 

+ Sm, ((*, -ii„y+ (s, - s„y + («, - s„n 

erms in the last summation on the right-hand aide are all 

ivs that 

« the result stated. 

ission of the curvature of a path in terms of generalised 

3 shewn* that the curvature of a kinematically possible path 
iynamical system with n degrees of freedom can be expressed 
derivates of the n independent coordinates which define the 
lystem. 

• Joumal/ilr Math. uiin. p. 823. 



252 The FrincipleB of ffamiUon and Qnnss 

^^ (9i< 9» '■■< 9n) t>c the coordiDates; let (^\, q^, .... q^) be tht 
tions of these coordinates in any kinematically possible path 
(910. ?». •■■. 5m) be the accelerations in the actual trajectory wh 
spends to the same values of (g,, 5,, ..., q„, ji, j„ .,., g„). Using a: 
any one of the three rectangular coordinates of any particle m^, and J 
the corresponding component of force, the Gauss-Hertz curvature o 
is 'S.m, (£, — X,jm^' ; and it haa been shewn in the last article thf 

be written in the form 

Sm^ {st„ — X^jm^f + 2m, {it, — i„)*. 

The first of these summations is the same for all the paths considi 
it depends only on the actual trajectory : we can therefore omit 
causing the whole expression to lose its miniraura-property, and v 
the remaining summation 2Tn,(i, — in,)" the curvature of the path. 

Let the kinetic energy be 

where the quantities «« are given functions of (g,, 5,, ..., 5,); let 
the determinant farmed of the quantities a^, and let An denote thi 
ati in this determinant. 



From the equation 



Xm^i* — 2 lauqiqi 



we have 
Now 



-9*31, 



3?t Hi ' 

r dqidqi ^ 

and consequently, since the coordinates and velocities are the same 
paths considered, we have 

i^-i„ = Sg^^(gt-5to). 
But if we write 

dt \3ji/ dqii r^qt ^ ' 

since this expression is zero for the actual trajectory, we have 

St = the difference of the values of -j- (^ ] for the path cons: 
the actual trajectory, 
or St = SfflH iqi - j'to) (Jfc= 1, 2 

whence we have 91-4*.= ^ S^hS/ (k = l,2 



r 



106, 107] The Principles of HamUUm and Gauss 263 

and consequently 

D k loqu 
The curvature, 2m^ {x^ — x^, is therefore 

r 

or Tfa 2 2 2 tatiAjaAijSiSj. 

But by a well-known property of determinants, we have 

2 2a«-4«-40= jD^y, 

i k 

and therefore finally the curvature can he expressed in terms of the coordinates 
(?i> 9a> •••» 9n) «wci ^Aeir derivates in the form 

-f.XXAijSjSi. 

107. AppelVs equations. 

The Gauss-Hertz law of Least Curvature is the basis of a form in which 
Appell has proposed* to write the general differential equations of dynamics. 
This form, as will be seen, is equally applicable to holonomic and non- 
holonomic systems. 

Consider any djmamical system ; let 

Atkdqi+A^dq^-^- ... + -4^dqr„ + Tjtd^ = (A; = l, 2, ..., m) 

be the non-integrable equations connecting the variations of the generalised 
coordinates Ji, J21 •••, ?n; in holonomic systems these equations will of course 
be non-existent. 

Let 8 denote the function ^2^^ {xj? -h yj? + z/), where m^ typifies the mass 

k 

of a particle of the system, whose rectangular coordinates at time t are 
{^k> Vki ^k)' By means of the equations which define the position of the 
particles at any time in terms of the coordinates (gi, 53, ..., jn), it is possible 
to express 8 in terms of (51, 52* •••» ?n) and the first and second derivates of 
these variables with respect to the time. Moreover, by use of the equations 
of constraint we can express m of the velocities (?i, gi, ..., gn) in terms of the 
others: let the coordinates corresponding to these latter be denoted by (pi, 
jPi, ..., pnr^' By differentiating these relations we can express g'l, g'a, ..., 

gn, in terms of the quantities pi,pi, ...,iJii-m, A» P«» ...,Pn-m, g^ ga, ••., gn, 
and hence S can be expressed in terms of this last set of variables. 

* Journal fur Math. cxxi. (1900). 



\ 



264 The Principles of Hamilton and Gauss 

Now aoy small displacement which is consistent with the ca 
can be defined by the changes (Spi, ^i, ..., Sp^,^ in the q 

(p,,Pt, ...,pn-ni); let S PfBpr denote the work done by the ezten 

in such a displacement. As in § 26, we have 

Let the equation which expresses the change in ar^ in terms of th( 
in (piipi, ..■,Pn~m) he 

where (vi. v,, ..., Vn-m) ^^ known functions of the coordioa 
equations of this type are of course non-integrable. From this 

Sxtldpr = "Tr, and so the equation which expresses it in terms of 

ipi,p„ ....p,.-™) 
will be of the form 

it = S iTrP, + a, 

where a denotes some function of the coordinates. Differentia 
equation, we have 







whence 




It follows that 






--^(^.^-^t-^'g 




..».(..|.^|..|; 




as 



and therefore the equations of a dynamical system, whether kolonom 
can be caressed in the form 

dp, ' \ . . . 

where S denotes the function ^trat^Xj^ -^ yi^ -y z^), and (pi.pt, --■,; 
coordinates equal in number to the degrees of freedom of the system. 

It is evident that the result is valid even if the quantities p,, 
are not true coordinates, hut are quasi-coordinates. 



107, 108] The Principles of Hamilton and Gauss 



255 



Example, Obtain from Appell's equations the equations 

Jffwj — (C— -4) o>3a>is jtr, 

for the motion of a rigid body one of whose points is fixed ; where (coj, o^, wg) are 
the components of angular velocity of the body resolved along its own principal axes 
of inertia at the fixed point, (A^ B, C) are the principal moments of inertia, and (Z, My N) 
are the moments of the external forces about the principal axes. 

108. BertrancCs theorem. 

A theorem in impulsive motion, which belongs to the same group of 
results as the least-curvature principle of Gauss and Hertz, is due to 
Bertrand* and may be stated thus ; If a given set of impulses are applied to 
different points of a system {whether holonomic or non-holonomic) in motion^ 
the kinetic energy of the resulting motion is greater than the kinetic energy 
of the motion which the system would acquire under the action of the sam£ 
impulses and constraints and of any additional constraints due to the reactions 
of perfectly smooth or perfectly rough fixed surfaces, or rigid connexions 
between particles of the system. 

For let m be the mass of a typical particle of the system, and let (u, v, w)y 
{u, v\ w'), (u^, v,, Wi) denote the components of velocity of this particle before 
the application of the impulses^ after the application of the impulses, and in 
the comparison motion, respectively. 

Let (X, F, Z) denote the components of the external impulse acting on 
the particle : {X\ T\ Z') the components of the impulse due to the con- 
straints of the system : and (X' + X^, F'+ F^, Z' + Zi) the components of 
the impulse due to the constraints in the comparison motion. 

The equations of impulsive motion are 
m{u'-u)^X + X\ m(t;'-t;)=F+F', m(w'- t(;) = Z+Z', 

m(wi-w) = X + X' + Zi, m(vi-v)= F+F'-hFi, m{w^'-w) = Z -^ Z' -^-Z^, 

Subtracting, we have 

m{ui — u') — Xiy m{vi'-v') = Fj, m{wi'-w)^ Zj. 

Multiply these last equations by u^yVu w^ respectively, add, and sum for 
all the particles of the system ; we thus have 

2m [{ui — u') Ui + (vi — i/)Vi-\- (wi — w') Wi] = 2 (Xi u^ + YiVi + Z^ w^). 

Now from the nature of the constraints, it follows that fiinite forces 
acting on all the particles of the system and proportional to the impulsive 
forces (Jfi, Fi, Z^, would on the whole do no work in a displacement whose 

* Bertrand's notes to Lagrange's M^c, Anal, 



266 The Principles of Hamilton and Gauss 

components are proportional to the quantities (Ui, «,, Wi); and then 
have 

or Sm E(mi - w') «i + (v, - v') u, + (w, - w') w,] = ; 

this equation cim be written in the form 

2m (w"" + 1/" + «)'•) - 2m («,' + «,» + «-,') = 2m [(«' -«,)' + (p' - r,)" + (w 
which shews that 

^2m («'' + v'* -Y w'") > J2m («,' + v,' + tt>i'), 
and 8o establishes Bert rand's theorem. 

Tbe following result, due to Lord Kelvin aod generallj known as Thoimon't 
can easily be eatabliahed hy a. proof of the same character as the above : If any n 
point* of a dynamical tffttem are tuddenlp set in motion tdtk praeribtd veUx 
kinetic energy of the reralting motion is leu than that of any other kinematieaU; 
motion wAi'cA the tyttem can take tnith the prescribed velocities, the excess being the 
the motion which must be compounded with either to produce the other. 

Example. A framework of (n — I) equal rhombuses, each with one diagoDi 
same continuous straight line, and two open ends, each of which ta half of a rhi 
formed hj 2n equal rods which are freely jointed in pairs at the corners ot 
rhombuses. Impulses P perpendicular to and towards tha line of the diag( 
applied to tbe two fi-ee extremities of one open end ; shew that the initial 
parallel to the diagonal, of the extremities of tbe otber open end is 



where in is the mass of each rod, and 2a is the angle between each pcur of 
tbe points of crossing. {Camb. Matb, TripOH, Part I, 

Miscellaneous Examples. 

1. If the problem of determining tbe motion of a particle on a surface who 
element ie given by the equation 

de^-Bdu^+SFdudv+Odv*, 
under the action of forces such that tha potential energy is V(a, v), can be aolv 
that the problem of determining the motion of a particle on a surface vrhoi 
element is given by 

d**- K(tt, i!){Edu^+2Fdiidv+Odv'), 
under forces derivable from a potential energy 1/F(k, »), cau also be solved. 

(Dar 

2. If in two dynamical systems in which the kinetic eueigies are resj 
Soujiji and sfiujij*, and the potential energies are respectively f and V, the tra 
are the same curves, though described with different velocities, so that tbe i 
between the coordinates (q„ q„ ...,qj are the same in the two problems, shew tb 

yl7+i' 
where o, j3, y, B, are constants, and that 

iba,dgtdqt=(yU+»)2attdg,dq^. (Ptu, 



te Principles of Hcmiilton and Gauss 257 

'AJoctoriea of & particle id a plane, deacribed uoder forces euch that the 
r the particle is V {x, y), with a value h of the constant of energy, are 
sformation 



conjugate functions of (;r, y), shew that the new curves so obtained are 
1 particle acted on bj forces derivable from the potential energy 

[7»(z,n+wn)-»]((g)'+(|4)'), 

of the conatant of energy. (Ooursat.) 

'denote reepectively the kinetic and potential energies of a. dynamical 



.i{(^.»3V(»,.-)V(->'^)} 



ib does not 'involve the accelerations ; and hence that 






iim(i»+i--«+i-«) 



m the occelerationa have the values corresponding to the actual motion, 
I all motions which are conaiatent with the constraints and satisfy 
of energy, and which have the same values of the coordinates and 
nstant conaidered. (Fttrster.) 



CHAPTER X. 

HAMILTONIAN SYSTEMS AND THEIE INTE0EAL-INVAEIAN1 

109. Samilltm't form of Ihe eqmtims of mutton. 
We shall now obtain tor the difterential equation, of motion of 
«,rvati.e holonomic dynamical system a form whioh was introdi 
Hamilton" in 1835, and which constitutes the basis of most of the a 
theory of Dynamics. 

Let{o o« 5„)hethe coordinate8andi(g„5„ ...,g™,5i,i., ■ 
'I- ''^1 the kinetii pjitential of the system, so that the equation, of molioi 

/■■" Lagrangian form are 

'*fyi^_"_.0 (r-1,2, . 



^L 
Write 5J-?' 



(r-1,2, 
(r = l, 2, 



so that P' 9^, 

From the former of these sets of equations we can regard eithi 
sets of quantities (i.g, 9-) " (P-P ?"> " '^°'='"'°' "'"" ' 

If S denote the increment in any fiinction of the variables 

(g„5, 5„j),.y. p.) or (5„ <h 9". ?" * *' 

due to small changes in these arguments, we have 



= l(y,S?,+;),!?r) 

- 6 s p,9, + i <iMr - i'W. 

' PMl. Traiu. 1835, p. 95. 



109, 110] Hamiltonian SystemSy etc. 269 

n 

Thus if the quantity 2 prqr — i, when expressed in terms of 

r=l 

\Ql> Qii • • • > ?»»> Pli P21 • • • J Pny *)i 

I 

be denoted by H, we have 

SJ3'= S (grSpr-^rSgr), 
r=:l 

yAe motion of the dynamical system mxiy he regarded cw defined by these 
equations, which are said to be in the Hamiltonian or canonical form ; the 
dependent variables are (ji, q^, ..., qn, Pn p^, '•'» Pn)> and the system consists 
of 271 equations, each of the first order ; whereas the Lagrangian system 
consists of n equations, each of the second order. 

When the kinetic potential L does not involve t explicitly, the Hamiltonian 
function H will evidently likewise not involve t explicitly, and the system 
will possess (§ 41) an integral of energy, namely 

* r=l Oqr 

where A is a constant. This equation can be written 

and this is the integral of energy, which is possessed by the dynamical system 
when the function H does not involve the time explicitly. For natural problems, Cfi*^) 
it follows at once from § 41 that H is the sum of the kinetic and potential 
energies of the system. 

Example, Shew that the equations of motion of the simple pendulum are 

dt" dp' dt~ dq' 

where 

ff=ip^-gl~^coaq, 

and where q denotes the angle made by the pendulum with the vertical at time t, I is the 
length of the pendulum, and the mass of the bob is taken as unity. »^ 

110. Jacobis theorem on equations arising from the Calculus of Varia- 
lions. 

From the preceding chapter it appears that the whole science of Dynamics 
can be based on the stationary character of certain integrals, namely those 
which occur in Hamilton's principle and the principle of Least Action: 
similarly the diflferential equations of most physical problems can be regarded 
as arising in problems of the Calculus of Variations. 

Thus, the problem of finding the state of thermal equilibrium in an isotropic 
conducting body, when the points of its surface are kept at given temperatures, can be 

17—2 



Hamiltonian Systems arid 



formulated as follows : to fibd, among &11 functions V having given values at t 
that one which makes the value of the integral 

int^rated throughout the surface, a mioimum. 

Jacobi has shewn that all the differeniial equations which ai 
problems in Ike Calcuius of Variations, with one independent variah 
expressed in the HamiUonian form. 

Suppose, for clearness, that there are two depeadent variables; I 
is equally applicable to way number of variables. 

Let L (i, y, y, y, ■■■, y, z, z, a, ..., e) be a fuDctioa of the ind 
variable (, the dependent variables y, z, and their denvates up to on 
respectively. 

The coiiditionB that the integral 



fL{i.y,y.- 



,y,z, z. 



,z)dt. 



may be stationary, can, by the ordinary procedure of the Calculus i 
tious, be written in the form 



3j dtKly)*- 
3i_i/3i\ 



K-1)' 



d" /9£\ 

de'\ 










.. + (-!)" 


_, d"- IdLX 

d^' la'jj 




i)- 


.+(-1)- 


_ (i— ■ pL\ 
dr^Vyl 


Pm - 






11 


81 d [BL\ 




. • + (-!)"- 


dr-Ai"l' 


dL 




. + (-1)- 


dr-\3t/ 


P,M = 






7.- 



1] their IntegrcU-Iiivarianta 261 

B 

?i = J/. ?i = y, — . 3m = y. ?«+■ = «. ?>»+, = « ?m+n = ^. 

if 

f is supposed expressed as a function of (f, <f,, ..., qm+n.P\, ■■;Pm*-n), 
itities y and z being eliminated by use of the equations p„ = dLjdy, 
L/dt) and if S denote an increment due to small changes in the 
ts?», 3i. .-■. qm+n,Pi,p„ ....pnifn, we have 
.„ "^iSZ J. dL J- »^^dL . ■ dL J."' 

oH = — i -^ S^r+i SiOy- i — S^m+r+l Si *^ 



dL . dL . SL . dL 

imeS SJEf := — ^ Pr^r + 2 5rSpr- 

, if if is expressed in terms of the variables 

(*,pi.p.. ....iWft.?i.?i. ■■■.?«.+«). 

do, 3JS" dpr dH , , n , \ 

lifferential equations of the problem are thtts expressed in the Hamil- 
trm. 

ijrstems of differential equations which arise in the problems of the 
of Yariationa are often called isoperimetricai systems. 

Integral-invariants. 
nature of Hamiltonian systems of differential equations is funda- 
■ connected with the properties of certain expressions to which 
has given the name integral-invarianta.*^ 
ider any system of ordinary differential equations 

dar, _ _ <^ _ y- '^n _ T 

W"'^" ~dt~'^" ■■■• dt~^'" 
I, X„ ,,., X„, are given functions of jr,, a;,, ...,Xn,t. We may regard 
oations as defining the motion of a point whose coordinates are 
. , iE„) in space of n dimensions. , , 



r 



262 HamUtonian Systems and 

If pow we consider a group of such points, which occupy a p-dime 
region fo at the beginning of the motion, they will at any subsequent 
occupy another ^-dimensional region ^. A ;>-tuple integral taken o\ 
called an integral-invariant, if it has the same value at all times 
number p is called the order of the integral-invariant. 

Thus, in the motion of an incompressible fluid, the integral wbicl; 
sents the volume of the fluid, when the integration is extended over 
elements of fluid which were contained initially in any given regioi 
integral-invariant ; since the total volume occupied by these elemen 
not vary with the time. 

Example 1. Consider the djDataical problem of determining the motion of a 
in a plane under no forces : let (x, y] be the coordinatea of the particle, and ( 
componenta of velocity. The equations of motion may be written 
x—u, ^=r, li-O, i-0. 
The quantity 



=/<. 



where the integration is taken, in the four-dimensional space in which (x, 
are coordinates, along the curvilinear arc which ia the locus at time ( of points wh 
initially on some given curvilinear arc in the space, is on integral-invariant 
solution of the dynamical problem is given by the equations 
u=a, v=b, x=at+e, y=bt+d, 
where a, 6, c, «f are constants : and therefore we have 
I^Ut\a+»c-tia) 



and this ia independent of t. 

Example 2. In the plane motion of a particle whose coordinates are (x, y) an 
velocity-components are {u, v), under the influence of a centre of force at the origi 

attraction is directly proportional to the distance, shew that 

ia an integral-invariant. 

112. The variational equations. 

The integral-invariants of a given system of diCferential equations 
integrals of another system of differential equations which can be 
from these. 

For let the given system of equations be 

W 

Let (xi, Xa, ..., x„) and (a^ -f Sa;, , a:, -f &f,, ..., a!„ + &c«) be the vb 
the dependent variables at time ( in two neighbouring solutions of thi 
equations; where (Sxj.&c,, ..., £a;„) are inflnitesimal quantities. Them 

^( 



.jr,(«„«, «,„«) (>-=i,2,. 



j-(Xr+ Bxr) =Xria^ + Sx,, x, + tx, x„ + Sx„, t) (r = l,2. .. 



111-113] their Integral' Invariants 263 

and consequently 

j^S«, = ^'S^, + ^'&ri+...+g'&r„ (r = l. 2,. ..,«). 

These last n equations together with the original n equations, can be 
regarded as a set of 2n equations in which (a?i, x^^ ..., x^, Sa^, 8x2, ..., Bxn) 
are the dependent variables. 

Now if 

jXFr{Xiy X2, ..., Xn)BXr 

denotes an integral-invariant of the original system, the quantity 

yz \2Fr(Xi, a?2, ..., Xn)BXr- 

must, since the path of integration is quite arbitrary, be zero in virtue of 
precisely this extended system of differential equations ; and therefore 

S-Pr (^ > ^2 > • • • » ^n) ^r = COUStaut, 

r 

must be an integral of these equations : so that to an integral-invariant of 
order one of tiie original system of equations there corresponds an integral of 
the extended system of equations^ and vice versa. 

If a particular solution (a?i, a?9, ..., a?n) of the original equations is known, 
we can substitute the corresponding values (a?i, x^, ..., «;«) in the extended 
differential equations, and so obtain n linear differential equations to deter- 
mine (&Ci, &rs, ... , Sa?»), i.e. to determine the solutions of the original equations 
which are adjacent to the known particular solution. These n equations are 
called the vwriatioTial equation s. 

113. Integral-invariants of order one. 

Let us now find the conditions to be satisfied in order that 



/' 



where (ifi, M^, ..., -Mn) are functions of (a?i, a?,, ..., Xn, 0> ™ay ^ *^ integral- 
invariant of order one of the system of differential equations 

-— — Xrix^y X2, ..., Xn, t) (r= 1, 2, ..., n). 

We must have 

J (MiBxj^ -h M^Sx^ -h . .. + MnBxn) = 0, 

where the derivates of (Sa?i, 8x2, ..., 8xn) are to be determined by the ex- 



264 HamUtonian Spsterm and [ch. x 

tended system of differential equations introduced in the last article; and 
therefore 

Since (&B,, &Ei, ..., &r„) are independent, the coefficient of each quantity 
Sxr in this equation must be zero: and consequently the conditions /or 
integral-invariancy are 

Corollary 1. If an integral of the differential equations, say 
F{xi, Xt, ..., ain, t) = constant, 
is known, we can at once determine an integral-invariant. 
For we have 

d(dF\^ " d /dF\^ ^ « dFdX,_ d (dF ^ = dF „\ 
at UJ ^ »1 a^ la^J '^^ + *.. ai* 1^ " 3^, laT + *r, a^ '^ V 

= 0, 
and therefore tiie expression 

. . ; . ^^.s^') 

w an inf^j^roZ-tntiartont. 

Corollary 2. The converse of Corollary I is also true, namely that if 
\\ 2 ^— BXf\ is an integral-invariant of the differential eqaaHona, where U is 
a given function of Hie variables, then an integral of the system can he found. 

For we have 

dt XdxJ t=i dxic\dxj j=i dxic dxr 

dxr\dt i^iaaii / 
and consequently the expression 

dt toi a^t 

which is a given function of (ir,,a;, a:„, t), is independent of (ic,,a^, ...,ar„); 

let its value be ^(f): this is a known quantity. 



113, 114] their Integral'Invariants 265 

Then we have 

or U '-\^{t)dt = constant ; 

and this is an integral of the system. 

114. Relative integral-invariants. 

Hitherto we have only considered those integral-invariants which have 
the invariantive property when the domain of the initial values, over which 
the integration is taken, is quite arbitrary; these are sometimes called 
absolute integral-invariants. We shall now consider integrals which have the 
invariantive property only when the domain over which the integration is 
taken is a closed manifold (using the language of n-dimensional geometry) ; 
these are called relative integral -invariants. 

The theory of relative integral-invariants can be reduced to that of 
absolute integral-invariants in the following way. 

Let {{Miixi + ilfaSara + . . . + M^hx^ 

be a relative integral-invariant of the equations 

-jf=^r (^=^1,2, ...,n), 

where (Mi, M^, ..., if„, Xi, X9, ..., X„) are functions of (a^i, x^, ..., a?^, t)\ so 
that this expression is invariable with respect to t when the integration is 
taken, in the space in which (^1, ^2) •••> ^n) ^^^ coordinates, round the closed 
curve which is the locus at time t of points which were initially situated on 
some definite closed curve in the space. 

By Stokes' theorem, this integral is equivalent to the integral 

where the integration is now taken over a diaphragm bounded by the curve ; 
this diaphragm can be taken to be the locus at time t of points which were 
originally situated on a definite diaphragm bounded by the initial position of 
the closed curve : and since the diaphragm is not a closed surfece, this integral 
is an absolute integral-invariant of order two of the equations. 

Similarly, by a generalisation of Stokes' theorem, any relative integral- 
invariant of order p is equivalent to an absolute integral-invariant of 
order (p + 1). 



L. 



266 Hamiltonian Systems and 

116. A relative inte^at-invariant which is poeeeased by all Ham 
syBtems. 

Consider now the case in which the ayatem of differential eqnati 
Hamiltonian system, eo that it can be written 

dt dpr ' dt^dqr ' ' ' 

where H is a given function of (ji, q^, ..., q„, pi,pt, ■■■,Pn, 0- 
For this sjmtem let 

(Ldt 



n=JLi 



denote Hamilton's integral, bo that L is the kinetic potential ; let 

(«..a, a,./9„/3„...,^0 

be the initial values of the variables 

respectively, and let S denote the variation from a point of one orbii 
contemporaneous point of an adjacent orbit. By § 99, we have 

Sn= 2 PrSg^- S ^,Sa,. 

Let C, denote any closed curve in the space of 2p dimensions i 
(q,, qt, .... ?„, Pi, Pi, -...Pb) are coordinates, and let C denote th 
curve which is the locus at time t of the points which are initiall 
Integrating the last equation round the set of trajectories which p 
Ct to C, we have 

( i PrSq.= l 2 0M, 

JCt=1 JC,r=l 

and this equation shews that the quantitt/ j 2 p^Bq^ is a relative i 
invariant of any Hamiltonian system o/ differential equations. 

116. On systems which possess the relative integral-invariant jSp 

We shall next study the converse problem suggested by the i 
the last article, namely that of determining all the systems of dil 

equations which possess the relative integral -invariant I £ pr^> 

(9i> 9i> -■■> ?«) ^i^ h^lf ^^^ dependent variables, and (pi, p,, -..iP,,) 
other half 

Consider then a system of ordinary differential equations of ore 



their Integral-Invariants 267 

les can be separated iDto two sets, (q„ 9,, ..:, q„) and 
ich that 

ral-invariant of the equations, and conaequently by Stokes' 



//« 



;gral- invariant. 

3 of differential equations be 

t=«„ t-p, (-1.2 "). 

Q„, P], Ft, ..., P^) are given functions of 

{qi.q= 9n.Pi.Pa. ...,Pn, *)• 

f integration of the absolute integral-invariant is of two 
a suppose that each point in it is specified by two quantities 
J not vary with the time but are characteristic of the tra- 
ihe point in question lies. The absolute integral-invariaat 
nitten in the form 

not vary with the time, we must have 

d 5 3(gi,P.) „ 
dt i^id iX, It) 

(ft, P,) , 3Q. a (P.. ft) . HP, 3 (;,. ;.) 8P, d (},, p.) ) 
) (>, c) "^ 3pj 3 (\, ,i) "^ 3„ a(x, ,1) * 3}! 3 (\, /i) J "■ 

complete arbitrariness of the domain of integration and 

d^ the coefficients of ^'fe, |? ^' , and |-* |* in thi. 

lish separately. We thus obtain 

3^ + ?^* = 

3P(_3Pt^ 

3a_3e.. 

3pt 3pi 



(i,h- 



(r-1.! 



HamUtonian Systems and 

36 equations shew that a function H{qi, q^ gn.pi.P 

ucfa that 

Spr dq, ^ 

IB we have the result that i/a system of equations 

» ihe relative integral-invariant 

j{pM>+piSqt+ — +i>,%»)> 
3 egualto?u Aare (Ae HamUtonian form 

dqr^dH ^Pr^_9fi 

(ft 9pr ' dt dqr 

the converse of the theorem of the last article. 
oUary. If 

j{Pi^,+IhSq^+...+pn^n) 
ative integral-invariant of a system of equations 

dt ^" dt ' ^ • 

k is greater than n, it follows in the same way that the e( 
•"I ?».pi. P>. ■■■•P^ fomi a HamUtonian system 

f is a function of (5,, j,, .,., 5„, pi,p,, ...,pni o°ly. n( 
r»rt, ..., qt.pa+i, ■■-.pk)- 

'. The expression of integral-invariants in terms ofintegra 
rhe solution of a system of differential equations 

~=Xr{,Xi,X,, ...,x^,t) (r = l, 

Evn, the absolute and relative integral-invariants of the 
be constructed. 

as, let 

Ci. c,, ..., Cn are constants, be n integrals of the system; t 
il-invariants of order one are evidently given by the formu 



jiN,iif, + N,Byt+,..+NJ 



■»). 



their Integral-Invarianta 269 

''„) are any functions of (yi, y,, ■..,yn) which do not 
ative integral- invariants of order one are given by the 



ion of (x,, Xj, ..., Xn, t), since the term JBF vanishes 

integration is closed. 

.is that any system of differettiial equations possesses an 
UtUe and relative integral-invariants of the first order. 

of Lie and Koenigs. 

lilts enable us to establish a theorem due to Lie* and 
ction of any system of ordinary differential equations to 

^.X, fr=M,...,«, 

f equations, and let 

J(fi8ah + f,&c,+ ...+ftSart) 

isolate integral-invariant of order one of this system, 
ire given functions of the variables : we have seen in 
\ infinite number of such integral-invariants exist. 

ential form 

f,S^-l-f,&ii+... + fi8ari 

onical form 

yi8y, + p,fiy, -I- ... +p„S5„ - 6n, 

0>i.p.. ..-.pn.gi. ?s. ■.■.9», fi) 

tions of (X|, iCi, ..., xt), in number not greater than k, 
zeroj. Let («,,«,, .... Ui_„) be a set of other functions 
I that («,,«,, ...,Uk-n,qi.qt. ....Jn.ih.pj. ...,p»)are 
nt functions of (ir,, x,, .... x^); and suppose that the 

NatttT., 1877. 

gsibility of this reduction (vhiuh bowevei requires in generftl the 
rdiniiry differentia! |eqiutiotis) will be foncd in aoj treatiie on Pfaff'B 



270 ffamiltotiian Systems and 



system of differential equations, when 
as independent variables, becomes 

dt '^" 

dt ' 


expressed 

.u, 


in terms of tile; 
(««1,2, 


J., ft e. 


,P,.P.. • 


...i".. 


u„ u,.... 


, U,^) are fui 


expression 












fc,8j. 


+ p,Sq 


,+ ■■■+?, 


Sg.) 



tegral-in variant (relative or absolute) of this system, s 
cy is a property unaffected by such trsnaformatious 
2d: and consequently it follows (§ 116) that the first 
9 form 

dt dp/ dt " dqr ^^ 

7 is a function of (q„ 3,, ..., q^, p,, pt,' ■■■, Pn. onl, 

•f differential equatioTUi is thus reduced to a Hamilton' 

., together with the {k — 2n) additional equations 

du, 

dt ' 



'V, (a = 1, 2, 



The Last Multiplier. 
re proceeding to discuss integral-invariants of higher on 
considered, we shall introduce the conception, due to i 
Itiplier of a system of equations. 

dxi _ dxj _ _ dxn _ dx 

r,, X„ ,,., X„,X)aregivenfunction8of the variables (a;,, 
en system of equations: and suppose that (n~l) intt 
ire known, say 

/,(ar„ir„ .... a:„, w) = a^ {r~l,2. 

I these equations let (xj, x,, ..., ^n-i) be expressed as fu 
then there remains only the solution of the equatioi: 

dxn _dx 

X^'X" 
fected; in which accents are used to denote that (iE, 
n replaced in X„ and X by the values thus obtained. 



118, 119] their Integral- Invariants 

We shall shew that the integral of this equaUon is 
'' da:„ — X„' dx) = constant, 



/f(^'^ 



where M denotes any solution of the partial differential equation 



9 /iii-v^ 3 
[aiJi-tf-t- ... I- ' M >- 1 ■ 

and A denotes the Jacofnan 



^^(MX,)*^iMXi+...+^(MX,)^l-JMX) = 0. 



3 (/../. A-.) 

3(«,.«, «_,)■ 

The function JIf is called the Last Multiplier of the system of diEFc 
equations. 

For the proof of this theorem, we shall require the following lemm: 
If a system of differential equations 

^.X, (,.1,V. 

\s transformed by change of variables into another system 

where D denotes the Jacobian 

9(a^, a^, ..., x^ 

9(yi,y., ...,yn)' 
To prove this, we have 

„] ox^ r=\ OX, \i=i dyki 

r=i .=1 *=i ^x, \ dy,dyii dy, SyJ 

In this expression the coefficient of dYtldy, ia 2 ^—ir-^, which 
T=iOx,dyic 
or unity according as s is different from, or equal to, k Also dy,/dxr = 
where il„ denotes the minor oldxjdy, in the determinant i): so thecoi 
of Ft in the above expression, which ia 

r=\ ,=\dxrdy,dy)i' 



272 


Haim 


lltOi 


nian Sy^t 


erm and 




can be written 












i!..l/ 


"Sy,di/t' 


1 


V 9(ai. iPt. 


...,«„,8« 
3 to, ft, 


■,/8»,». 


or 






1 W 

Diyt' 






We have therefore 












m- 


. i 
1 


? 8(cr.) 
.r, 8ft ' 






which establishes the lemma. 










Now io the <: 


mgioal probli 


3m 1 


write 








d^_d^_ 


. _<^_ 


-•i?-^. 





and consider the change of variables from 

{a^.Xt, ...,Xn.x) to (Oi, a,, ...,a„_i,a^,a!): 
by the lemma, we have 

so the quantity J/, which is a solution of the equation 

jtf d( 3a;i 8a:, *" 3a;, dx ' 
satisfies the equation 

J. dM d^fXj^\ 3/X'\_ 

3 /Z.'if'N 3 /X'M'\ „ 

which shews that the espressioD 

~(X-dx,-X,'da;) 

is the perfect differential of aoiae function of a;„ and x ; this ef 
theorem of the Last Multiplier. 

BolUmann and Larmor't hydrodynamical repraentation of the Lait Jl 
The theorem of the L&at Multiplier can also be made apparent bj 

HideratioDs. For simplicitj we shall take the number of variables to be th 

differential equations can be written 



/; 



119, 120] their IrUegral-Invariants 273 

where (ic, v, w) are given functioDB of (4?, y^ z) ; and the last multiplier M satisfies 
the eqiiatioD 

3l(i^«)+|(jr«)+|(iA.)=0. 

This equation shews that in the hydrodynamical problem of the steady motion of 
a fluid in which {u^ v, w) are the velocity-components at the point (^, y, z\ the equation of 
continuity is satisfied when M is taken as the density of the fluid at the point (^, y, z). 

Now let 0(a?, y, «)=C 

be an integral of the differential equations ; then the flow will take place between the 
surfaces represented by this equation ; thus we can consider separately the flow in the 
two-dimensional sheet between consecutive surfaces C and C+ hC, The flow through the 
gap between any two given points P and Q on C must be the same whatever be the 
arc joining P and Q across which it is estimated : and since the flow across arcs PR and RQ 
together is the same as that across PQy we see that the flow across an arc joining P 6knd Q 
must be expressible in the form /($) -/(P)* So if ds denote an element of this arc, and 

T the (variable) thickness of the sheet, so that r={(80/9x)2+(9<^/8y)*+(9<^/3«)*}~*. dC, and 
if I denotes the velocity-component perpendicular to ds, we have 

80 that M(rds is the perfect differential of a function of position. But it is easily seen 
that this expression can be written in the form AfbC (vdx-u dy)/d<l)/dz; and consequently 

Jfivdx—udy) 

d^/dz~ 

is a perfect differential : this is the theorem of the last multiplier for the case con- 
sidered. 

120. Derivation of an integral from two multipliers. 

Suppose now that two distinct solutions Jlf and N of the partial differential 
equation of the last multiplier have been obtained, so that 

and 

Subtracting these equations, we have 

but this is the condition that the equation 

log (M/N) = constant 
shall be an integral of the system 

dxi __ cfecg _ dxn dx 

and we have therefore the theorem that the quotient of two laM multipliers of 
a system of differential equations is an integral of the system. 

w. D. 18 



( 



274 ffamiltonian Systems and 

The reader who is acquainted with the theory of infinitesimal trai 
&b1e to prove without difficult; that if the equation 



^.l^^-l.-- 


-'-I*-|-» 






*..l-«.^- 


...+l..|.+f,| 


then the reciprocal of the deteiminjint 






jr, J,. 


...JC. J! ■ 




ft, la- 


-f,. £, 








l« U- 


....I- 1. 1 



t multiplier 

121. Application of the last multiplier to Samiltoniar 
mile's theorem. 

If the system of differential equations considered is a Hai 
we have evidently ^hX,jdxr = 0, and consequently ^ = 1 is ( 
partial differential equation which determines the last multi 
TtiuUiplier of a Sa/miltontan system of equations is unity. 

From this result we can deduce a theorem due to I 
enables us to integrate completely any conservative holor 
Kvifc'^ hJ*^' system with two degrees of freedom when one integral is k: 
^ . If-I to the integral of energy. 

Let the system be 

dq, dq, dp, dp, J, 

dp, dpt dq, dqt 

and in addition to the integral of energy H (q^, q„p,, pt) — I 
y^iqi<^iipi<Pt) = c fee known. From the theorem of the I 
follows that 

is another integral ; where, in the integrand, p, and p, are 
replaced by their values in terms of g, and g, obtained f 
integrals ff and V. 

But if we suppose that the result of solving the equa 
V=c forpi andp, is represented by the equations 
rp.=/i(9i. 9i.*. c), 
\pt=A{quqt,h,c), 
• Journal At Sloth, t. (1840), p. 861. 



120, 121] their Integral-Invariants 276 

then we have identically 







dHdf.^dHdf, Q 
9pi 8c 8pj 3c ' 




« 


, dpi dc 3p, dc ' 


and therefore 






dH dH 




8/i 9pj ?/i 3pi 




9c diV.HY dc d{V,Hy 


« . « ^ 





80 ^&e theorem of the last multiplier can he expressed by the statement that 

is an integral. 

This result leads directly to the theorem of Liouville already mentioned, 
which may be thus stated: If in the dynamical system defined by the equations 

dqr_dH dpr^^dH 
dt'dpr' dt " dqr ^ - ' ^' 

the integral of energy is H{qi, g,, pi, p^^K and if V{qi^ q^, pi, 2>8) = c 
denotes any other integral not involving the time, then the expression 
Pidqi-^Pidq^y where pi and p^ have the values found from these integrals, 
is the exact differential of a function 0{qi, ja, h, c); and the remaining 
integrals of the system are 

de , de 

;r- = constant, and ;^f = ^ + constant. 
dc oh 

This amounts to saying that if any singly-infinite family of orbits is 
selected (e.g. the orbits which issue from a point ji = ai, ja = Oa) which have 
the same energy, so that to any point (ji, q^) there correspond definite values 
of pi and Pa (namely the values of pi and pa corresponding to the orbit which 
passes through the point qi, q^ and belongs to the family), then the value of 

the integral Ipidgi+PscZga taken along any arc joining two definite points 

(?io> 320) and (ju, jai) is independent of the arc chosen. 

To complete the proof of Liouville's result, we bave on diflferentiating the 
equations H^h and V=^c, 

9?i 3pi 3?i 3pa 3?i 

V 9gi 9pi dqi dpz 9ji 

18—2 



and consequently 


8(P„P.) 


8(7, ff) 

8 to, ft) 


But since V=k 


; is an integral, we liave 




or 


iy.jy.jy-jv. 


•0, 



and therefore -i ' — ^ = f*- 

This eqaatioQ shews that/idg,+/,dga is the perfect differential 
fiiQCtioD 5(g„ 5„ A, c) : and the result derived above from the theoi 
last multiplier shews that dd/dc = constant is an integral. 

Moreover, we have 
and therefore 

But obtaining d/l/dA and df^dh in the same way as dfxjdc and 9/ 
found, we have 

dv dv 

Conseqaently di = ^dqi + ^ dg,, 

or ( = iTv + constant, 

which completes the proof of Liouville's theorem. 

Example. In the problem of two centres of gravitation (§ 63), if (r, r^ d 
radii vect«res to the centres of force, and {6, ff) the angles formed by r, r' 
line joining the centres of force, obtain the integral 

rV^^ - 2c (n cos d +/ ooH *)= constant, 
and hence complete the solution by Liouville's theorem. 



their frUegreU-Tnvariants 277 

tegral-invariants whose oi-der is equal to the order of the 

ty of the laBt multiplier of a aystem of differential equations is 
ith that of the integml-invariants whose order is equal to the 
lystem. 

t-^' (-'.^ '). 

%, ..., Xk) are given functions of {xnX^ xt, t),he a system 

lifferential equations ; and let ue find the condition which must 
1 order that 



///•••/* 



r,£a^ ... &ct 



tegral -invariant, where ^ is a function of the variables. 

, . . . , Ci) be any set of constants of integration of these equations, 
olving the equations, (x„ x,, ..., Xk) can be expressed in terms 
Ci, *). Then we have 

JMl^S^...S.,.jjj...fM\f^^^-^;-^^Sc.Sc,...i,,. 

: the condition of integral-invariancy is 

d_ ij^ d(x„Xt arQ ) _ ^ 

dt\ 3{o„c„ ...,Ci)J 

L.a=a. ■•■,«>) ,J^ 4 ^('^' '^' ■■■' ''r-i, -^r. Jr+i Jg*) _ q 

i,c„...,ct) ,=i a(c,.c„ ...,Ct) " ' 

dM d(w^,X, Xt) .^ISXrd (X„ X,, ..., Xt) ^ Q 

dt 3(0,, C,, ..., Ci) r=l 3«r 3(Ci, C,, ...,Ci) 

dt r-l oXt 

that M must be a last multiplier of the system of equations. 
It gives immediately the theorem that for a dynamical system 
is determined by the equations 

dq^dH dpr__dH^ 

lA'dfr' 'dt dqr (J-^l. 2. ...,n), 

nyfuncHon ofiq^, ?,, ..., q„,pi,pt, ..., Pn, t), the expression 



jjj—j^Sg, ... 83nSp,%...Sp„ 



^rinvariant ; since in this case unity is a last multiplier. This 
importance in the applications of dynamics to thermodynamics 



278 HamilUmian Systems and 

JExaiaj^. For a sjatem with two degrees of freedom, let the energy-inte 
solved for p, take the form 

Shew that, for tr^ectories wbiob oorrespoad to the same value of the o 
energy, the quaDtitj 

is iodependent of ( and also of the choice of coordinates : and hence ebev 
tr^jectoriee of the problem can be represented as the atream-lines ia the stea 
of a fluid whose density ia 'dH'jik. 

123. Reduction of differential eqitatwns to the Lagrangian foi 
Another question to which the theory of the last multipliei 
applied is the following : To tind under what conditions a given a 
ordinary diflfereutial equations of the second order 

?*-/*(?., 9. 9»> 9i, qt 9«) (*-l. 2, 

is equivalent to a Lagrangian system 

1(1) -I- <'-.^. 

where £ is a function of (jt, q^ fm fn fi- •■■■^n. 0- 

If these two systems are equivalent, the equations 

must evidently reduce to identities when the quantities q^ are rep 
the expressions ft ; and therefore the required condition is that aft 
shall exist satisfying the simultaneous partial differential equations 

ioh«re ($1, (^1, ..., qn, qi,q3, •■■, 9n> ') ixre regarded a^ the independent < 

When n = 1, the question can be solved in terms of the last m 
For the equation satisfied by Z is then 

d^'^^'^dqdq^'^dqdt dq ' 
from which we have 



"hqKd^^} dqXdqdq'^'^ dqdt dq) 



dfdq ^ dfot ' 
and therefore if we write 8'Ljd^ = M, the function M satisBea the eq 



122-124] their IntegraUInvariants 279 

but this is the equation defining the last multiplier M of the system of 
equations 

and therefore when n = 1, the determination of the f unction L reduces to the 
determination of the last multiplier of the system, 

124. Case in which the kinetic energy is quadratic in the velocities. 

When n > 1, the most important case is that in which each of the functions /^ 
consists of a part Fr which is homogeneous and of the second degree in (^j, ^,, ... , g^^) and 
a part Or which does not involve (^j, q^, ..., ^»), and it is required to determine whether 
the equations 

qr^Fr + Or (r=l, 2, ..., n), 

are equivalent to a system 

iC€)~Wr^' (r-l, 2, .... «). 

where T is homogeneous and of the second degree in (^i, ^2» ••m Sin) ^^^ ^^^ involves the 
variables {q^ ^2» •••> ?!»)> ^^^ (Qu %> •••> 60 are functions of (y^, q^, ..., qn) only. 

The value of T is clearly not dependent on (Oi, O^y ..., O^), and therefore we can 

consider the problem in which ((?|, O^, ..., O^) are zero, i.e. the problem of finding 

a function T such that the equations 

qr^Fr (r«l, 2, ...,«), 

are equivalent to the system 

d /dT\ dT ^ /ION 

dtWy^r^ (r=l,2,...,n). 

The condition for this is the existence of a function T satisfying the partial differential 
equations 

n 927' i» gJTT 97» 

- ° Y^*+ 2 ^4^^*-^=0 (r=l,2,...,n). 



*=l ^qr^k k=l ^r^qk ^ H 

n 

Since Fk is homogeneous, we have 2 q,dFjildq,=2Fiiy and therefore 



i» 92>7» » n 9/»^ 927» 

•=i^r Vk^i 9?. 9?*/ .-r*' "fc-l 9^f9^r 9^* 

But since dF/dq^ is homogeneous, we have 

dFM_ » . a^/jfc 
dqr~i=i^*dqrdq,' 
and therefore 

k=^idqrdqi, * •-! *9^r \ it-l 9?. W *=l9^r9^fc' 

The equations to be satisfied by T may consequently be written 

M^i^^qr V k=i ^q, ^qkJ k^i^qr^qk —i^qr^q, ^qr 

B=rdqr y k^idqt dqi, dqj V k^x^r ^qk ^qJ 



/ 



280 HamiUonian Systems and [ch. x 

and evidently these can be replaced hj the equations 

KlWri^i" (-1, 2. .... n). 

Thus, writing /,. for (Fy+ Or), we have the theorem that if the system of equations 

gr=fr (r»l,2, ...,n), 

where fr consists of a part which is homogeneous of degree two in the velocities and a 
part which does not involve the velocities, is reducible to the form 

^©•"§1°^' (r-l, 2, ...,«), 

then T must be an integral of the system 



Miscellaneous Examples. 

1. In the problem of two centres of gravitation, the distance between the centres of 
force is 2<;, and the semi-major axes of the two conies which pass through the moving 
particle and have their foci at the centres of force are (q^y q^). Writing 

^i~ ji«-c* dt ' ^* c«-j,a dt * 
shew that the equations of motion are 

dqr_dHr dpr__dHr . . ^. 

dt'dpr' dt~ dqr ^ • ^' 

where «^_l_2l!z^ « 24.1.^1:2^ „ 8_ Jh ^- 

and f4 and fi^ are constants. 

2. Shew that 

^^i^Pi^qj^Pj, 



m 



where the summation is extended over the ^n (n- 1) combinations of the indices i and^', 
is an integral-invariant of any Hamiltonian system in which (qi, q^, •", q^ Pn P^i •••iPw) 
are the variables. (Poincar^.) 

3. In the problem defined by the equations 

dt dpr' dt^^dqr ^'■-A» *;» 

where ^^QiPi-^tP^-^i-^^^y 

shew that ^ — -**= constant 

is an integral ; and hence by LiouviUe's theorem (§ 121) obtain the two remaining 
integrals 

(?i<?2 —constant, 

1 ^^ 9i "^ ' + constant 



I 

t 

} 

t 



their Integral-Invariants 

is a laat multiplier of a ajatem of diSbrential equations 
dxj dx^_ dx^ dx 

! equation 

/(*), x^, ..., x„ a;)=Conetant 

[it«gral, and if an accent auueied to a function of ^j, x^, ...,z 

; x^ baa been replaced in the function bj its values found froD 

'l(?ffix^' is a last multiplier of the reduced system 

dx\ dxa dx^^i dx 

U), u,, ..., Ua) be n dependent variables, and let /„ /j, ..., 
intial eipreesioDS defined hy the equations 

, ..., Vn) are functions of I such that 

differential, shew that the functions (fi,«,, ..., vj satisfy a 
jquiitioDa, which will be called the system adjoint to the sj 

4-0 (r. 

Qotee the erpreaaion 







a (5?,, 


)-w. 








(.'■• 


my given 


function of (g, 


i.j,,..., 


i., ?i. 


9i. 


■...?,. 


,t\ 


fdiewtha 


sntial equations 
















m-^^^' 


.,) = 


=0 




(»■ 


> itself. 

















tt the converse of thia Utter theorem is also true. 



CHAPTER XI. 

THE TRANSFORMATION-THEORY OF DYNAMICS. 

126. Contact-transformations. 

We have seen in Chapter III. (^ 38, 42) that the integration of a 
dynamical system which is soluble by quadratures can generally be effected 
by transforming it into another dynamical system with fewer degrees of 
freedom. We shall in the present chapter investigate the general theory ' 
which underlies this procedure, and, indeed, underlies the solution of all 
dynamical systems. 

Let (gi, gaj •••, ?n> Pi* Pay •••» Pn) be a set of 2n variables, and let 
(Qii Qai •••» Qm -Pi> -^21 •••, -Pn) be 2n other variables which are* defined in 
terras of them by 2n equations. If the equations connecting the two sets of 
variables are such that the differential form 

PidQi + PidQi + . . . + PndQn -pidqi-pidqi- .,.- p^dq^ 

is, when expressed in terms of (gi,ga, ..., S'nj Pi> i^> •••>l>i») and their differ- 
entials, the perfect differential of a function of (gi, g,, ..., g»,2>i» />«> •••> p«), 
then the change from the set of variables (ji, q^y ..., qniPitP^* -"jPh) ^o ^-he 
other set {Qi, Qa, ..., Q^, P^, Pj, ..., P„) is called a contact'transformation , 

(Q \J j(tu\l^(ffil '^^ contact-transformations thus defined (which are the only kind used in Dynamics) 
* * ' / u yj^ *^ * special class of Lie's general contact-transformattoTis, which are transformations from 
^,^7 ' ' a set of (2n + l) variables (q^, q^, ..., ^n, Pi, ..., p», z) to another set (§i, §2> •••> $»» 

Pi, Pj, ..., Pn> ^> for which the equation 

dZ- P^dQj^-P^dQi-,,. - PndQn=p {dz-p^dq^-p^dq^- ... -/Jnrfj'J 

is satisfied, where p denotes some function of (qnq^^ '•'> qnjPu Pa •••>P»» *)• y 

If the n variables {Q^ Qj, .... Q„) are functions of (gi, ga* •••»?») only, 
the contact-transformation from the variables (?i, ..., S'nj Pi> •■•! Pn) to the 
variables (Qj, Qa» •?•> Qni-Pi» •••>-?»») is coiled an extended point-transformaHony (^M^'vh 
the equations which connect (ji, ja* •■•» ffn) with (Qi, Qa> •••! On) being in this /** -^ 
case said to define a point-transformation. 



V;Kj7.». <^ f ' -•^-' "*^//"-> ■ 



/ 



f 

I 

i 

/ 



126, 126] The TransformaiionrTheory of Dynamics 283 

The definition of contact-transformations may be thus expressed : a con- 
Uict'traTisformation leaves the differential form Xpr^r invariant^ to the Tnodulus 
of an exa^t differential. 

From the definition it is clear that the result of performing two contact- 
transformations in succession is to obtain a change of variables which is itself 
a contact-transformation ; this is generally expressed by the statement that 
contact'transformatione possess the group-property. It is also evident that if 
the transformation from (ji , ft, • • • » ?ni Pi> • • • » Pn) to (Qj , Qj, . . , , Q», Pi, . . . , PJ 
is a contact-transformation, then the transformation from (Qi, Qa, .... Q», 
Pi, Pj, ..., P„) to (ft, ft, ..., ft, Ply ..., jp») is also a contact-transformation; 
this is generally expressed by saying that the inverse of a contact-traTisforma' 
Hon is a contact-transformation. 

Example 1. Shew that the transformation defined by the equations 

^ = (29)*c*cofljt>, 

.P-(2y)*«-*8inp, 
is a contact-transformation. 

In this case we have 

PdQ - pdq^{2q)^ sin p {(2^)~* cos pdq - {2q)^ Binpdp} -pdq 
—d(q ainp cos p - qp\ 
which is a perfect diflferential. *^ 

Example 2. Shew that the transformation 

e^ioggsinp), f>cta^t^^j^-^j(^t^^h] 

[P^qcotpy 
is a contact-transformation. ^ 

Example 3. Shew that the transformation 



is a contact-transformation. 



|«-log(l+j*co8^), T^.u^o^ cll^^lr'^t-h) 

(P=2(l-hj*cosji)2'*sin;?, ^ ' 



126. ThA eaplicit expression of contact-transformations. 

Let the transformation from variables (ft, ft, ..., ft > Ihi •••>!>«) to variables 
(On Qiy •••> Qi»» Pi> •••, Pfi) be a contact-transformation, so that 

i (PrdQr-prdqr)=-dW, 
r=l 

where d IT is a complete differential. 

From the equations which define (Qi, ..., Q^, Pi, ..., P^) in terms of 
(ft , ft , . . . , ft, jpi, . . .,p^) it may be possible to eliminate (Pi, Pa, . .., PmPi , • -Mi^n) 
completely, so as to obtain one or more relations between the variables 

vWi» yli9 •••> vln» ft* •••» ft) J 



t^'-t^- 


... -.fi- 


<i,.4|.*«, 


+ 


must have 










P, 


=!!-■ 


an. 


^, an. 




P, 




v'"' 
"-8^- 


, an. 






284 The Tran^ormation-Tkeory of Dyimmica 

let the number of such relations be k, and let them be denoted by 

"rta, q, 9.. 0. «n)-0 (r -1, 2 k 

Since the variations {dq^, dq,, ..., dqn, dQ, dQ„) in the equ 

I {P,dQ,-p4qr) = dW 
are conditioned only by the relations 

(•■-1,: 



.")! 



where (X, , X,, . . . , X^) are undetermined multipliers aod where W ie 
°f (?i. 9st •••. 3n. Qi. Qi, ■■■. Qn)- The equations (A) and (B) a 
equations to determine the (2n 4- k) quantities 

(Q„...,Q„,P„...,P„,X„...,Xt) 

in terms of (ji qn.Pi, ■■•,Pn)- These equations may therefore I 

as explicitly formulating the cotUact-traneformaiion, in terms of ti 
(W, il,, n„ .... Hfc) which characterise the transformation. 

Conversely, if (IT, fli, fig, ..., fit) are any (i + 1) fuoctions oft! 
(?i. ?i. ■■-I ?n. Qi. ■". Cn). where k^n, and if 

(Qi,e Cn.-Pl. - ,-Pn,X., -At) 

are defined in terms of ($i, q^, .-, qmPii ■■■• pn) by the equations 
(nAq>,q. ?„«.,e. Q,) = (r-1, 

dq, Sq, dq, 

then the transformation from {q,, g„ .... }„, p,, ..,, p„) (o (Qi, 
Pi, ..., Pn) i* o amtacKrona/ormaiitm ; for the expression 

i (P,dQ,-y,d3,) 

becomes, in virtue of these equations, dW, and so ts a perfect di9< 



126, 127] The Transformation-'Theory of Dynamics 285 

Example. If C=(2^)*it"*cos/>, P^{2q)^k^mip, 

shew that P=^, ^ -g^, 

where W^\Q{^h-Jc^Q^)^^-qGO%'^ {**§/(2^)*}, 

so that the transformation from {q^ p) to (Q, P) is a contact-transformation. ^ 

127. TAe bilinear covariant of a general differential form. 

Now let {wi, ajj, . . . , a?„) be any set of n variables, and consider a differential ^^^MtS^fffit^ 
form ^*%0t- 

Xidxi 4- X^dx^ + . . . + XndXny 

where (X,, Xj, ..., Xn) denote any functions of (iCi, x^, ..-,a:n); a form of this 
kind is called a Pfaff's expression in the variables (ooi, x^, ..., Xn). Let this 
expression be denoted by da, and write 

where S is the symbol of an independent set of increments. Then we have 

B0a - d^3 = 8 {X^dx, + X^da:^ + ... +Zncirn)- d(ZiSiCi -f-Z2&ra+ ... + Z„&cJ 

=^BXidxi + .,. -{'SXndxn + XiSdxi+ ... + X^Sdx^ 

— dXi Bxi — ..." dX^Bx^ — JTid&i?! — ... — X^dBx^. 
« 
Using the relations BdXr = dBx^y which exist since the variations d and B 

are independent, and replacing dX^, BX^ by 

_'d^, + ... + _rd^„, ^' 5^. + ...+_'Sa.„ respectively, 

we have i'^A;^/'^^*"*aS^ci — d^«*= 2 2 ai^dxiBxu 

where Oi; denotes the quantity dXijdxj — dXj/dxi. 

Let (yi, ya> •••» yn) be a new set of variables derived from (a?i, x^, ..., x^) 
by some transformation ; let the differential form when expressed in terms of 
these variables be 

Fidyi + Fadya + . . . + F„dy„, 

and let the quantity dYi/dyj — dYj/dyi be denoted by bij. Then since the vd.fn^SOt 
expression B0d'-d0s has obviously the same value whatever be the variables 
in terms of which it is expressed, we have 

n n n H 

2 2 oudxiBxj^ 2 S bijdyiByj. 
The expression 2a{j(£r{&t7j is, on account of this equation, called the 

n 

bilinear eovariant of the form 2 Xrdxr. 



286 The Transformation-Theory of Dyna/m 

128. The conditions for a contact-transformation exprest 
the bilinear covariant. 

Let (Qi, Q, Q„ P,, .... P^) be variables connected 

9n. Pi, ■■•, Pm) by a contact-transformation, 30 that S P^t 

X Prdq, by aa exact differeotial. 

It is clear from the last article that the bilinear covariant 
form is not affected by the addition of an exact differential t( 
it depends only on the quantities dXi/dxj— dXj/da^i, which a 
the form is an exact differential : and we have shewn tl 
covariant of a form is transformed by any transformation i 
covariant of the transformed form. It follows that the bilini 

the forms X PrdQr and 2 Prdqr are equal, i.e. that 

i iSP,dQr-dPrBQr)= i (Sp,dqr-Sq,dp,); 
BO that if the transformation from 

(quq,.-..q..p,.-.p.) to (Q,, Q„ .... Q». A, -. 
IS a contact-transformation, the expression 

2 (Sprdq, — Sqr:dpr) 
is invariant under the transformation. 

Example. For the transformation defined hj the equations 

we have 

1dP-=(ig)-^ i* ainpdq+m)* i^ coapdp, 
B§'«(2g)~*i"*co9f a9-(2y)*)f*8inp«p, 
3P-(2y)"*i*sinpSy+{2y)li'co8f «p, 
dQ ={2g)'^i~^ COB pdq-[2q)^t-hiap dp. 
By multiplication we have 

dPSQ- SPdQ= - sin* p(dq6p- ISqdp)+CM^p{dp iq - flp. 
= dptg-Spd<i, 
and consequently the transformation is a contact-transformation. 

129. The conditions for a contact-transformation in tern 
b racket-expressions. 

We shall now give another form to tbe conditions that g 
from variables (ji,?,, ...,5„,/>,, ...,/>«) to variables (0,, Q„ ... 
may be a contact-transformation. 



The TraTigformation-Tkeory of Dynamicg 287 

i -■-. 9ii. i*!) ■■■>Pn) ftfs ^''y functions of two variables (w, v) (and 
ly number of other variables), the espressioD 

Lagrange's bracket-expression, and is usually denoted by the 



(^i, 9„ .... 9h, Pi, .... j)„) are any functions of 2n variables 
)„, Pi, .... P„), then in the expression 

S (dp^hq^ — hprdq,) 

;e dpr by 

' for the other quantities ; we thus obtain, on collecting terms, 

r-l *.l 

lummation on the right-hand side is taken over all pairs of 
, «,) in the set (Q„ Q Q„, P,, .... fj. 

e traDsformation from the valuables (j,, j,, ..., q„, pi, ..., p^) to 
) (Qi, Q , Qh, P,, ..., P.) is a contact-transformation, we 

is for all types of variation S and d of the quantities ; comparing 

ve equation, we have therefore 

([P.-,Pt] = 0, [ft,QJ=0 ii,k = 1.2.....n), 

[Qi.Pk\ = {i,k = 1.2 n;t%k), 

[Qi.Pi] = l (i = l,2,...,n). 

iTf be regarded as partial differential equations which must be 
h' ?»' ■■■> 3b' Pi. •■■•Pn)< considered as/unctions of 

(«„«. «.,-p, -P.) 

! the transformation from one set o/ variables to the other may be 
nsformatian. These equations represent in an explicit form the 
nplied in the invariance of the expression 

S {dp^Zq^ - Sprdq^). 



288 The TransformationrTheory of Dynamics 

130. Poisson'a bracket-eaipreaaions. 

We shall next introduce another claaa of hracket-expression) 
intimately connected with those of Lagrange. 

If u and V are any two functions of a set of variables {q„ 
Pi> •■■> Pn)> *'h6 expression 

5 /du dv du Sv\ 

is called the Poisson'a bracket-expression of the functions u an 
denoted by the symbol (m, v). 

Suppose now that (ui, «,, ..., Um) are 2n independent fund 
variables {qi.qt, ...,qn<pi. ■■■.p»). so that convei-sely (},, q„...,q, 
are functions of («!,«,, .... «»,). There will evidently be sonn 
between the Poisson-brackets (u,, u,) and the Lagrange-brackt 
this connexion we shall now investigate. 

We have 

(=1^ •■ "" i-i (■.! j-i VS^i cip( dpi dqiJ \dut du, du 

Now multiply out the right-hand side, remembering that 

'ip'ji and %'^^£' 
(-1 oji &u, (=1 opi du, 

are each zero if * £ j and unity if i = j ; and that 



'^^ and 2 



3u( 9' 



3j),- du, 



(=1 3?i 3«t 
are each zero ; the equation becomes 

,=i "■ " i=i\dpidu, dqidu,/' 

and consequently 

Sn 

2 («t, «r) [«ti M.] •= when r < «, 



while 



£ (M(, Wr)[«t, «r] = l- 



But these are the conditions which must be satisfied in ord 
two determinants 

[w,, M,] [u,, u,] ... [m,, w^] and (it,, u,) {m,, it,) ... (u^ 



[w».«i] [«»."».] 



(Wl. «*,) («*.. 



130, 131] The Transforrnxxtion-Theory of Dynamics 289 

may be conjugate, i.e. that any element in the one should be equal to the 
minor of the corresponding element in the other, divided by this latter deter- 
minant ; the product of the two determinants being unity ; and thus the 
connexion between the Lagrange-brackets and the Poisson-brackets is expressed 
by the fact that the determinants formed from them are conjugate. 

Example 1. If /, <^, ^ are any three functions of {q^^ q^, ..., q^ Pu •>? > Pn) shew that 

Example 2. If Fj ♦ are functions of (/i, /j, ...,/*), which in turn are functions 
of (?i» ?8> — > ?n, Pi9 —. Pn)> shew that 

where the summation is taken over all combinations fn fi> 

131. The conditions for a contact-transformation expressed by m^ans of 
Poisson's bracket-expressions. 

Now let (Qi, Qa, ..., Qn> -Pit •••> -Pn) denote 2n functions of 2/i variables 
(?ii ftj •••, 9ny Pif "'iPn)t ^0 shall shew that the conditions which must be 
satisfied in order that the transformation from one set of variables to the other 
may be a contact-transformation may be written in the form 

(Pi, Pj) = 0, (Qi, Qj) = (i, j = 1, 2, . . . , n). 

{Qi,Pj) = (t,i = l, 2. ....n;t5j). 

(Qi.Pi) = l (i = l,2,...,n). 

For we have seen in § 129 th»t the conditions for a contact-transformation 
are expressed by the equations 

[Pf,P,] = 0, [Qi,Qj] = (i,j=l,2, ...,n), 

[QuPj]=^0 (i,i = l,2,...,n;t<j), 

[QnPi] = l (i=l, 2, ...,n). 



Hence the relations 



2n 

2 (ut, Ur) [wt, w,] = - (r > s\ 



<=i 



of the last article become 



while the relations 



(Qi,Qj) = 0, (Pi.Pj) = (i,j = l,2....,n). 

(Pi,Qi) = (i.j = 1.2.....n;i>j). 



2n 

2 ('Ut,Ur)[Ut,Ur] = l 
t=l 



give (Qi,Pt) = l (i = l, 2, ...,7i); 

the theorem is thus established. 

w. D. 19 



\ 






s ' 



290 The Transformation- Theory of Dynamics 

Example 1. If ($,, $„ .■•.^m Pi,.-,P^asaaoDaa'Aa&wi\h{qi,qt, ■■■t7i>if 
\>j a contact-traDBformatioD, shew that 

" /?* 3^' _ 9* 3^\ ■ /a^ 3f 3$ &+\ 
„A3«r3A Sv5eJ"r:tV9?r3pr"3pr¥J' 
BO that the Poisson-brackets of an; two functiana ^ and ^ with respect to the ti 
Tariabtes are equal 

Example 2. If [Q,, $„ ..., Q.) are given functions of (?,, jg, ..., Jhi Pi> — 
satisfj the partial differential equations 

(«r, ft)-0 Cr,.=l,2, 

Bhew that n other functionB (P„ P„ ... , /"„) can be found such that the transl 

from (y,, 5„ ,.,, y,, p, p,) to (§,, §„ .... 9«i A' — i ^J is » oooti 

fonnatioD. 

132. The sub-groups of Mathieu trangformationa and etctendei 
trameformations. 

If within a group of transformations there exists a set of transfoi 
9uch that the result of performing in eucceseion two transformation 
set is always equivalent to a transformation which also belongs to the 
set of transformations is said to form a sub-group of the group, 

A sub-group of the general group of contact-transformations is e 
constituted hy those transformations for which the equation 

is satisfied. These transformations have been studied by Matbieu*. 

They are essentially the same as the transformations called " homogeneouc 
tr ansformations in {q^, g,, .... y„ «,, .... y,)" by Lie. 

In this case, we see from § 126 that <Q„ Q„ .... Q„, Pi, ..., P„) a 
obtained by eliminating (X,, X,, .,., X*) from the (2n +t) equations 

n,fe,?, ?.,«, «.)-0 (r-1,2,. 

<r-l,2, . 

(r.1,2,. 

From the form of these equations it is evident that if {p\,pa, ■■■ 
each multiplied by any quantity ft, the effect is to multiply (Pi, P,, 
each by /* ; and therefore (P„ P,, .... P„) must be homogeneous of 
degree (though nob necessarily integral) in (pi.pt, ...,Pn)- 

A sub-group within the group of Mathieu transformations is con 
by those transformations for which (Pi, P„'..., P^ are not only home 
• Journal de Math. in. {1871). 



!..?. ?.. ft. ■ 


.•,«.)-o 


, m. da 


^. an. 


Bg, "• Bg, 


30, 



131-133] The Transformation-Theory of Dynamics 291 

of the first degree in (pi, jOa, ••• > i>n) ^^t also integral, i.e. linear, in them ; so 
that we have equations of the form 

n 

^r = 2 Pkffkiqu 52, ..., gj (^ = 1, 2, ..., n). 

Substituting in the equation 
and equating to zero the coefficient of ^jt, we have 

n 

2 /r*(?i» ?i» •••» qn)dQr^dqk (A=l, 2, .... tl), 



n 



80 (ji, g,, ..., ?n) ^6 functions of (Qi, Q«, ..., Q^) only, and 

/r* = g^* (r, A:= 1, 2, ..., n). 

It follows that transformations of this kind are obtained by assigning 
n arbitrary relations connecting the variables (^i, ?«, ..., qn) with the variables 
(Qu Qj* •••> Qn)* ci'i^d then determining (Pi, P^, ..., Pn)from the equations 

P,= 22>4X* (r = l,2, ...,n). 

These transformations are e xtended point-transformation s (§ 125). 

i» n 

Example. If 2 PrdQr— 2 Prdgry 

r=l r=l 

shew that 2 pj, ^=0, 2 p.^^Pr. 

133. Infinitesimal contact-transformations. 

We shall now consider transformations in which the new variables 
(Oil Q21 •••, On, Pi, •••, Pn) differ from the original variables (g,, gg, •••, qn> 
Pit ••.,jPn) by quantities which are infinitesimal. Let these diflFerences be 
denoted by (Ag^, Aq^j, ..., Aj^. Api, ..., Apn), where 

Ap, 
and A^ is an arbitrary infinitesimal constant ; so that 

Qr=qr + £iqr=-qr + <f>r^t) / ^i o n 

Pr=Pr + ^Pr=Pr+irr^t) ^r - 1, ^ ..., n), 

and the transformation is specified by the functions 

(<f>if <f>2> •••» <f>n, V^i, V^aj •••, V^n). 

• 

Now suppose that the transformation is a contact-transformation. Then 
we have 



>r = '^r(?i, 92' •••'?»», Pi, ...,Pn)A« J ^^ " / > •••.»;> 



2 {PrdQr-prdqr) = dW, 






19—2 



V 



292 TJie Transformation'Theory of Dynamics [en. xi 

where W is some function of (^i, q^, ..., Jn»l>i, •••i/>n); ov 

n 

or A^ 2 {'^rdqr+Prd(f>r) = dW, 

It is evident that the function W must contain A^ as a factor : writing 
Tr= Z7Af, where J7is some function of (5, , Jj, ..., ?n»l>ii •••,i>n) the equation 
becomes 

n 

2 {ylrrdqr+Prd<f>r) = dU, 
Hence' we have 

2 (irrdqr - <^rdpr) = d U7- 2 Pr(f>r ) 
r=l \ . r=l / 

= -dJK'(3i, ga> ..., ?n, Pit •.-, Pn) Say, 



and therefore 



•^'=1^' ^'=~aY, (r=l, 2, ....«). 



Thus ^Ae most general infinitesimal contact-transformation is defined by the 
equations 

Q, = g, + g- A^, P^^p^^ — M (r = l, 2, ...,7i), 

where K is an arbitrary function of (q^ g,, ..., qn, Pi, •-^tPn), CL^d A^ is an 
arbitrary infinitesimal quantity independent ©/'(q'i, Ja, •••, ?n> jPn ••• » Pn)- 

The increment in any function /(^i, ?2, ..., ?n» Pi, "MPn) when its argu- 
ments (5i, ja, ..., qnyPu '",Pn) a^c subjected to this transformation is 






or (f.K)At; 

on this account the Poisson-bracket (/, J^) is said to be the s ymbol of, 
most general infinitesimal transformation of the infinite group which consists 
of all contact-tr ansformations of the 27i variables (ji, q^, ..., qn,Pi, ...,/>n)« 

134. The resulting new view of dynamics. 

The theorem established in the last article leads to an entirely new 
conception of the nature of the motion in a conservative holonomic dynamical 
system. For the motion is expressed (§ 109) by equations of the type 

dq^ JH dp dH 
' dt dpr' dt dqr ^ i.,i,...,n), 

and from the last article it follows that we can interpret these equations as 
implying that the transformation from the values of the variables at time t 



133-135] TTie Transformation'Theory of Dynamics 293 

to their values at time t-vdt is an infinitesimal conta,ct-transformation. The 
whole course of a dynamical system can thus be regarded as the (gradual self - 
unfolding of a conta^-transformation '. This result, taken in conjunction with 
the grouprproperty of contact-transformations, is the foundation of the 
transformation-theory of d}niamical systems. 

From this it is evident that if (ji, ?a, ..., qn^Pi, --MPn) are the variables 
in a dynamical system, and («!, o,, ...,«»» A> •••»^n) are their respective 
values at some selected epoch ^ = ^o» the equations which express (q^i, q^, ..., ?nt 
Ply •",Pn) ill terms of (oi, a,, ..., On, A» •••> fim t), (and which constitute the 
solution of the differential equations of motion) express a contact- transforma- 
tion from (fli, Ota, ...,an, A/-->/8n)to(gi, g„ ...,qn,Pi, ...,i>n); in this t is 
regarded merely as a parameter occurring in the equations which define the 
transformation. 

136. Helmholti^s reciprocal theorem. 

Since the values of the variables (ji, Jj, ..., ?n, Pu ••.,i>n) of a d)niamical 
system at time t are derivable by a contact-transformation from their values 
(ai, ffg, ..., On, /8i> •••! /8n) at time t^, we have (§ 128) 

S (A;)iSji-Sp,Ag<)= i (A/8«8ae - S/9,.Aac), 

where the symbols A and h refer to increments arrived at by passages from 
a given orbit to two diflferent adjacent orbits respectively. 

Now suppose that h refers to the increments obtained in passing to that 
orbit which is defined by the values 

(ttj, tta, ..., a„, ^1, ^a, ..., ^r-i, fir + ^^r^ ^r+n •••» fin) 

at time ^o; and let A refer to the increment obtained in passing to .that orbit 
which is defined by the values 

(?i» Jai •••, ?n,Pii '"yPi-iiPM+^PifPi+i, ''-,Pn) 

at time ^ ; then the above equation becomes 

Ap,Sg, = -S^^Aa,., 

so the increment in q^ due to an increment in fir (when Oj, Oj, ..., a„, 
^1, '"7 fir-it fir+i, -"t fin are not varied) is equal to the increment (with sign 
reversed) in Or corresponding to an increment in p, (when gi, Jj, ...,?n,.Pi, ..., 
Pt-i, Pf+i, •••» Pn are not varied) equal to the previous increment in fir. 

This result can for many systems be physically interpreted, as was 
observed by Helmholtz*; for a small impulse applied to a system can be 
conveniently measured by the resulting change in one of the momenta 
(Pit ••• , Pn\ and the change in Or due to a change in p, can be realised in the 
reversed motion, i.e. the motion which starts from some given position with 

* Journal fUr Math, c. (1886). 



294 The Transformation- Theory of Dynamics [ce. xi 

each of the velocities coirespooding to that position changed in Bign, so that 
the subsequent history of the system is the same aa its previous biator)', but 
performed in reverse ofder. We can therefore state the theorem broadly 
thus : the change produced in any iiUerval by a small initial impulse of any 
type in tJie coordinate of any other (or of the same) type, in the direct motion, 
is equal to the change produced in the same intenal of the reversed motion in 
the coordinate of the first type by an equal small initial impulse of the 
seccmd type*. 

Example. In elliptic motion under a centre of force in the centre, if a smalL velocity 
&v in the direction of the normal be communicated to the particle as it is paasing through 
either extremity of the major axis, shew that the tangeiitial deviation produced after 
a quarter-period is fi~' iv, where /i is the constant of force. Shew alao that a tangential 
velocity An, communicated at the extremity of the minor axle, produces after a quarter- 
period an equal normal deviation |i~' iv. (Lamb.) 

136. The transformation of a. given dynamical system into another 
dyTiamical system. 

It appears from § 116 that if a Hamiltonian system of dififereotial 
equations 

d, 8ff dp,__dH 

a d^,' dt- dfr * ■ '■ 

is transformed by change of variables, the system of differential equations so 
obtained will still have the Hamiltonian form 



dt dPr' dt dQ, 



(r-l, 2, . 



/P,S«,K 



provided the new variables (Qi, Q„ ..,, Q„, Pi P„) are such that 

?, + P^<2.+ ... +P„SQn 
is an integral-invariant (relative or absolute) of the original system. 

A transformation of this kind is, in general, special to the problem 
considered, i.e. it transforms the given Hamiltonian system into aoother 
Hamiltonian system, but it will not necessarily transform any other arbitrarily 
chosen Hamiltonian system into a Hamiltonian system. Among these 
transformations however are included transformations which have the pro- 
perty of conserving the Hamiltonian form of any dynamical system to which 
they may be applied : these may be obtained in the following way. 

We have seen {§ 115) that 

' Cf. Lftmb, Proc. Lond. Math. Soc. la. (1898), p. 144. 



136, 136] The Transformation'Theory of Dynamics 296 

is a relative integral-invariant of any Hamiitonian system. Let (Qi, Qj, . . . , Qn, 
Pi, -.., Pfi) be a set of 2n variables obtained from {q^y q^, ..., qn,pi, ...,i?n) 
by a contact-transformation, so that 

i PrdQr- i Prdqr=-dW, 

where dTT denotes an exact diflFerential. The equations which define the 
transformation may involve the time, so that (Qi, Q,, ..., Q„, Pj, ..., P^) are 
functions of (ji, ?2» •••» ?nii>i» •••! JPm OJ but in the variation denoted by d 
in this equation the time is not supposed to be varied*: if ^ is supposed to 
vary, the equation becomes 

2 PrdQ^ - i p^dqr=^dW+ Udt, 

r=l r-l 

where U denotes some function of the variables. 

Now the variation denoted by S in the integral-invariant is a variation 
from a point of one orbit to the contemporaneous point of an adjacent orbit ; 
if therefore we regard the variables as functions of (oi, a,, ..., Ojn, 0» where 
(Oi, Oa, ...i Oan) ^^^ the Constants of integration which occur in the solution of 
the equations of motion, the variation 8 is one in which (ou a^, •••, ^m) ^^ 
varied but t is not varied : we have consequently, as a special case of the last 
equation, 

2 PMr- iprSqr^BW, 
r=l r»l 



and therefore 



f 2 PrBQ, 



is a relative integral-invariant; so the transformed system of differential 
equations, in which (Qi, Q^, ..., Qn, Pi, .>.,Pn) are taken as dependent 
variables, will have the Hamiitonian form and can be written 



dQr_dK 
"dP/ 



dP. 



dK 



dt dP/ dt dQr 
where K is some function of (Qi, Qj, ..., Qn, Pi, ..., Pn, t). 



(r = l, 2, ...,n), 



Hence a contact-transformation of the variables (qi, q^, --', qn, Pi, *", Pn) 
of any dynamical system conserves the Hamiitonian form of the equations of 
the system. In the case of an ordinary "change of variables" in the dynamical 
system, in which (Q,, Qa, •••» Qn) are functions of (ji, qt, ..., qn) only, the 
contact-transformation is merely an extended point-transformation. 

Example, Shew that the contact-transformation defined by the equations 

y=(2Q)*it-*C08P, ^-(2§)*ir*8in P, 

changes the system 

dt"'^' cU dq' 




296 Tke Transformation-Tkeory of Dynamics 

where ff=i(p*+i»3'}, 

into the STstem 

rf§ 3A" rfP JK 

dt'SP' dt~ 5^' 
where K-tQ. 

137. Repreaentaiion of a dynamical problem hy a differential ft 

The reason for the importFance of coDtact-traneformations in 
with dynamical problems ia more clearly aeen by the introduction ol 
differential form which is invari&ntively related to the problem. 

Let any differential form with (2n + 1) independent variableB ( 
aw+i) be 

XidiCi + Xjcic, + ... +Xm+id*»i+r, 

we have seen (§ 127) that its bilinear covariaot 



where a^j denotes the quantity {dXi/dx^ — dXjjdx^, is invariantively related to 
the form. If we equate to zero the coefficients of hxx, Sx^, ..., Sx^^+i, we 
obtain the eyetera of (2n + 1) equations 

Sa+l tmt-1 £■+! 

2 aiidxi = 0, 1 aiadxi = 0, ..., 2 a,-m+i{£ir,- = 0. 

t-i i-i 1=1 

Since the determinant of the quantities a^ is skew-symmetric and of odd 
order, it is zero, and these equations are therefore mutually compatible. 
They are known as the first Pfaff's ayatem of equations corresponding to the 

differential form S X^dx^, and from the mode of their formation are in- 

variantively connected with it ; that is to say, if any change of variables is 
made, the new variables (y, .yt, .... yn+i) being given functions of {x„Xt, .,., 
Xta+i), and if the differential form be changed by this transformation to 



and if 2 biidyt = 0, 2 6ijdyi~0, .... 2 6i,iwi<^y» = 0. 
be the first Pfaff's system derived from the differential form 

'"i ' ¥,dy„ 
then this system is equivalent to the system 



136-138] The Transformation- Theory of Ihfnanmca 

Consider now the special diEfereotial form 

Pirfg, + p^t + . . . + p,dqn — Sdt 
in the (2n + l) variables (g-,, 5,, ,.,, ^bi pi, --.Pn, 0> where fi"ia ai 
of (?!> 9i> •■•< 9n>Pi> ■•■>Pn> ')■ Forming the correspondiag quaiiti 
find that the first FfafiTs system of differential equations of this 
form is 

'" a?, 

at 
Of these the last equation is a consequence of the others : and tb< 
system of equations can be written 

dq^JH dp^^JH ^ 

dt dp, ' dt dq^ v •- 

but these are the equations of motion of a dynamical system in 
Hamiltonian function is H. It follows that the dynamical ayt 
Hamiltonian fumsUon is H ig invariantively connected with the 1 
form 

Pidqi + p^qt + . , . + p„dq„ - Sdt, 

inasmucli <u the equations of motion of the dynamical system, in te. 
variables (jCi, «,,..., a:^ t) whatever, are the first P fag's systi 
differential form 

X,d<r, + X^, + „ . + X^dxn + Tdr 
which ia derived from the form 

Pidqi+pidqj+ ... +pndq„-Sdt 
by Vt£ transformcUioR from the variables (5,, 5,, .... g„, p,, ..., p, 
variables Qe,, w,, ..., ii^, t). 

138, The Hamiltonian function of the transformed equations. 
The result of the last article furnishes another proof of the tb< 
the equations of dynamics 

di dp/ dt 9g, ^^ ' 

conserve the Hamiltonian form under all contact-transformations of 
9n.pi> •■■iPii)> and moreover it enables us to find the Hamiltonia 
K of the system thus obtained, 

dt ~dP/ dt " 3Q, *'" '" 



The Transformation-Theory of Dynamics 

let the coDtact-traDsformatioQ be defined by the equation 
|n,-0 (r-1, 

an, 
I air so, an, an. 

1,, fl,, ..., n*, F) are any functions of the variables (5, 

... «., t). 

1 these equations we have ideutically 

ce (the symbol d denoting a variation in which all tl 
g t, are changed) 

-•"'^-ar' 

air 

" '■ a< J 

perfect dlEferential dW on the right-hand side can b 
does not affect the first Pfaff's system of the differentia 
e contaet-tram/orviation transforms tits system of equatioi 

dq,_dH iip,^_SH 

dt Sp, ' di dqr 

item 

dQ, IK dP, dK 

di'dPr' dt 3Q, 



(r-1 
(r-l, 



'-^— ¥ -,;/'"ar' 

supposed expressed in terms o/iQi.Qt Qn, Pi Pi 

Transfonnations in which tlie independent variable is ch 
result of § 137 also enables us to determine those tranaf 
le set of (2»i+ 1) variables (j,, qt, ■■■.qniPn ■■■•PmO **> ° 
..., Qb, Pi, ..., P„, T) by which any Hamiltonian system 
dqr^dH dpr^_dH^ J 

dt dpr ' dt dqr 

brmed into a system of the Hamiltonian form 

dQ^_dK dPr 

dT 3i>/ dT BQ, 



7 



138-140] The Transformation' Tfieory of Dynamics 299 

For this is the same thing as finding the transformations which change the 
differential form 

Pidqi +Pidqi + . . . +Pnd'qn + Adi, 

where the variables (ji, ?a, ...,?«, Pi, ...jPn* *> ^) ^^^ connected by the 
equation 

B(qi, }a» ••-, ?n»Pl» '">Pn> t) + h = 0, 

into the differential form 

PidQi + PjdQa -f- ... + PndQn + fcd^ + a perfect differential, 

where the variables (Qi, Qa* •••, 0», ^i, -Pa, •••, -P»i 2^, k) are connected by 
the relation 

^{Qi* Qif •••! Q»» Pit •••» -Pi», X ) H-A; = 0. 

But any contact-transformations of the (2n + 2) variables (ji, ?ai •••, 9n, ^> 
Pi, •••»i>i», A) to new variables (Qi, Qj, ..., Q^, T, P^ Pj, ..., P,, A?) will satisfy 
this condition ; when the transformation has been assigned, the function K 
is obtained by substituting in the equation 

the values of (qi, jj, ..., q^, t,pi, .... |?n, A) as functions of (Qi, ..., Q„, T, 
Pi, ..., Pn, A;), and then solving this equation for k, so that it takes the form 

K(Qu Qa, ..., Qn, Pi, ..., P», ?') + A = 0; 
the required transformations are thereby completely determined. 

140. New formulation of the integration-problem. 

We have seen (§ 137) that if any change of variables is made in the 

dynamical system 

dqr dS dpr dH 

dt dp/ dt dqr ^r-l,z,...,n;, 

the new differential equations will be the first Ffaff's system of the form 
which is derived from 

Pi dqi + p%dq2 + . . . + jp»rf}» — Hdt 
by the transformation. 

Suppose that a transformation is found, defined by a set of equations 

?r = ^r(Ql, Qa, ..., Qn, Pi, •••, P|», Ol , , « 

(r = l, 2, ..., n) 

Pr'^'^riQuQ^. ...,QntPu ...,P«, OJ 

which is such that the above differential form, when expressed in terms of 
the new variables, becomes 

PidQi + PadQa+... + PndQ»-dr, 



300 The Trans/ormeUion- Theory of Dynamics 

where dT is the perfect differential of some fimction of the 
(Qu Qs. ■ ■. Qn. Pi, ■■■, -f.. 0; the corresponding first Pfaff's 
equations is 

dQr = 0. dPr = (r = l, 5 

and the integrals of these equations are 

Q^ = Constant, i*^= Constant (r = 1, 2 
so the equations 

?,-*-(«„ ft Qn.p, p„,m 

P,=^.{Q,.Q, Q..P, P..t)i "" ' 

conetHutp the solution of the dynamical system, when the ^antities { 
Q„ Pi, ..., P.) are regarded as 2n arbitrary conatajUa oj integratic 

The integraticm-problem is thus reduced to the determination i 
formation for which the last term of the differential form become- 
differential. 



Miscellaneous Examples. 

1. Shew that the transformatiou defined by the equations 

ie a contact- traDsfonnatioo, and that it reduces the dynoiaical eyatem whose 
function is J(Pi'+/>i*+X~'y,*+X~*jt') to the dynamical Hyatom whose 
function is Q,. 

2. If (^i, x^, —, ;?„) denote an; functions of (9,, q^, ..., q^ipi, ■■■>fit)i ^ 

if moreover a,„ denotes dX^x^—dXJdx„, D denotes the detarminaDt fe 
quantities 0^1,, Aa denotes the minor of a^ in D, divided by D, and u a 
arbitral; functiona of the variiibles, shew that 

r=i V^i^ ^ ^ 3y^/ (-1 *-i " 3*( 8** ' 
a Shew that for any Hamiltonian system the inte^iral-invarianta 

(jl...J6QiiQt... t^^tPi ... SPn, 

extended over corresponding domains, are equal if (^i, q,, ..., g^, p,, 
(Qi> ^i> "■> ^B' -^i) •■-' -''■•) ^-^ connected by a contact-tranaformation. 



140] The TramforrruUion-Theory of Dynamics 



301 



4. Prove that the contact-transformation defined by the equations 

hi -Xr* (2Q,)* cos Pi +X2"* {2Q^)^ cos Pj, 
?8 « - Xj - * (2ft)* cos P, + \f * (2ft)* cos Pj, 
ft-i (2Xift)* sin Pi +i (2Xjft)* sin P„ 

l^,= -i(2X.ft)*8inPi+i(2Xjft)*sinP„ 



changes the system 

where 

into the system 



where 



dt 
A'=Xift + X2ft. 
Integrate this sj^stem, and hence integrate the original system. 




/dqrJ^H dpr^dff 

/ dt dpr* dt bq^. 

dQr^dK dPr^dK 

dt ZPr' dt ■" Dft 



(r=l,2), 



(r=l, 2), 



i 



t 



CHAPTER XH. 

PROPERTIES OF THE INTEGRALS OF DVNAmXCAL S 



141. Reduction of the order of a Hamilionian ays^ 
inteffral of energy. 

We have shewn in § 42 bow the LagrangiaQ equatio^ 
conservative holoooiriic system can be reduced in order by usi 
of energy of the system. We shall require the corresponding t 
equations of motion in their Hamittonian form ; this may 1 
follows. 

Consider a dynamical system with n degrees of freedom 
Hamiltonian function H does not involve the time explicitly, s 

H-\-h = 0, 
where A is a constant, is the integral of energy of the system. 

Let this equation be solved for the variable pi, so that it a 
^{Pt,V»> ■■-,?.. ?it •■..9,.. ^)+i'i = 0. 

The differential form associated with the system is 
pidji+PidyiH- ... +p„d9, + Ad(, 

where the variables (ji, gj S., Pi-p». ■•■.?■. A, i) are con 

last equation : the differential form can therefore be written 

P)dg'i+pjdgi + ..■ ■^pndq^ + hdt — K {pi, pt, ...,p,, g-i, .... 
where we can regard {q^, j„ ..., 5,,p,, ..., p», A, () as the (2ji-t 

But the differential equations corresponding to this form ai 

dqi dp,' dq, dq, 
<^_SK rfA_ 
dqi ~ dh ' dq. 

The last pair of equations can be separated from the rest 
since the first (2n — 2) equations do not involve (, and A i 



(r: 



en 



141,142] Properties of the Integrcda of Dynamical 8y 303 

The original differential equations can therefore be replaced by the reduced 

system 

dqr^dK dp,_ dK 

d^^'df/ d^r Wr (^-2,3,...,n), 

which has only (w— 1) degrees of freedom. 

This result is equivalent to that obtained in § 42, as can be shewn by 
direct transformation. 

Example, Consider the sjBtem 

^L<^_S dpr dff 

where 

/i being a constant ; these are easily seen to be the equations of motion of a particle >»-** A^fHi-L ji'^'^^''^ 
which is attracted to a fixed point with a force varying as the inverse cuhe of the distance : 
q^ and qi are respectively the radius vector and vectorial angle of the particle referred to 
the centre of force. 

Writing H= — A, and applying the theorem given above, the equations reduce to the 

system 

dq2_dK dp2_ dK 

dqi ap,' dqi" dq^' 
where 

Since K does not involve q^ the equation ir= Constant is an integral of this last 
system, and we can therefore perform the same process again : writing K=^ - h, we have 

and the system reduces to the single equation 

dq^ih q^*\qi ^) ' 
the integral of which (supposing fi<i^)iR 

where c is an arbitrary constant. This is the equation, in polar coordinates, of the orbit 
described by the particle. 

142. The Hamilton-Jacobi equation. 

It follows from § 138 that if a contact- transformation defined by the 
equations 

^'=="aQ/ ^'•^a^ (r=l,2,...,n), 

where W denotes a given function of (}i, Jj, ...,?», Qi, Qj, ..., Qn> 0» is 
performed on the variables of a dynamical system defined by the equations 

dqr_dH dpr__dH . ^ ' 



k 



• 



304 Properties of the Integrals of 

the resulting syatem is 

dQ^ SK dP, _ BK 
dt "dPr' dt 30, *'■" 

where K = S+^-~. 

If the fuDCtioQ K is zero, the ayatem will be said to be tran: 
the equilibrium -problem. Now the function K will be zero, pre 
function such that 

li^(<l:9. 1..Q. 0..0 + ff(9.,?. !..?, P 

i.e. provided W, considered as a fuoction of the variables (q,, 
aatisfiea the partial differential equation 



f--('. 



dW dw dw 
' ^q\ ' 3?s ' ' 3?ii ' 



This is called the Hamilton-Jacobi partial differential eqvatic 
with the given dynamical ayatem. 

Suppose that a " complete integral " of thia equation, i.e. a i 
taining n arbitrary constanta in addition to the additive constat 
Let (Oi, a», .... O be these arbitrary constants, so that the soli 
\vritten W(qi, 5,, .... ?,, «!, «», ..., a,*, t); and perform on 1 
dynamical system the contact-transformation from the variable: 
qni pi' ■■-. Pn) to variables (Oi, a„ ,.., a„ ft, /3», ,.., /3J, defi 
equations 

Since W aatisfiea the Hamilton-Jacobi equation, the Hamilton 
of the new system is zero, and consequently the equations of the 

Eo that (a,, a,, ..., a,, ft, ..., j8„) are constant throughout the 
follows that if W denotes a complete integral o/tke Samilton-Jaa 
containing n arbitrary/ constanta {Si, a,, ..., a,), then the equations 

consHtvte the solution of the dynamical problem, since ths^ express t 

(,q„ ji qn.Pi, ■••,Pn) in terms oft and 2n arbitrary constants 

a„ ft ft). In this way the solution of any dynamical aysi 



Dynamical Systems 

om is made to depend on the solution 
aon of the first order in (n + 1 ) imlepeudi 

ider the sjstem 

dt" Zp' cU~ Sj' 



The Hamilton -Jacobi equation correspoDdiog ' 






of this equatioD mny be found iu the followiog n 
iincUmis of their respective arguments : then we 

o-/'(i)+J {*'(?))'- J. 

ui be ntUfied by writing 

mt ; which gives 

'(0=^, *(5)-(8,«.)*«n-'{yM*+!2rt(o-?)/ 

ir=^ + (a|.o)t 8iu - > (g/a)» + {2;^ (o - y)/<.|» 

the original problem is therefore given by tbe e 
ind {9 are the two conatAntH of integration. 

on's integral as a solution of the HamiUon 

infinite number of complete integrals 
ferential equation ; and each one of them I 
am the variables (g,, q,, .... q^.pi, ■■■,pn, 
es (a,, Oj, ..., a„, A, ...,y3J, (the transforn 
uations of motion of the system when < 
8], ..., /9,) become the equations of the eq 
(a„ Oj, ..., B,, )9,, ..., ^t) are constants. 

nfinite number of transformations there 
that in which the quantities (cti, cii, ..., 

of (jj, (/,, ..., 5,, p, p,) respectively, 

I taken as an epoch from which the motioi 
ind in an explicit form the corresponding 
Tacobi partial differential equation. 



306 Properties of the Integrals of 

For consider Hamilton's integral 

(' Ldt. 

where L denotes the kinetic potential of the system. Supposi 

a variation due' to small changes {8a,, So, So,, S/9,, ..., Sff 

conditiona 

Then (§ 99) we have 

It follows that if the quantity I Ldt, when the integratii 

be expressed in terms of (g,, g, 9.1 Q^d •, Oai'X (^^ suppo 

Le. we assume that it is not possible to eliminate {$,, ft, ... 

from the relations connecting (a,, .... a„, /S,, ..,, ^„, q^, ..., q^,pi, ..., J),), so 

as to obtain relations between {q, 9,, oti, .,,, a,)) and if the function thus 

obtained be denoted by W {qi, q„ ..., q^, a,, ,.., a,, (). then we shall have 

-^=P., 3^— A (r = 1,2, ...,«), 

and therefore t he transformatvm from 

(?..?■ ?.,?, P.) to («■,". o.,ft /3,> 

is a contact-tram formatiop , fl^'^ *>'" inUaral of thjt hinftit^ pnfftnt.ial iji tjitt 
d etermim no function of the transformat ion. 
Also we liave 

dl ~ dl * „, 8s, di ' 

and therefore the integral o/ttie kinetic potential aatisJUa the equation 
Sir, „/ IW dW \ „ 

which is the Hamilton^ Jacoln equation- 

,, a., ^,, ,.., /9J be the initial values (at time (g) of 
eepectively, in the dynamical eystom represented bj tbe 

di S^/ di ay, [r-i,l!,...,«J. 

Suppose that ftom tbe relatione coDoecting (a,, a,, ..., a,, 0,, ,.., 0,) with 
(?it 9i' '"> 9>»?ii ■■■>fii)i'- '^ poaaible to eliminate Oi, /9i, —1 AiiPii .-..p.) entirelj, eo 



Example. Let (d 


1. "1. 


(?i. ?». ..-. ?., Pi, 


.. PJ 


equations 





namical Systerm 

relations exist between (?, 
so as to take the form 





,?..<■-+ 1... 


,..,a., i; 


l-a, 


g»l 


'/I- 






n. 


of (ft- ?.. 


-- ?-, 


-i. 


"-^/.l,^' 


3/; 




A = 




an 





' ; and shew that the functii 

ve= F+ s x./» 

rential equation 

BTF BIT \ 

integrals with infinti 

dpr ' dt dqr 

nical system, and let 
■. 9n,Pi, —>P*. = Coi 
stem ; .we shall shew th. 
particular solution of i 

OD for hq, is 

9p, c>pi9p^ 9^1 
j,=i dt dq^J ii^i dqt 3j 
dt \dpr) dt \dpj 



/ 



308 Properties of the Integrals of [ch. xn 

and hence the variational equations for (Sq^t Sq^, ..., S^J are satisfied by the 
values 

Sqr^e^, 8p^=-€g^ (r = l,2, ...,n), 

where 6 is a small constant. Similarly the variational equations for 

can be shewn to be satisfied by these values ; and hence the equations 

8?r=e^. 8Pr = -6^^ (r = l,2,....n). 

where e is a small constant and (f> is an integral of the original equations, 
constitute a solution of the variational equations. 

This result can evidently be stated in the form : The infinitesimal contact- 
transformation of the variables (ji, ?j, ..., J«, jpi, -..li^n)* which is defined by 
the equations 

&?,= e^^. 8j,, = -.|^^ (r = l;2,...,n), 

transforms any orbit into an adjacent orbit, and therefore transforms the 
whole family of orbits into itself. Adopting the language of the group- 
theory, we say that the dynamical system admits this infinitesimal contact- 
transformation. We have therefore the theorem that integrals of a dynamical 
system, and contact-transformations which change the system into itself .are 
substantia Uy the same thing ; any integral 

^(?i> ?2. ...f qnyPu '"> Pny = Constant 

corresponds to an infinitesimal transformation whose symbol (§ 133) is the 
Poisson-bracket {<f>, f). 

It will be observed that the ignoration of coordinates arises from the particular 
case of this theorem in which the integral is pr=CoT\Bta,nt, where gr is the ignorable 
coordinate ; the corresponding transformation is that which changes q^ without changing 
any of the other variables. 

146. Poisson's theorem. 

The last result leads to a theorem discovered by Poisson in 1809, by 
means of which it is possible to construct from two known integrals of a 
dynamical system a third expression which is constant along any trajectory of 
the system, and which therefore (when it proves to be independent of the 
integrals already known) furiiishes a new integral of the system. 

Let j> (g'l, ?a, . . . , ?n. Pi, . . . , i>n, = Constant 

and '^{<lu 9i. ..., ?n, JPi, ...,i>n, = Constant 

denote the two integrals which are supposed known. Consider the in- 
finitesimal contact-transformation whose symbol is the Poisson-bracket 



/ 



V 



144-146] Dynamical Systems 309 

(/, '^); since '^ is an integral, this (§ 144) transforms every orbit into an 
adjacent orbit. 

The increment of the function <^ under this transformation is e (<f>, '^), 
where 6 is a small constant ; but since <^ is an integral, <f) has constant values 
along the original orbit and along the adjacent orbit : the value of (<^, yfr) 
must therefore be constant throughout the motion. We thus have Poisson's 
theorem, that if <f> and '^ are two integrals of the system^ the Poiason-hracket 
{(f), ^Ir) is constant throughotd the motion. 

If (<f>, yjr\ which is a function of the variables (q^, q^y ..., qn^Pn •••, J^n* 0» 

does not reduce to merely zero or a constant, and if moreover it is not 

expressible in terms of <f>, y^ and such other integrals as are already known, 

then the equation 

(<^, '^) r= Constant 

constitutes a new integral of the system. 

The following example will shew how Poisson's theorem can be applied to obtain new 
integrals of a dynamical system when two integrals are already known. 

Consider the motion of a particle of unit mass, whose rectangular coordinates are 
(?i> 9'2> 9zi *^d whose components of velocity, are (p^, p,* Z's)* which is free to move 
in space under the influence of a centre of force at the origin. The integrals of angular 
momentum about two of the axes are 

i^3 S'a - 9'si'2 = Constant, 

and jE?i ^3 - S'l jtJj = Constant. 

Let these be taken as the two known integrals ^ and ^ ; the Poisson-bracket (^, ^), 
which is 



becomes in this case 
and in fact, the equation 






Pi 9i ~ 9iPi = Constant 

is another integral of the motion, being the integral of angular momentum about the 
third axis. 

146. The constancy of Lagrange's bracket-expressions. 

The theorem of Foisson has, as might be expected, an analogue in the 
theory of Lagrange's bracket-expressions. 

Let Ur^ar (r = 1, 2, ..., 2ri) 

denote 2n integrals of a dynamical system with n degrees of freedom, con- 
stituting the complete solution of the problem : the quantities Ur being given 
functions of the variables (^i, gj, ..., qn* Pn "-» Pny 0» *^^ ^^® quantities a^ 
being arbitrary constants. By means of these equations we can express 

(?i> ?2» •••> ?n, J5i» ...f jPn) as functions of (o^, a,, ..., Ojn, 0> ^^^ '^^^ ^^^ 
Lagrange's bracket-expressions [ar, a J, where ar and a, are any two of the 

quantities {ai, a^, ..., cutn)' 



810 ProperUea of the Integrals of 

SiDce the transformatioQ from the variables {q,, j,, ..., ^n,;* 
time t to their values at time t + dtiaa contact-transformatioD, we 

I £^(i,,8p,-85,ap,)-0, 

where the symbols A and S refer to iodepeodeDt displacemeol 
trajectory to aa adjacent trajectory. If now we take the symh< 
to a variation in which Of only is varied, the rest of the <]uani 

(a,, a,. ...,aj„) 
remaining unchanged, and take S to refer to a variation in whi( 
varied, the last equation becomes 

dt r.i \dai 9of doj ddi) ' 

which shews that the Lagravge-bracket [ot, Oj] has a constant value during the 
motion along any trajectory ; this theorem was given by Lagrange in 1808. 

Lagrange's result, unlike Foisson's, does not enable us to find any new 
integrals ; for we have to know all the integrals before we can form the 
Lagrange's bracket-espressioos. 

147. Involution-syHenu. 

Let (u,, u,, ..., Ur) denote r functions of 2n independent variables 
(qi,qt,—,qn.pi, ■-.Pn); 
if it is possible to express all the Poisson-brackets (uj, u^) as functions of 
{vj, u,, ..,, Ur), the functions (m,, u,, ..., itr)are said to form & function-group* . 
Any function of (u,, «,, ,,., u,) belongs to this group. 

If the quantities (uj, iij) are all zero, the functions (ui, tt,, .... u,) are said 
to he tn involution, or to form an involution-st/stetn. 

Now suppose that (u,, Ug, .... u,) are functions in involution: and let 
0^0 and w— be any two equations which are consequences of the 
equations 

u, = 0, ti,-0, ..., «r = 0; 
we shall shew that v and w satisfy the relation {y, «») = 0, 

For since (u,, u,, ..., u,)are in involution, each of the equations 
w, = 0, «,=-0, .... u,= 
admits each of the r infinitesimal transformations whose symbols are 

* Lie, Math. Ana. Tin. (1876). 



ion ti — 0, being a consequf 
iformations ; that is to say 

(ut, o) = 
luations 
,=0, «, = 0, ...,H, = 

ansformatioQ whose symb 
equence of these equatio 
Jmit this transfonnation, e 



1 = 0, t), = 0, ..., t;^=0 

•tions 

,=.0, u,-0, .... w, = 0, 

, Vr) are in involuHon. 



itablished for systems with 
oded to systems with an; 
lich was given by Liouviili 
lis 

.9j, ■■-.9«,Pi. ■■.,Pi.,0 = 
irbitrary constants, are h 

it "dpr' dt dqr 

ncHon of {q,, q„ ..., qn,p 
re in involution, then on S( 
lin them in the form 

[?i, 3», ■■, gn, Oi, Oj, ■■-. Oi 
/,) respectively for (pi.pi 
i+p,dqt+ ...+p„dqn — B 
a a perfect differential : de 
\, qt, ■■-, 9n.Oi. o.. ■". a«. 

Journal dt Math. ix. p. 137. 



312 



Properties of the Integrals of 



[oh. xn 



the remaining integrals of the system are 

dv 



da^ 



= ftr 



(r = l, 2, ...,n) 



or 



(r, 5 = 1, 2, ...,»), 
(r, 5 = 1, 2, ...,n). 



Also 



and consequently 



where (6i, tat •••! K) are arbitrary constants. 

For since the functions <^i— c^, <^ — flj, ..., <f>n''Cin are in involution, it 
follows by the last article that the functions pi —/i, p^ —/a, . . . , pn — /«• are iii 
involution, and therefore 

dH _dpr^dfr 
dqr dt dt 

dt fsi dqg dt 

3< ,=1 dqr dp, ' 

dfr dH_ ^dHdJ. 

dt dqr ,-idp,dqr 

__dH, 
dqr- 

where Hi stands for the function JT when expressed in terms of the arguments 

vlii 9a» •••> 9n> tti> •••> ^> t). 

The equations 

dqr dq, ' dt dq^ ' 

shew that fdq, +f^q^ + . . . +/ndqn - ^i^^ 

is the perfect differential of some function V(qi, q^, ..., 5n, (h, ••., «», ^); 
which establishes the first part of Liouville's theorem. 

If now the symbol d denote the total differential of the function V with 
respect to all its arguments, we have therefore 

dV 
dV=f,dqi ->rf4q% + . . . -^fndqn - H^t + 2 ^— da^. 

r oa^ 

In this equation replace the quantities a,, by their values <^^ : we thus 
obtain an identity in (ji, g-,, ..., j^,^!,^,, ..., p„, t\ namely 

dV 
^^"^^ d<f>r=Pidqi+p4qi+ ... +Pndqn-Hdt, 

where on the left-hand side of the equation we suppose that in rfK' and 

dV 

g— the quantities (oi, a,, ..., an) are replaced by their values (^i, ^. ..., ^). 



1' 



Dynamical Systems 

I that the differeatial form 

p,d^i + pidqt + , . . + pndq„ — Sdt, 
tenoB of the variables {q,,qi, ■■■,9n, ^ 

- il-dAr + dV. 

reutial equations of the original dynamic 
■st Pfaff's system of this differentiat form, i 

3F/3a, are therefore constaot throughout 

I are new arbitrary constants, are integrals 
iroof of Liouville's theorem. 

notion of a body luder no forces with one point 
i&n angles which specify the position of the bod 
le filed point, {A, B, C) the principal momenbs 
ut, a the constant of energy, a, the angular i 
I Oy the angular momentum about the normal 
(»„ ^,) denote ^TjhS, iTjo^,, ZTjd^ respectiv. 

[(V-V-''.')'/«,}-tan-'{(V-^i*-V)*/'hl. 

tial of a function V, and that the remaining 

>itrarj constants. 

tCa theorem. *J 

B estahliBhed a connexion between the 

ad certain families of particular solutions o 

system in which some of the coordinates 
be the ignorable and {qm+i> •■-. 9n) the 
t Z denote the kinetic potential. 
■ UtTid. dell' Ace. dei Lineti (1901), p. 3. 



314 Properties of the Integrals of [oh. xn 

The integrals correspondiag to the ignorable coordinates are 

or . 

^r-r = Constant (r = 1, 2, . . . , m), 

and corresponding to these integrals there exists a cUiss of particular solutions 
of the system, namely those steady motions (§ 83) in which (ji, g,, ..., q^ 
have constant values which can be chosen arbitrarily, while {qv^i, ^m+ti •••> ?n) 
have constant values which are determined by the equations 

or 

^ =0 (r = m + l, m + 2, ...,n); 

there are oo ^ of these particular solutions, since the m constant values of 
(?ii ?2» •••> 9m) and the m initial values of (gi, q^, ..., g„^) can be arbitrarily 
assigned. The theorem of Levi-Civita, to the coDsideration of which we 
shall now proceed, may be regarded as an extension of this result. 

Let ^^^A dp,_^dH /^^i2 n>i 

^^ dt^dpr' 'dt' d^r (r-l,2,...,n) 

be the equations of motion of a dynamical system, the function H being 
supposed not to involve the time explicitly. 

Let K(qu 9a, ..., 9«»Pi» ...,Pn) = (r = l, 2, ..., m) ...(A) 

be a system of m relations, which when solved for (pi,pj, ...,j?m) take 
the form 

Pr-fr(qu 9«>---> 9n>Pm+li •.., J»n) (r=l, 2, ..., m)...(Ai), 

and which are invariant relations with respect to the Hamiltonian system, 
i.e. which are such that if we differentiate the relations (Ai) with respect 
to t, we obtain relations which are satisfied identically in virtue of the 
Hamiltonian equations and of the equations (Aj) themselves. These 
invariant relations include, as a particular case, integrals of the system: 
in this case, they will involve arbitrary constanta • 

Since the relations (Ai) are invariant relations, we have 

^^~ = "Jj7 = - 2 5^v-+2 5^x- (r = l, 2, ...,m), 
oqr dt j^m+idpjdqj j^idqj dpj 

and writing 

ir.w)- i Ci\--f\-). 

i-m+1 ym 9* 9© 3f!;/ 



this becomes 



^■^^^••^'^^1^1;=' <'•='•' "*>•••('>' 



this equation becomes an identity when for each of the quantities 
(Pij i>9i •••,J>m) we substitute the corresponding function/^. 



elftt 
pres 

y.\- 

froir 






f eq 



iioD! 

r a 

L? 

), 



316 



Properties of the Integrals of 



[CH. xn 



now taking account of (B), we have from (3) 

dpr *=i dp, SpJ 

and hence equations (6) become 



dt [dpj Jti dp, [dprdq, "^ tap, • '^'\\ 
d (dK\ ^^dH [^K_ (dK n 

dt [dqrJ sii dp's Idqrdqs "^ [dq^ ' n] ' 



(r = mH-l, m + 2, ..., n), 



d/dK\ ^ d/dK\ ^ . 



or by (7), 

m + 1, m + 2, ..., n) 

which proves that the system of equations (A) and (B) is invariant with 
respect to the Hamiltonian equations. 

Now from the equations (A) and (B), let the variables 

be determined in terms of (ji, jj, ...,*9«i): from the invariant character ol 
(A) and (B) it follows that on substituting these values in the Hamiltonian 
equations, we shall obtain m independent equations, namely those which 
express {dqi/dty dqjdt, ...,dqf^dt) in terms of (ji, jj, ..., gm), the others being 
identically satisfied: and the general solution of this system, which will 
contain m arbitrary constants, will give oo^ particular solutions of the 
Hamiltonian equations. The solution of this system can, by making use 
of the integral of energy, be reduced to that of a system of order (m— 1): 
and thus we obtain Levi-Civita's theorem, which can be thus stated : To any 
set of m invariant relations of a Hamiltonian system, which are in involution, 
there corresponds a family of oo^ particular solutHms of the Hamiltonian 
system, whose determination depends on the integration of a system of or,der 
(m - 1). 

If the invariant relations (A) are integrals of the system, they will contain 
another set of m arbitrary constants : and hence to a set of m integrals of a 
Hamiltonian system, which are in involution, there corresponds in general 
a family of oo^ particular solutions of the system, which are obtained by 
integrating a system of order (m — 1). 

Example. For the dynamical system defined by the Hamiltonian function 

ff= g'l Pi - q^Pt - aqi* + bqt\ 
shew that the Levi-Civita particular solutions corresponding to the integral 

( ft ~ l^%ilq\ = Constant, 



Dynamical Systems 

lations 

j, = 0, 9j = «"'*', p,-ae-'+', p^-b«-'*'' 
%ry constant. 

vikick possess integrals linear in the momenta. '^ 
proceed to the coasideration of (systems w] 
D special kinds, 
h dynamical Bystem, expressed by the equations 

dt dp/ dt~ Bq^ ^'" 

lich is linear and homogeneous in (Pi,pt, --..p 
/Pi+/iPi+ ■■• +/nP» = Constant, 

*"„) are given functions of (g,, q^ 5„). 

^stem of equations 

dqi dqt _ dqa 

n — 1) ; suppose that the (n — 1) integrals whic 

QAq^qi. ...,gn)= Constant (r = l, 2, .. 
nction defined by the equation 

igrand the variables (<;,, g, j„) are suppoa 

in terms of (9,, Q„ Q,, ..., Ob„i) before the 

liable!; change in such a way that (Q,, Q,, ..., ( 
iiries, it follows from the above equation that 

.., Q„) are regarded as a set of new variables 
f„) can be eiipressed, we shall have 

thttt we consider the coutact-transfonnation v 
Dint- transformation from the variables {q„ q„ .. 
■ ■■< Qn). SO tliat the new variables (Pi, P,, 
y the equations 

^'-J/'at- <' = >• 



318 Properties of the Integrals of 

By this transformation the dlSerenbial equations of the dye 
are changed into a new set of Hamiltonian equations 

dt dPr' dt dQr 
and the known integral becomes 

P„ = Constant. 

Since dPJdt = 0, we have dKjdQn = 0. so the function K d< 
Qn explicitly : and thus we obtain the result that when a dyt 
possesses an integral which is linear and homogeneous in (pi,pi 
exists a point-tranaformation from the variailes {c[i,q,, ..■, Jn) tc 
(Qij Qt. ■•■> Qn)> which is such that the transformed Hamiltc 
does not involve Q,. The system as transformed possesses 

ignorable coordinate, and we have the theorem that the oi-^ -^ — -- 

systems which possess integrals linear in the momenta are those which possess 
ignorable coordinates, or which can be transformed by an extended point- 
transformation into systems which possess ignorable coordinates. 
The converse of this theorem is evidently true. 
This result might have been foreseea from the theorem (§ 144) that if 
^(9it 7)1 ■■■< ?■• Pii ■■■> Pnr O'C^onstant 
is an integi^ of the system, tben the difierentlal equations of motioD admit the 
infiniteaimal transformation whose symbol is(^,/}. For when i^ is linear aod homc^neoua 
in (Pii Pit •■■< P")' ^^^ transformation is (§ 132) an extended point- transformation : if 
this point-transformation is transformed by change of variables so aa to have the aymbol 
S//dQ., it is clear that the Hamiltoniaa function of the equations after transformation 
cannot involve Q„ explicitly. 

Considering now in particular systems whose kinetic potential consists 
of a kinetic energy Tiq,, qj, ..., qn,qi, ■.■,qn) which is quadratic in the 
velocities {q,, j,, .... 9„) and a potential energy V{jj, g,, ..., q„) which is 
independent of the velocities, we see that in order that an integral linear in 
the velocities may exist the system must possess an ignorable coordinate, 
or must be transformable by a point-transformation into a system which 
possesses an ignorable coordinate. But in either case the functions T and V 
evidently admit the same inftniteaimal transformation, namely the trans- 
formation which, when the coordinates are so chosen that one of them is the 
ignorable coordinate, consists in increasing the ignorable coordinate by a 
small quantity and leaving the other coordinates and the velocities unaltered ; 
and conversely, if T and V admit the same infinitesimal transformation, then 
there exists an integral linear in the velocities. This result is known as 
Levy's theorem, baving been published by L^vy* iu 1878. 

* Complei Rertdut, Liiivi. 



151, 152] Dynamical Systems 321 

Equating to zero the terms of the second degree in x and y in equation 
(A), we have 

9y ' 9a? * dx dy ' 
from these equations we deduce 

S^mx-^p, T = — my + q, 
where (m, p, q) are constants. 

Equating to zero the terms independent of x and y in (A), we have 

ay ox 
or gr (^wa? + p)- — (my-g) = 0. 

This equation shews that if (m, p, q) are different from zero, the force is 
directed to a fixed centre of force, whose coordinates are —p/m and q/m\ we 
shall exclude this simple particular case, and hence it follows that the con- 
stants (m, j}, q) must each be zero, so that the integral contains no terms of 
the first degree in i, y. 

Equating to zero the terms linear in x and y in (A), we have 

dx * dy dx ' 

dy dx dy 

Differentiating the former of these equations with respect to y, and the 
latter with respect to a:, and equating the two values of ^-^ thus obtained, 
we have 

2P?I+2^-^+0^^+?^^ = 2JJ ^ + 2^^^ + ^^+0^i^ 
dxdy dx dy 9y" 9y dy dxdy dx dy dx dx da^ * 

and replacing P, Q, R by their values as found above, we have 

[B^-^)('-^^y^^'v-^'''^^)+^B^sy^^y"-^ 

+ |^(6ay + 36) + ^^(- 6ax - 36') = 0. 

Darboux* has shewn that this partial differential equation for the function 
V can be integrated in the following way. 

* Archives NSerlandaises, (ii) vi. p. 871 (1901). 
w. D. 21 



Pr(^erties of the I 

g the particular case in whict 
lange of axes reduce the given i 
i (asy - y£y + c£'+c'y' + 
its to supposing that 

a=J, 6 = 0, b' = 
we replace c~c' by ^, the p 

Tate this equation, we form i 
;s 

xy(df-da^) + (a?-j/' 

s equation we take a^ and y' 
uation : we thus find that its ii 
{m + l)(mj?-y')- 
lOtes the arbitrary constant. E 
is integral in the form 

^, y' _ 

irbitrary constant is now a, 
act that the characteristic ci 
! two families of confocal coni< 
;hen as new variables a and /! 
hyperbolas, so that 

rom the general theory that tbi 
B are functions of a and ^ ; in 

"' <^-<^,^^^'£ 

I immediately integrated, giving 

(,.-«7./(<». 

1^ are arbitrary functions of the 

ike motion of a particle in a plat 



152, 153] Dynamical Systems 

farces, which possess an integral quadratic in the velocities other tf. 
integral of energy, are those for which the potential energy has the form 
/(«)-0(g) 

where a and ff are the parameters of confocal ellipses and hyperbolas. 
Since by differeDtiation we have 

the kinetic energy is 

Eiiid an inspection of the forms of T and V shews that these problem. 
Liouville's class (§ 43), and are therefore integrable by qyuidraturea. 

163. Oeneral dynamical tyttami poeteuing inteyraU quadratic tn the vdodtict. 

The complete detanninatioD of the explicit form of the most general djnamicj 
whose equations of motion poeeegs an integral quadratic Id the velocities (in 
to the integral of energy) has not yet been effectedj^Vlt ia obvious from § 43 
dynamical ayatems which are of Liouville's type, or which are reducible to this t 
point-transformation, possess such integrals ; and several more extended typee li 
determined. 

Example iT/ Let **i(?*) {t, 1 = 1,2, 

be n' functions depending solely on the arguments indicated, and let 

*= 2 if>H*ti (^=1, 2, 

denote the determinant formed by these Unctions. Shew that if the kinetic en 
dynamical system is reducible to the form 



and the potential energy is eero, there exists not only the int^ral of energy, 

but also (n - 1 ) other integrala, homogeneous and of the second degree in t 
namely 



where (a,, a,, ,.., a.) are arbitrary constants: and that the problem is s< 

quadratures. (S 

ExampU 2. Let the equations of motion of a dynamical ayst«m with two < 
freedom be 

dt\^qj % " ^' 
where ^=i('^i'+2Ay,y,+6j,«), 

t, 21 



Properties of the Integrt 

17 fuscUoDB of the coordinates (q^, f,] : 

a' ji* + ih'qiq^ + 6' j,' — Constaii 
elocities and distinct &om the equatioii 
»ordinatfie. If A and &' denote (06— 

'•-j(|.)"(«V+»rt.'+« 

or dqrjdi, shew that the equations 

■elations between the coordin&t«a (f,, 
tt one set of equationa can be traasfon 



MiSCELLAHEOUS ExaHPL: 
al system is defined by its kinetic ene 

i*(¥^ *¥■+■■■*¥■)• 

the determinant 

«ii 



1^ <^ . 



^ ♦ill 0M 

Die of the ;Hh line are functions of jjt onl 
ential Niergy 



t denotes a function of }* only. Shew 

r="-o,(+ s ■{(■i0(i + <ii4><t+-"+<ii.4>i 

1,0 Ai^ arbitrary conatanta. 

■^(Ji. ?i. -, ?., Pi> -.-, ;>■„ ()=< 
iTnamicat eystem which posaesaee an inb 

^=ConBtant, ^=ConBtant, etc., an 



CH. xn] Dynamical Systems 826 

3. A system of equations 



dt 

^^ = ^r(?l> S'2> •••» ^»» A» •••>i'«> 



(r=l, 2, ...,») 



is such that if <f> and ^ are any two integrals whatever, the Poisson-bracket (<^, ^) is also 
an integral. Shew that the equations must have the Hamiltonian form 

dOr dff dpr dH / 1 o \ 

(Korkine.) 
. 4. If Ox = Constant, o^^ Constant, ..., 0*= Constant, 

/3js= Constant, /Sg^ Constant, ..., /3fc= Constant, 

are any 2k integrals of a Hamiltonian system of differential equations, the variables being 
{?!) S'2> •••» ^i** Pu •••> Pn)j sbew that 

S 2 + |^|?2...|^*^...|^» = Constant 

is also an integral. (Laurent.) 

5. Let the expression 

^^" ^' '"•^" ill d(x,i,x„,...,x^)' 

where j&j, -fiTg, ..., H^ are functions of the nv variables Xji(j = ly 2, ..., n ; i=l, 2, ..., v) 
be called a PoiMon-bracket of the nth order. If (?,, (?2, ..., C^j^y are hv functions of 
yii,yi2» -"^yhv'i ^11, ^u» ...j^^; «i> «2> •••> a^^, where (A+^=n), and if 

F^O-) (t-1, 2, ..., (^J)) 

denotes all the Poisson-brackets formed from every n functions G, shew that 

i',((?-)=0 (i=l, 2, .... (*^'')) 

represents the necessary and sufficient conditions that the functions 

yrt=-^rt(^ii» ^'i2» •••♦ ^fn^'f ^i> ^> ••'» «*»') (*=!> 2, .... A; <=1, 2, ..., v) 

arising from the equations 

Gi=0 (1=1, 2,.... Av) 

shall satisfy the simultaneous partial differential equations of the first order 

i'iCy*. ^=0 (i=i, 2, ...,(*;)), 

where Pi{f^, F) denotes the expression which is obtained when we replace h of the 
functions F m Pi {F^) by as many ys. (Albeggiani.) 

6. A particle of unit mass whose coordinates referred to fixed rectangular axes are 
{x, y) is free to move in a plane under forces derivable frt^m a potential-energy function 
/(a?, y), the total energy being A. Shew that if the orthogonal trajectories of the curves 



'iesofihe IiUegrah of Dynamiccil Systems [oh. xn 

rential equations of motioD of the particle poesees an integral linear and 
e velocitiea {±, y). 

yaa of motion of a free system of m particles are 

exists of the form 

£ /.it-C^-CoDatant, 

I ^snt ^''^^ O IB a constant, shew tbat thia 

S i^,+ x' Or,(ar,ir-*^.)-C*"Coiiatant, 

jes k, and a^, are constants. (Pennacchietti.) 

ilea move on a anrfoce under the action of difierent forces depeoding 
pective positions : if tiieir difierential equations of motion have in 
Tal independent of the time, shew that the surface is applicable on 
ution. (Bertrand.) 



1 



CHAPTER XIII. 



THE REDUCTION OF THE PROBLEM OF THREE BODIES. 



164. Introduction, 

The most celebrated of all dynamical problems is known as the Problem 
of Three Bodies^ and may be enunciated as follows : 

Three particles attract each other according to the Newtonian law, so thai 
between each pair of particles there is an attractive force which is proportional 
to the product of the masses of the particles and the inverse square of their 
distance apart : they are free to move in space, and are initially supposed to be 
moving in any given manner ; to determine their subsequent motion. 

The practical importance of this problem arises from its applications to 
Celestial Mechanics: the bodies which constitute the solar system attract 
each other according to the Newtonian law, and (as they have approximately 
the form of spheres, whose dimensions are very small compared with the 
distances which separate them) it is usual to consider the problem of deter- 
mining their motion in an ideal form, in which the bodies are replaced by 
particles of masses equal to the masses of the respective bodies and occupying 
the positions of their centres of gravity*. 

The problem of three bodies cannot be solved in finite terms by means 
of any of the functions at present known to analysis. This difficulty has 
stimulated research to such an extent, that between the years 1760 and 1904 
over 800 memoirs, many of them beai*ing the names of the greatest mathema- 
ticians, have been published on the subject f. In the present chapter, we 
shall discuss the known integrals of the system and their application to the 
reduction of the problem to a dynamical problem with a lesser number of 
degrees of freedom. 

* The motions of the bodies relative to their centres of gravity (in the consideration of which 
their sizes and shapes of coarse cannot be neglected) are discussed separately, e.g. in the Theory 
of Precession and Nutation. In some oases however (e.g. in the Theory of the Satellites of the 
Major Planets) the oblateness of one of the bodies exercises so great an effect, that the problem 
cannot be divided in this way. 

t Gf. the author's Report on the progress of the solution of the Problem of Three Bodies in the 
British Association Beport of 1899, p. 121. 



The Reduction of the 

differential eqvaUons of the problem. 

R denote the three particles, (wi|, »?ia, m,) i 
leir mutual distances. Take any fixed rectai 
' ?>)> (?» 3" 9>)> (9r> 3>< ?b) be the coordinates □ 
dnetic energy of the system is 

"i (?i' + ?.* + 9.') + i «^ (9«° + ?.* + ?-•) + 4 "4 (?7' ■ 
attraction between mj and m, is /:'Tn,Tn,r„~' 
ttraction : we shall BUppose the units so chose 
attraction becomes miwtari,-', and the corre; 
energy is -7nim,ru~'. The potential enei: 

■j^ mtiTit nijWi, jn,jn. 



- m,m, 1(97 - Si)" + (?.- 9,)' + {?.- 9.)"!- 

- w,m. Kg, - g.)" + (g, - J.)" + (}, - j.)"!" 
tioQS of motion of the system are 



mtqr- 



dV 



lotes the integer part of J C" + 2)- This s 
equations, each of the 2nd order, and the s; 



m^qr^pr 



i take the Hamiltonian form 

dqr^dH dpr__dH 
dt dpr' dt " dqr 
t a set of 18 differential equations, each of th< 
n of the variables {q,,g„---,qa,Pi,Ps, ■■•,?>)■ 

ewn by Lagrange* that this system can be n 
of the 6th order. That a reduction of this kiu' 
Irom the following considerations. 

■st place, since no forces act except the mutual 

t pilcet qui ont remporti Ut prii de I'Acad. de Pari; ii 
^aoe tbs sjatem to the Hamiltoaian form. 



7 



\ 



156, 166] Problem of Three Bodies 829 

particles, the centre of gravity of the system moves in a straight line with 
uniform velocity. This fact is expressed by the 6 integrals 

m^qz + wij^e + w*s?9 - (Pj 4-l?6 + ;}») ^ = a«, 

where Oi, a,, ..., ag are constants. It may be expected that the use of these 
integrals will enable us to depress the equations of motion from the 18th to 
the 12th order. 

In the second place, the angular momentum of the three bodies round 
each of the coordinate axes is constant throughout the motion. This fact 
is analytically expressed by the equations 

I JlPs - q^Pl + ?4l>5 - q^Pi + 97P8 - 98^7 = <h» 
?1 Pj - ?8Pa + ?5P6 - q^Pi + q%P^ - 9»1>8 = (hy 

^q^Pi - 9ii^s + 96l>4 - 94P6 + ?»P7 - 97P» = a», 

where a^, Os, a^ are constants. By use of these three integrals we may 
expect to be able to further depress the equations of motion from the 
12th to the 9th order. But when one of the coordinates which define the 
position of the system is taken to be the azimuth (f> of one of the bodies 
with respect to some fixed axis (say the axis of z), and the other coordinates 
define the position of the system relative to the plane having this azimuth, 
the coordinate (}> is an ignorable coordinate, and consequently the corre- 
sponding integral (which is one of the integrals of angular momentum 
above-mentioned) can be used to depress the order of the system by two 
units ; the equations of motion can therefore, as a matter of fact, be reduced 
in this way to the 8th order. This fact (though contained implicitly in 
Lagrange's memoir already cited) was first explicitly noticed by Jacobi* in 
1843, and is generally referred to as the elimination of the nodes. 

Lastly, it is possible again to depress the order of the equations by two 
units as in § 42, by using the integral of energy and eliminating the time. 
So finally the equations of motion can he reduced to a system of the 6th order. 

156. Jacobi's egtuUion, 

Jacobi t, in considering the motion of any number of free particles in speu^ which 
attract each other according to the Newtonian law, has introduced the function 

* Joum. fiir Math. xxvi. p. 115. t Vorlesungen Uber Dyn., p. 22. 



The RedwHi 

ad tOj are the DonsBea of two typical 
em at time t, if ia the total mnaa of tl 
re of particles in the system. This I 
the stability of the system, will be 



11 suppose the ceotre of gravity of th 
1 of the particle nij referred to filed i 
The kinetic energy of the system is 

[uently we have 

summation on the rigbt-hand side is 
id we have 2miii=0, in virtue of th< 

■e have ^~23f ^ "^"^ K*i-*>)* 

ienotes the velocity of the particle m 
same way we can shew that 

V denotes the potential energy of th 
i by the condition that T is to be zt 
rom each other, we have 



nations of motion of the particle m^ 

«l(*'*=-g^. '>k!/i=- 

ly these equations by Xt, yt, z^, re 
if the system: since V ia homogenc 



I called Jacobit egvatim 



166, 167] Problem of Three Bodies 331 

167. Reduction to the 12th orders by use of the integrals of motion of the 
centre of gravity. 

We shall now proceed to carry out the reductions which have been 
described*. It will appear that it is possible to retain the Hamiltonian form 
of the equations throughout all the transformations. 

Taking the equations of motion of the Problem of Three Bodies in the 

form obtained in § 155, 

dqr_dH^ dpr_ dH ^ 

dt^dpr' dt "" dqr ^r-i,z,...,y;, 

we have first to reduce this system from the 18th to the 12th order, by use 
of the integrals of motion of the centre of gravity. For this purpose we 
perform on the variables the contact-transformation defined by the equations 

dW dW 

where W^p^q^ -{-p^q^ + p,?,' ^p^ql -^Pf^q^ +p^q% + (Pi +^4 +P7) qi 

+ (Pa +P6 +P8) q% + (P8 + P« +P») ?»'. 

Interpreting these equations, it is easily seen that (5/, q^, q^) are the 
coordinates of twi relative to tw^, (g/, q^, q^) are the coordinates of m^ relative 
to wij, {qjy js'i 9»') are the coordinates of m^, (p/i Pa', Ps) are the components 
of momentum of mj, (p/, Ps', Pe) are the components of momentum of m,, and 
(jOy', ^g', p^) are the components of momentum of the system. 

The differential equations now become (§ 138) 

dq;_dH dpr'__dH 
dt'dpr" dt ~ dqr' Kr-i.^.-.V), 

where, on substitution of the new variables for the old, we have 

+ — {Pi P/ +P^'P^' + P/P^' + hh'' + iPs'' + iP^^'-Pr' (Pi' + P/) 

-P8'(p;+P5')-p;(p,'.+p;)} 

- 7n,W3 {q:^ + ?;» + 9«'«) -* - m,m, {?/» + g,'« + g,'"}"* 
-rn.m^ {(g/ - q/f + (3,' - q.J + (?,' - ?«?} "*. 

Since qj^ q^, 3/ are altogether absent from H, they are ignorable 
coordinates : the corresponding integrals are 

p/ = Constant, p^' = Constant, p^' = Constant. 

* The oontact-transformatioii used in § 157 is due to Poinoar6, C,R, cxxm. (1896) ; that used 
in § 158 is new, and appears worthy of note from the fact that it is an extended point-trans- 
formation, which shews that the redaction could be performed on the equations in their 
Lagrangian (as opposed to their Hamiltonian) form, by pure point- transformations. The second 
transformation in the alternative reduction (§ 160) is not an extended point-transformation. 
Another reduction of the Problem of Three Bodies can be constructed from the standpoint of 
Lie*s Theory ot Inyolution-systems and Distinguished Functions : cf. Lie, Math, Ann. viii. p. 282. 



I 



T?ie Reduction of the [ch. xm 

[tbout loss of generality suppose these constants of integration 
this only means that the centre of gravity of the system is 
; rest : the reduced kinetic potential obtained by ignoration of 
ill therefore be derived fix>m the unreduced kinetic potential 
p,', p^, pf by zero, and the new Hamiltonian function will be 
H in the same way. The system of the 12(A order, to which the 
■otion of the problem of three bodies have now been reduced, may 
ritten (suppressing the accents to the letteis) 

dt'dpr' dt 5qr t'-=l. Z, ....ftj. 

- Mim, {{q, - q^y + (3, - q,y + {q, - q,y] "*. 
m possesses an integral of energy, 

H = Constant, 
igrals of angular momentum, namely 

f ?>;>. - qtPi + q,p, - gtP. = -^i 

j 9^ - q,pi + q,p, - q,p* = -d, 

I q,p, - q,Pi + qtp> - q>p4 = ^ . 

At are constants, 

uction to the 8th order, by use of the integrals of angular 
d elimination of the nodes. 

n of the 12th order obtained in the last article must now be 
e 8th order, by using the three integrals of angular momentum 
ating the nodes. This may be done in the following way. 
the variables the con tact -transformation defined by the 

^' = Wr- ^'=9?/ <^=1.2,...,6), 

q,' - q,' COS q,' sin 9,') +^ {3/ sin 9,' + jj'cosg/ cos g/) +piq,' sin g,' 
?>' — q* cos Jb' sin 9,'} +p» (i^j' sin g/ + q^' cos q^ cos Ji')+P«9«' ^^^ ?« ■ 
[y seen that the new variables can be interpreted physically as 



157, 168] ProUem of Three Bodies 

In addition to the fixed axes Oxyz, take a new eet of moving axes 0«' 
Oaf ia to be the intersection or node of the plane Oxy with the plane 
three bodies, Oy' is to be a line perpendicular to this iD the plane < 
three bodies, and Oz' is to be normal to the plane of the three bodies. 
iti' ?J ) *•* ^^^ coordinates of m, relative to axes drawn through wi, p 
to Ox', Oy ; (jj', 9,') are the coordinates of m, relative to the same ax 
is the angle between Ox' and Ox ; 9/ is the angle between Oz' and ( 
and pt are the components of momentum of mi relative to the axes Oct 
Pi and pt are the components of momentum of m, relative to the same 
Pi and pt are the angular momenta of the system relative to the ai 
and Oaf respectively. 

The equations of motion in terms of the new variables are (§ 138) 

dt Bp," dt ~ a?,' ^ '.'•■■■ 

where, OD substitution in H of the new variables for the old, we have 

+ Pi'qi cosea q,' + p 

+ p;q,' cosec q,' + p 
1 r ' ' , ■ ' 1 

KPi '?j' — Pi'Si ' + Pa 9i' — i>«'?»') q* cot 5,' + j),'^/ cosec j,' -(- ; 
KPiV - Pt'qi + PaV - Pt'qt) ffi' cot 9/ +^,'},' cosec 5.' + j 

- Tft,m, (9,''+ g-/' )"' - '».'fti{?.'' + ?»'')■* - ^.m, ((?,' - 9,')' + (?»' - 9 
Now 9,' does not occur in H, and is therefore an ignorable coordinate 
corresponding integral is 

p/ = k, where A; is a const 

The equation dq,'jdt=dHISk can be integrated by a simple quadi 

when the rest of the equations of motion have been integrated ; the equ 

for qt and pi will therefore fall out of the system, which thus reduces 

system of the 10th order 

d^^SH d_ 

dt dpr' dt dq, 

where p,' is to be replaced by the constant k wherever it occurs in S, 

We have now made use of one of the three integrals of angular momt 
(namely p^ ~ k) and the elimination of the nodes : when the othe 



(r = ] 



168, 169] Problem of Three Bodies 

But since p,' = 0, we have ZHjZqi =Pt' = 0, and therefore 

da da ' 
in other words, we can make the substitution for 5,' in H before 
derivates of H ; and thus (suppressing the accents) the equations c 
the Problem of Three Bodies are reduced to the system of the 8tk or 
dqr^dH dpr^_dH 
dt dp,' dt dqr 

where 

[k'-(p,q,-p,q^ + p,qt- 

- »».»>. (?.' + ?.')"' - «hm, (9,' + 9,')-» - m,m^ ((5, - j,)* + (q. 

Many of the quantities occurring in S have simple physical inte 

thus (9,9, — 91^4) is twice the area of the triangle formed by the b 

m, + m, + m, t \2m, "*" 2mj) ^* "*" Uwt. 2m,/ *' m, '*' 
is the moment of inertia of the three bodies about the line ia 
plane of the bodies meets the invariable plane through theii 
gravity. 

It 18 also to be noted that this value of M difibrs from the value of ff wht 
terms which do not involve the variables p,, p^, PsiPt'. these terms in it cai 
regarded as part of the potential energy, and we can qaj that the nystem di 
corresponding system for which i ia zero only by certain modifications in 
energy. It can easily be shewn that when t is zero the motion takes place ii 

189. Seduction to the 6th order. 

The equations of motion can now be further reduced from tht 
6th order, by making use of the integral of energy 

H = Constant, 
and eliminating the time. The theorem of § 141 shews that in 
this reduction the Hamiltonian form of the differential equabi 
conserved. As the actual reduction is not required subsequently 
be given here in detail. 

The Hamiltonian system of the Qth order tiius obtained is, in 
state of our knowledge, the ultimate reduced form of the equations 1 
the general Problem of Three Bodies. 



The Reduction of th 

Itemative reduction of the problem from 

.1 now give another reduction • of the 
Bamiltonian aysteni of the 6th order. 

original Hamiltonian system of equi 
ned by the contact-transformation 
, dW BW 

U - 2i) + Pi' (9' - ?») + P^' (5« - ?") 



+!>/(< 






+ Ps' ("»!?> + WHS"! + mjj,) + 
igrals of motion of the centre of gravity 
variables, can be written 

?7' = ?.'-9.'=i>r' = P.'-p.'' 
lently the transformed system is only 
i accents in the new variables, it is 

dt 3p, ' dt dq, 

(p,» +P,' + p^') + —^, (p,> + p.' +p*) - V 
2ma 



%,m, ^q* + q,' + qj> - ^^^^ (q, ff. + 5, J, ■ 



m,mt , Ttit (nti + r, 



' variables may be physically interpret 
be centre of gravity of m^ and ma, 

* Due to B*daa, AnnaUt dt v£c. Norm. Sup. v. 



r 



160] Problem of Three Bodies 337 

projections of Tn^rn^ on the fixed axes, and (94, g,, q^ are the projections 
of (rmj on the axes. Further, 

l^^ = Pr (r = l,2,3), and /J' = ?>, (r = 4,5,6). 

The new Hamiltonian system clearly represents the equations of motion 
of two particles, one of mass /i at a point whose coordinates are (ji, g^i 9s)> 
and the other of mass fi at a point whose coordinates are (^4, q^, q^) ; these 
particles being supposed to move freely in space under the action of forces 
derivable from a potential energy represented by the terms in H which 
are independent of the ^^'s. We have therefore replaced the Problem of 
Three Bodies by the problem of two bodies moving under this system of 
forces. This reduction, though substantially contained in Jacobi's* paper of 
1843, was first explicitly stated by Bertrand"f" in 1852. 

We shall suppose the axes so chosen that the plane of o:^ is the invariable 
plane for the motion of the particles fi and fi\ i.e. so that the angular 
momentum of these particles about any line in the plane Oxy is zero. 

Let the Hamiltonian system of the 12th order be transformed by the 
contact-transformation which is defined by the equations 

*^ = ^/ ^^=V (r = l,2,...,6), 

where 

W = (p2 sin g/ + pi cos je') ?i' cos 5/ 4- 9/ sin g,' {(pa cos q/ - pi sin q^y + p^]^ 
+ (p5 sin 3«' 4- p4 cos ?«') q^ cos q^ + q^ sin q^ {(pa cos q^ - p4 sin q^Y 4- Pe'}*. 

The new variables are easily seen to have the following physical inter- 
pretations: g/ is the length of the radius vector from the origin to the 
particle /x, 5/ is the radius from the origin to fi\ q^ is the angle between q^ 
and the intersection (or node) of the invariable plane with the plane through 
two consecutive positions of g/ (which we shall call thQ plane of instantanecms 
motion of /*), q^ is the angle between qz and the node of the invariable plane 
on the plane of instantaneous motion of /x^ q^ is the angle between Ox 
and the former of these nodes, q^ is the angle between Ox and the latter of 
these nodes, pi' is ftg/, p/ is fi'q^, p,' is the angular momentum of /i round 
the origin, P4' is the angular momentum of fju round the origin, pa' is the 
angular momentum of fi round the normal at the origin to the invariable 
plane, and p^ is the angular momentum of ^i round the same line. 

The equations of motion in their new form are (§ 138) 

dt~dpr" dt~ dq; Kr-L,i,...,Ki). 

* Journal fUr Math. xxvi. p. 115. t Journal de math. xvu. p. 893. 

w. D. 22 



160, 161] Problem of Three Bodies 339 

and it is therefore allowable to substitute for p^' in H before the derivates of 
H have been formed The equations of motion are thus reduced to a system 
of the 8th order, which (suppressing the accents) can be written in the form 

dq^_dH dpr_ dH 9 q ±\ 

where, effecting in H the transformations which have been indicated, we have 

The equations of motion can further be reduced to a system of the 
6th order by the method of § 141, using the integral of energy 

H = Constant 

and eliminating the time. As the reduction is not required subsequently, it 
will not be given in detail here. 

161. The problem of three bodies in a plane. 

The motion of the three particles may be supposed to take place in a 
plane, instead of in three-dimensional space ; this will obviously happen if the 
directions of the initial velocities of the bodies are in the plane of the bodies. 

This case is known as the problem of three bodies in a plane : we shall 
now proceed to reduce the equations of motion to a Hamiltonian system of 
the lowest possible order. 

Let (gi, 52) be the coordinates of mi, (gs, q^ the coordinates of TWg, and 
(9b> Je) the coordinates of w,, referred to any fixed axes Ox, Oy in the plane 
of the motion ; and let p^ =■ muq^ where k denotes the greatest integer in 
\{r+\). The equations of motion are (as in § 155) 

dqr_dU dpr dH 

dt~dp/ dt" dqr i^-i, A...,o;, 

where 

^ = 2^ ^^' "^ ^'^ ■*■ i ^P»' "^ P'^ ■*■ 2^ (Pfi' + P6")-^2^ {(?3-?b)H(?4-?6)'}-* 

- WlsWi {(^B - g,)» + (?6 - ?2)'}"* - ^1^ {(?l - q%y + (^2 - ?4)'l"*. 

These equations will now be reduced from the 12th to the 8th order, by 
using the four integrals of motion of the centre of gravity. Perform on the 
variables the contact-transformation defined by the equations 

dW , dW 

5' = ^' ^'=9^ (r = l,2....,6), 

22—2 



The Reduction of the [ch. xm 

^hat (^i', qa) are the coordinates of m, relative to axes 
to the fixed axes, (q,, j/) are the coordiaates of m^ 
axes, (q,', q,') are the coordinates of m, relative to the 
) are the components of momentum of m,, ipt>Pi') w* 
lomentum of m,, and (p,', p,') are the componeats of 
'Stem. 

equations for q,', q^, ps, p, disappear from the system ; 
accents in the new variables) the equations of motion 
1 of the 8th order, 



dqr^dE^ dpr^_SH 

dt dpr' dt dq^ 



(r = l, 2, 3, 4), 



;w that this system possesses an ignorable coordinate, 

aible a further reduction through two unit<i. 

jrstem the con tact- transformation defined by the equa- 

«'=a7,' P^=S^' (r = 1.2,3,4). 

«n q,'+p,(qi coa q^'~ q,' sin 9/) +pt (q,' sin q,' + q,' coa 9/). 

pretation of this transformation is as follows : qi is the 
q,' are the projections of m^mf on, and perpendicular to, 
between m,m, and the axis of x ; p,' is the component 
ong msTn, ; p,' and p^' are the components of momentum 
rpendicular to m^mt ; and pt is the augular momentum 

uations, when expressed in terms of the new variables, 
dt 9p/' d( Sy, ^ 



161, 162] Problem of Three Bodies 341 

Since 9/ is not contained in H, it is an ignorable coordinate ; the corre- 
sponding integral is p/ = k, where A; is a constant ; this can be interpreted as 
the integral of angular momentum of the system. The equation 7/== 9ir/3p/ 
can be integrated by a quadrature when the rest of the equations have been 
integrated ; and thus the equations for p/ and 5/ disappear from the system. 

Suppressing the accents on the new variables, the equations can therefore 

be written 

dqr^dH^ dpr^_^ (r=I, 2, 3), 

dt dpr ' dt dqr 
where 

- fThm^qr^ - wiim, {(q^ - q^Y -h Js'}"*. 

This is a system of the 6th order ; it can be reduced to the 4th order by 
the process of § 141, making use of the integral of energy and eliminating 
the time. 

162. The restricted problem of three bodies. 

Another special case of the problem of three bodies, which has occupied a 
prominent place in recent researches, is the restricted problem of three bodies; 
this may be enunciated as follows : 

Two bodies S and J revolve round their centre of gravity, 0, in circular 
orbits, under the influence of their mutual attraction. A third body P, 
without mass (i.e. such that it is attracted by S and J, but does not influence 
their motion), moves in the same plane as S and J] the restricted problem 
of three bodies is to determine the motion of the body P, which is generally 
called the planetoid. 

Let mi and m^ be the masses of 8 and J, and write 



Wi m^ 



SP^JP' 

Take any fixed rectangular axes OX, OY, through 0, in the plane of the 
motion ; let (X, Y) be the coordinates, and ( J7, V) the components of velocity, 
of P. The equations of motion are 

d«Z dF d»7 dF 



17 > 



dfi dX' df dV 

or in the Hamiltonian form, 

dX_dH . dr_dH dU__dH dV^_'^B: 
dt dU' dt~dV' dt~ dX' dt dY' 

where H = ^{U*+V')-F. 



162] Problem, of Three Bodies 

where u denotea a current variable of integration. These equatioi 
written 



.=(- 



Pi" 



Pi" p.'' Pi'V 



and it is easily seen that q,' is the mean anomaly of the planetoii 
ellipse which it would describe about a fised body of unit mass 
projected from its instantaneous position with its instantaneous vek 
is the longitude of the apse of this ellipse, measured from OJ; pi ii 
p,' is fa(l — 0*)]^ where a is the semi-major axis and e is the eccen< 
this ellipse. H does not involve t explicitly, so if i> Constant is an 
of the equations of motion, which are now 

dt "dp," dt " dq; ^''" 

If we take the sum of the masses of S and J to be the unit of n 
denote these masses by 1 — ^ and f* respectively, we have 



B-i{l 



^^h"'^~~SP~7P- 



w+f, -"P.- 



This is an analytic function of p,', p,', q,', g,', /i, which is periodic in q, 
with the period 2ir. Moreover, to find the term independent of ft h 
suppose ft to be zero ; since SP now becomes qi, we have 

rr ,/2 1\ , 1 1 

Thus finally, discarding the accents, the equations of motion of the r 
problem of three bodies can be taken in the form 

dq^^dS '^Pr__aH" , 

dt dp, ' dt " dq, °" 

where H can he expanded aa a power-aeries in fi. in ike form 

and Ha = — er~i— "/''■ 

while H,, Ht, .... are periodic in q, and j,, with the period 2ir. 

The equations of this 4th-order system can be reduced to a Ham 
system of the second order by use of the integral H = Constant and 
tion of the time, as in § 141. 



-l.P-'T. (r=o,i,a,... 



163] Problem of Three Bodteg 

4. Applj the contact-tranBformation defioed by the equations 

9>' = {(94 - 9r)' + (ft - ft)* + (ft - ?»)*)*. 
=-[(?r-?i)*+(ft-?i)'+(?.-ft)1*. 
= {(9i - 5.)' + (ft - ft)*+ (ft - ?«)")*, 
=6,C?,+*?,)+6,(?,+iSs)+6,{?T+ift). 
=ei?s+M«+'%ft. 
="ii?i+'%?i+'nrfi. 
= 'nift + '»ift + "'sfti 

= °l( ft+'9i) + °l ( ft+*'ft)+''l(gT+'g 8) 
6l (?l + '9.) + *l (ft + »?6) + ^(?7+lJ») ' 

_ '■'^ 

(where V stands for V— 1 and Oj, Og, n|, 6), &,, 6], c,, c,, e, are any nine constant 
satisfy the equations 

o,+a,+o,=0, 6,+6,+6,=0, c,+e,+e,=0, Oj6,-<jA=1) 
to the Hamiltonian system of the 18th order which (^ 150) detonninea the motio 
three bodies. 

Shew that the int^rals of motion of the centre of gravity are 

?.'= ft' =ft' =P8' =P7'=Ps'-0. 

Shew hirther that when the invariable plane is taken as plane of ^, the varial 
aero, and that the integral of angular momentum round the normal to the invarial 

piqi=h, where i is a const 

Hence shew that the equations reduce to the 8th order system 

-dt=^;' -df- ^; (•■-0,1,2 

where 

*i^T(i 2m,mj^ },'j3' 2m, 

+» i; {Po' (-^ - 6i?»')+*M {j! («j - 6rfo') -g^ (",- fitfoO} - J "^ 
Reduce this to a system of the 6th order, by the theorem of g 141. (Bru 



164] The Theorems of Brans and Poincari 847 

^ s ^ , u s ^ . 

We shall write Ah = Ah — A4 == A^i /*4 = Mb =* Me = /*'> 

6 2 

so that r= S ^. 

Let the coordinates of the three bodies be (5/, g/, js')* (?/> ?b'> ?«')> (97'» Js'* ?»')> 
and let m^g/ = pr\ where k denotes the greatest integer contained in ^ (r + 2) : 
the integrals whose existence we propose to discuss are of the form 

where a is an arbitraiy constant and ^ is an algebraic function of its 
arguments. The formulae of § 160 enable us to express the variables 
(?/> ?«'j ..., ?»'i7>i', ..., p/) as linear functions of (ji, y2» .--i 96»l>i» •••jPe)- we 
shall therefore, on making these substitutions in the integral, obtain an 
equation 

/(9i, ?», ..., ?6,JPi, ...,P6)=a (2). 

If the integral is compounded of the integrals of motion of the centre of 
gravity, / will evidently reduce to a constant ; if not, / will be an algebraic 
function of the variables (91, ..., ?6i Pi> •••> Ps). We have to enquire into the 
existence of integrals, such as (2), of the equations (1). 

(iii) An integral must involve the momenta. 

We shall first shew that an integral such as (2) must involve some of the 
quantities^, i.e. it cannot be a function of (g,, jj, ..., q^ only. 

For suppose, if possible, that the integral, say 

/(?i» ?a, ...» ?6) = a, 
does not involve {pi, pa» •••» Ps)- DiflTerentiating with respect to t, we have 

r=l C??r r=l ^?r Mr 

and therefore the equations 

^^ = (r = l,2. ...,6). 

must be satisfied identically; that is, / does not involve (^n 9i» ...» 9e)> ai^d 
so is a mere constant. 



in I 
lodi 



r w< 
>f b 
le t 






M 
of I 
ofi 
1 ii 



I+. 
>n-t 

»si< 
DOl 



hei 



r 



164] The Theorems of Sruns and Poincari 3 

wherey ia a rational functioaof the argumeDts indicated. The form of j 
be further restricted by the followiDg observation. If in the equatioi 
motion we replace q^. Pr, t by 5^**! Prl^~^, and tl^, respectively, where k ie 
constaot, the equations are unaltered. If therefore these subatitutioni 
made in equation (6), this equation must still be an integral of the ayt 
whatever k may be. 

Now / is a rational function of its arguments : it can therefore bt 
pressed as the quotient of two functions, each of which is a polynomi 
(ji, q,, ..., qt,Pi- ■■■< Pt, s). When in these polynomials we replace q^, 
by qrk', Prk~', 3^, respectively, the function / will (on multiplying 
numerator and denominator by an appropriate power of k) take the 
, A,kP + A,k^' + ... +Ap 
•''" B,ki + B,k^' + ...+B, 
where (j1,. A,, .... B,) are polynomials in (9,, ..., j,, pi, ..., p„ a), i 
dfjdt is zero, we have 



(B.« + B.l^. + ...+i),)(^-t. + ...+''^J 



-u.i.+^,^.4...,+^,)f|-i.+...+§).( 

Now k is arbitrary, bo the coeGBciects of successive powers of k in 
equation must be zero ; and therefore 

iB, 

it "• dl 



A, 




dA, „dA,_ dB,_ dft 
dt * • di ' dt ' dt 






dt ' dt ' 
These (5 + J» + 1) equations are equivalent to the system 

\_dA,_\^dA^ _1<U,_J^AB._ 1 dB, 

At dt ~ Aj dt ^ '" ~ A^ dt B, dt '" B, dt ' 
from which it is evident that each of the quantities 

A, A, A, B, B, 

A,' A,''" • A',' A, A,- 

is an integral : and thus we have the result that any integral 9uch oaf a 
compounded /rom other inleffrale, which are of the form 

g-jg- l-'f' f-'\ . Constant, 

6.(g. S.,Ji P;') 



164] The Theorems of Bruns and Poinca: 

It can be shewn in the same way that each of the other ii 
of <?, satisfies an equation of this kind. Denote the varioui 
■^'i 1^", ..., so that 

G.^^'-i^"' , 

and let the equations they satisfy be 

1 ^' = ' 1 ^' = " 
f dt ~'^' ^" dt " "•" 
then we have 

1 dG, u d^' V d^|r" 

G, dt ^ dt >fr' dt ' 

where Q) is a polynomial in (p, p,), of oider unity, 

(qi, ,.., qt, s). Thus 6] satisfies the equation 

and therefore (since (?i/6, is an integral), 0, also satisfies the 
do, „ 
-di-""'- 
As Q, and G^ satisfy the same di£ferential equation, we si 
^ to denote either of them: so is a real polynomial in(p,, ■• 
which satisfies the equation ^ = cd^. 

Now ^ is merely multiplied by a power of k when qr, p 
respectively by 5,^*, prAr-', si* : since 

^ dt r-\ \3?r f* ^r 3<Ir/ ' 

we see that w is multiplied by Ar~' when this substitution is 
that w cannot contain a term independent of (p,, ..., p,), si 
would be multiplied by an even power of J; ; ra is therefore ol 

w = <i>ipi + wjP, + ... + a),p,, 
where each of the quantities i»r is homogeneous of degree —1 
(?i, ..-,?., a)- 

Further, let one of the terms in ^ be of order min (pi, ... 
n in {q,, .... q,, s), while another term is of order m' in (j 
order n' in {qi, .,.,q,,a): since these terms are multipli 
power of k when the above substitution is made, we havi 

— m+2n = — m' + 2n', 
so m —m' is an even number. Hence ^ can be arranged in 

.<f> = 4>, + 4>»+^, + ■■■ , 
where 0o denotes the terms of highest order in (pi, ... , p,), 
of order less by two units in (p„ ...,p,) than these, and so 
the quantities ^^ is a polynomial in (pi, ...,p,, Qi, ..., qt, s), 
{pi, ...,p,) and also in (5,, ...,q,, «). 



r 



164] TJie Theorems of Brum and Poincar^ 

ProceediDg in this way, we ullimately arrive at the alter 
either 

or else a function ^ exists, which is polynomial in 51, ..,, q,, 

geneous in ji 2b and also in pi,pt, and is free from any fact 

mere powers of pi and p,, and which satisfiea the differential eq 

P;9f ^a^ 

Now let ^ = api' + bpi + cpi'-'pi + . . . ; 

equating coefficients of p,'"''' and p,'+' on the two sides of the 
we have 

Hiadqi' tt^bdqt' 

The quantities a, 6, c, .. . are polj-nomiala in (j,,^!, ...,5,): t 
a common polynomial factor Q, so that 

a = a'Q, b = b'Q, etc. 
Let ^' « a'pi' + b'p} + c'pi'-'p, + . . . , 

so that '^=0^'- 

Then 

= «i'Pi + Ws'Pi. aay, 

, 1 0o' ,1 dh' 

where oh — — , ^-, «» = — r, 5— . 

/iiO oji (lib oq^ 

The left-hand side of this equation is a polynomial in < 
P\i P))) hut if a contains 9,, then a>,' contains a', or some i\ 
a denominator. Hence ^' must contain a', or some factor of 
But this is inconsistent with the supposition that a', V, ..., ha 
factor. Hence a' cannot involve q^; and therefore w,' is ze 
(Ug' is zero. 

„. 1 30 1 3Q 

Thus *>l"rt~^ > Wa=y^— ^, 

and therefore 



164] The Theorems of Brujm and Poinec 

where to', a",... are the values of «> when the values of a 
0i>'> 0g"> •■■ respectively are substituted in it. 

Let 'I> = </),'<fr,"^"'.... 

Then we have 

= «'+<»"+ ... 

= n, 

where fl is a linear function of {pi.Pi, •••,pi), the coefficiet 
functions ot{q„ g,,..., qt)- 

Now $, from the manner of its formation, is a rat 
(9ii?t> ■■■,qt),^ot involving «: and it is clearly a polynomial 
So we can apply to O the results already obtained, whit 
multiplying 4> by some rational function of qi, q^, .,., q, 
therefore that ^ satisfies the equation 

r-l ft. 3?r 

This is a partial differential equation for ^: there a 
variables, and 5 independent solutions can at once be f< 

quantities (i^ -tB] (Ijll^IlP!). n fou.ws th. 

only of the quantities 

Now the factors of O differ from each other only 
roots 8 are used in their formation : so when such a relati' 

(9,, (f, q,) that two of these roots a become equal to 

two factors of <I> will become equal to each other ; hence if 
as an equation in pi, at least two roots will become eqi 
When this relation 

/(?..?. 9.) = 

exists between (91,5,,..., 5,), we shall therefore have 9<I>/3pi ^ 
d^jdpt, .... d^/dpt will each be aero. 

Since <^ is homogeneous in (^,p,, ...,p,), the equation 

^'3p, ^'dpt '" ^'dp,~ 
is equivalent toO = 0: ao$ = does not constitute an equ 
of the equations d^/dp^ = 0, . . . , 9*/3p, = 0. 



^heorems of Bruns and Pmneari [ch. 

8 are given to the variables which satisfy the e< 
emeote are cooaected by the equatioD 



k^l^-l 


8p,)-0; 




pi p.) satisfy the equations 9<I»/8p, t= 


0, this « 


kl^' 


-0, 




ween the increments iq^ must therefore be equiva 


k'iy- 


-0. 




3DS 






3//3,, dfliq, 
dfliq, 3«>/3?, ■ 


3//8?. 
■ M./S5.' 




le equations d^jdp,= 
values of ?„ 3., ... 


= ; and so, since S 
g.. Pi Pt which 


£?*isz 
ft 3?, 
satisfy tl 



r=l f^r^r 

•■ and S — i— = are therefore algebraically de 

r=l /*rO?, 

as d^/dpr'=0. Now the actual values of (ji, ,,., q,) 
this algebraical elimiDation ; so we can replace q, 
equations : and thus we see that the equations 





/(„ 


ft 


...,,.+ &<).o. 




ft9? 


/(?. 


ft 


....,.+&.).o. 




lences of the 


equati 


ons 






i.Pi.- 


.p.)- 


■ft 3?, 


«(?.. 


■ ■,q„p„- 


,p.)-o 

= 1, 2, .. 


f eliminating 


t between the 


equations 




K-*%'- 


..,,. + &.) = 


0, 




/('■-£'• 


...„..^.) = 


0, 





164] The Theorems of Srum and Foinea 

must be an algebraical cotubinstioQ of the equations 

5;*<«' '••'^ ''•'''ik'' 



■•i''>-i:l;*<9 «••?■• 



Mow one such algebraical combination of tbeae equations 

^(9i q>.Pu ...,p.)=0; 

for it can be derived by multiplying the equations by (pi, ... 
adding them. We shall shew that it is the eliminant whic 
mentioned. 

For let the eliminant in qnestion be denoted by ^ ; then 
4 /3^ . , 3* - \ „ 
,1(37/^' + ^,^"^ 
must be a combination of the equations 

r=l/trPJr r=l,=l/*rOq,Bq,\ I*, t^, I 

Since the latter equation involves it, we see that it canm 
combination : and so we must have 

s^/a?i yy/9g ,^ _ 3^/3g. ?^_1??! / 

3//9g. ~ 3//a?, ■■■ a//3?. ' elp, /*,32, ^'' 

The identity of these equations with those which ha' 
found for 4> shews that the equations = and '^ = are eqi 
4> = is the elimioant of the equations 



and 






Now the equations _/'(5i, 5,, .... gi) = 0, which are the con 
equation for a may have equal roots, can easily be written 
result enables us then to find all possible polynomials <E> 
fectorisation of*, to find all possible polynomials .^0, 

The 8 roots s are the 8 values of the expression ± rj 
''11 **» ^1 denote the mutual distances : so we may have two 
a result of any one of the equations 

r,=0, r, = 0. r, = 0, r,= ± r„ r,= + r„ r^= ±t-„ r 

The equation r, = gives 



r 

t 



164] TJie Theorems of Brums and Poincari 369 

When Tx is zero, this case reduces to that which was last discussed : and 
since the polynomial ^ is not resoluble in this special case, it cannot be 
resoluble in the general case. 

Thus finally, no real polynomials ^o» involving 8, can exist. 

Summarising the results obtained hitherto, we have shewn that any 
algebraic integral of the differential equations, which does not involve t, is 
an algebraic function of integrals ^, each of which can be written in the 
form 

where <^o is a homogeneous polynomial in the variables p, say of degree k, 
and a homogeneous algebraic function of the variables 9, say of degree I : 
^ is a homogeneous polynomial in the variables p, of degree {k — 2), and 
a homogeneous algebraic function of the variables q, of degree (2—1); ^4 is 
a homogeneous polynomial in the variables p, of degree {k — 4), and a 
homogeneous algebraic function of the variables q, of degree (i - 2) ; and 
so on. 

(viii) Proof that <f)o is a function only of the momenta and the integrals 
of angular momentum. 

We shall now proceed to shew than an integral 0, characterised by these 
properties, is an algebraic function of the classical integrals. 

The equation 



d<f>_ 



dt 



= 0, or 



r«l \Mr Oqr Oqr oprl 



gives on replacing by ^0 + ^a + ^4 • • • , and equating terms of equal degree, 

r«l oqr fir 



= 1 ^^Pr 4. ^^'"■^ ^ 

r=l 3?r f^ dpr dqr ' 

r^idprdq/ 

The first of these equations is a linear partial differential equation for 0o 
which can at once be solved, and gives 

where p Ml^M} (r = 2, 3, ...,6). 



360 The Theorems of Bruns and Poincari [ch. xiv 

Let the expression of ^, in terms of the variables ji, Pg, ...,P6> Pu •••»!>«> 

be ^j^/aCffi, Pj, Pzi ..., P^iPu --'tPe)* 



we have 



or 



Integrating we have 

SO there can be no logarithmic terms in I Xdqi, where 

X = 2 — ^— , expressed in terms of q'l, P,, ..., Pe, i)i, ...,|)«, 

r=l (?Pr ^3r 



\9pi .-2 9-P. /*l/ 3?1 r=a \9pr 3-Pr /V 

,=1 dpr dqr ,=4 3Pr V*l 9<?1 /*r 9?r/ ' 



du 



Jdq, 



) 



If F denotes the expression of U in terms of the variables 

9ii -^a> •••! -^«> Pi» ••• >P«> 

we have 

a^r /Lfci aP^ ^ ^ ajl 3?i r=2 aPr /*r 

The terms in JT which may give rise to logarithmic terms in IXdqieLre 
now seen to be 



SO 



the terms which may be logarithmic in jXdqi are 

r=2 AtrPi 3^r J r=2 #=2 3Pr MrM.Pl 9P J 



r=2 dPr flrfhoFJ^ 



Now. F is a sum of three terms, each of the form (A + -Bji + C??i*)""*. 



r 



164] The Theorems of Brum and Poincard 

Taking each of these terms separately, we have for the trsuscei 
of the last expreesiou 

r.. 3P, fyp, -r^"" (!•- iAC)i 

r.t.-idP/iLrh.p.iO-J^ClP,"" (B--iA< 

r-sSP, /Vft2CV^C3/'/'° {B--4ACr)>' 
Thus for each of the fractions {A + Bq, + Cqi)~^, we must ha 

r-. iP, I^P, r.I ... <lP, /•rftPi ' 3P. -. 3^- /»rft ' S- 
Now for the first of these fractioDS, namely (51' + 5,' + Jj')"', ' 

A.^(P.' + P,-). JB.*^ + *1^, C.l+e^ 
Pi' fcPi' ftp.' ft'P. 

so the first of the three equations will be 

\ ft'Pi' ft'Pi'/ r=* 3i*r MrPi r=B ^Pr ftPi Vft'Pi" /tj'Pi' 

KhP.I^'tJiP.I^'l 

V. Pr (if, l^p, 

IP, 1^ (SP, ft' 
or (since it^ = li^^Ht) 



I 8/. P, (d/.I^P,.d/, ftPA_n 



A-I- 

and oa solving this equation we see thatyo is a functioD of 

Pi.i>> p.. -P.. A, Cp*3.-p,9,). and (p.g'.-p,?.). 

Since the three expressions {A + Bqi + Cq^) are Unear fun< 
three quantities (qi' + q^ + 5,'), {q,q^ + 5,5, + qtq,), (qt + qs + q,'] 
our present purpose replace them by these three quantities : a 
expression {A +Bqi + Cq,') may be taken to be (q,q, + qtif+gigi 

,, (tf-' + "B ,,) + (^ +M!) (^. + ^P'*) + (S^- + MiV* 
\ Pi /*pi / \ Pi Pi/\pi f^Pi / V Pi P\ i\. 
80 for this expression we, have 

^^itE*j^ ^Ap* + ^'P.P. I i»-^*P^ , /*°f .p. 

Pi Pi' /*'Pi' Pi' ^'pi' ' 

/Pi /*'pi' m'Pi* ' 



164] The Theorems of Bruns and Poin&ird 

ao we have -^ = ^ 5 — ir= ^ ^r 's- ' 

at r-l OPr at r.i dpr Oqr 

and the equation for /, is 

/.-^(^ ^■■p' '"'-F.ilM' 

where Yr stands for dUjdq,, supposed expressed in terms of q^, P^, ... 
Pi, ...,p,. We have therefore 

M, da tdv, ^ • [dv fio iHv, de\, 

ft 3ft '^ +,rA3p, ft.ft 3ftJ dPr I ^' 

^ 3g ^' mjm, 

ft Sp, l(4+% + C5,')l 

I „,iB „»B _3^ , „_ 9A'l 

,.> \3ft /^ft 3p,/ ^ (i"-4.1(;)(jl +%, + (/'},")' 

where the symbol £ indicates summatiOD over the 3 values of the expre 
(A+Bq,+Cq,-). 

Now the term x(jP> -P*. ft ft) oaDnot give rise to t 

involving ( A + Bq, + Cj,») in the denominator : so the quantities multipl 
each of the expressions (A + Bgi + Cq,'')^ must themselves have the i 
character as ^, i.a they must be polynomial in (ft, •■-.p,) when exprt 

in terms of (j,, Jj, ,.., },. j) p,). We see therefore that the express 

SB 



— +2I fSg >^Pr !I0\ ~ 37',~"-°3P,'^*^ap,'^ ^'■3F, 



-J 

ii 3p, ^ 'r^j \3p, lirPi dpj B'-iAC 

must be a polynomial in (ft ft), when expressed in terms of (ft, .. 

q> ?.). Taking first 4 + ^j, + C?,» = 5,» + 3i' + g,'. this exprei 

becomes 

ft SO _ I /^_ftp,30-, 

ft 3pi r=8 \3pr /^ft 8p„J 
.. -yr(/»(P.'+J'.') + ?,(f,ft+P.ft)l+/',|y(f.ft + P,ft)+;.(ft' + ft'+i! 

2|ft"P."+ft'P,' + (ftP.-ftP,« 
or (omitting a factor p.) 



Pi 3Pi r-* V3pr 



-PrlPi(9»'+g.*)-p.gig.-p.glg.l + (grp|-prgi)(pigi + Pig«+Pig 

2pi i(?ip> -p^qiY + {q^p, - Ihqxf + (p,S, - p.^^"} 



164] The Theorems of Bruns and FoincarS 

But we have 

and therefore 

/, = xiPt.-.P:Pi p,)-mh(L,M,N)T^'U. 

Thus 

='k(L.M,N)(T"~mT^'U) + X(P* i'.,Pi,...,i'.) + ^< + * 

The integral can therefore be compounded from two other 
namely : 

1" the integral A (L, M, N) [T— U)'^, which is itself compoui 
the classical integrals, 

and 2° the integral ^', where 

and 0,' = x(-P.>—.^..Pi Pt). 

0; = 0. + m(m~l){m-^) ^ ^j^^ ^^ ^^ ^^^^^ 



But 0' is an integral of the same character as 0, except that it 
term, ^', is of order two degrees less in (p,, ..., p^) than the high 
00, of 0. Now we have shewn that can he compounded from the 
integrals together with the integral 0'. Similarly if>' can be con 
from the classical integral together with an integral ^" which has 
character as 0, but is of order less by 4 units than <f> in the va 
Proceeding in this way, we see that can be compounded of the 
integrals together with an integral 0*"', whose order in (pi,...,p,) 
unity or zero. If 0'"' is of order unity in (p,, ..., p,), then in the ec 

we must evidently have A; = ; in this case, therefore, ^'"' is compc 
the classical integrals. If 0i"i is of order zero in (p, , ,,., pj, it ia a 
of (q,, ..., q^ only : but we have already shewn that such integra 
exist : and so in any case <f> can be compounded algebraically 
claasicsl int^rals. Hence we have Bruns' theorem, that every 
integral of the differential equations of the problem of three bodies, v 
not involve the time, can be compounded by purely alg^aic processet 
classical integrals. 



164] The Theorems of Bruna and PoinmrS 

^"•"f.f f.f f'.-r.pl.c«ibytheir 

(01, E), ..., (-^1, ff), this equation muat become an identity: 
happen only if 

f =». -f - -f =». -f =« ' 

i.e. if each of the expresedons 

P, t'-<lh, «-^, .... t-<l>t, t-yjr„ .... t-^|rl 
is an integral. Hence any algebraic integral of the problem oj 
vihichinvolvest can be compounded (1) of a^ebraic integrals which 
t and (2) of integrals of the form 

t— ij> = Constant, 
where ^ia an algdtraic function o/{qi, q,, .... 5,, j>i, ...,p,). 
Now it is known that 

is an integral : hence any algebraic integral of the problem, wb: 
can be compounded of 

(1) algebraic integrals which do not involve t ; 

(2) integrals of the form 

A - «^?- + "^g« + "^g^ . Constant, 
Pi+Pt+Pr 

where <ft ie an algebraic function of (9,, ..., qg.pi, .... p,); and 

(3) the classical integral 

wiigi + T7itg«H- mtgT 
Pi+Pt + Pr 

But the integrals in classes (1) and (2) are algebraic integ 
not involve the time; and hence, by the result already obtai 
combinations of the classical integrals. 

Thus finally every algebraic integral of the differential eqi 
problem of three bodies, whether it involves the time or not, can b 
from the classical integrals. 

Bruna' theorem has been extended hj Paiolevi', who has shewn that 1 
the [a^>blem of n bodies which involvea the velocitiea algebraically (whethei 
are involved algebraically or not) is a combination of the claasical int^rale 

■ Bull Aitr. zv. (1S98), p. 61. 



tms of Bi-uns ant 



□other theorem od th< 
tilem of three bodies, 
,nd was discovered in 

ion of ths restricted p, 

of three bodies, the 
ten in the form 

dpr ' dt dq, 
n 9i. ?a. with period I 

dpiSpt dpt' 
jircitmataoce would 
we shall modify the 
the corresponding H 

' = A be the integral o 

J_afe" dp, _J. 
2hdp/ dt'^ 2i 
1 function H equal t 
restricted problem of 

^aff ^^_^^ 

dpr ' dt dq, 
rallies oi /i, H cao be 

1 npa . , 
4p,* p,' ^ 
ow zero, and (H,, 3t 

!S9. Nouv. anth. dt la Ml 



165] The Theorems of Bruns and Poincari 

(ii) Statem&it of Poincari'a theorem. 

Let * denote a fuoction of (gi, q^, p,, Pi, ft) which ia one-valu 
regular for all real values of g, and q,, for values of fi which do not 
a certaiD limit, and for values of p, aud p, which form a domain D 
may be as small as we please ; and suppose that ^ is periodic with res 
5, and q„ having the period Ztt. Under these conditions the functioi 
be expanded as a power-series in ft, say 

where 4>[,, 4>,, <t>,, ... are one-valued analytic functions of (ji, jj, 
periodic in qi and q,. Poincar4's theorem is that no integral of then 
proHem of three bodies exists (except the Jacobian integral of ener 
integrals equivalent to it), which is of the form 

4> = Constant, 

where 4> is afuncHon of this character. The proof which follows is ap 
to any dynamical system whose equations of motion are of the same 
those of the restricted problem of three bodies. 

The necessary and sufficient condition that 4> = Constant may 
integral is expressed by the vanishing of the Poisson-bracket (ff, 4>) ; 

{H,, *.) + fi {{ff, , <P,) + (H„ *,)1 + m' {(-ff.. *.) + (^i. *i) + iff.. *,)) H 
and therefore (ffo. *.) = 

(ff,.4>,)-)-(.ff., *,) = 0. 
(iii) Proof thai 4>, is not a function of H,. 

We shall first shew that 4>b cannot be a function of Hf For 
a relation of the form 4>, = ^ (fi,) to exist. From the equation ffo = £ 
we have on solving for p^ an equation of the form pi = (H^, p,), an' 
be a one-valued function of its arguments unless dH^jdp, is zerc 
domain D. Replacing p, by its value in the function <l>o {q,, 5,, p, 
have an equation of the form 

*« (?i, 9i. Pi. p.) = ■^ (9i. ?i. ^.. P>) ; 
and as 4>g is a one-valued function of its arguments, -^ will be a on 
function of (ji, q„ H,, p,); but by hypothesis, the function ^ depei 
on fig. It follows that ^ is a one-valued function of H^, so lonj 
variables p, , p% remain in the domain D, and provided dHsjhpi is not z( 
or more generally provided one of the derivates dH^ldpi and dS,/dj 
zero in D, a condition which is evidently satisfied in general. Sin( 
one-valued function, the equation yjr {H) = Constant will be a on 
integral of the differential equations, and therefore 4> — ^ (H) = i 
will also be a one-valued integral, and will be expansible aa a power- 



166] The Theorems of Bruna and PoincarS 871 

(v) Proof that the existence of a one^alv^d integral is inconsistent tnth 
the result o/(m) in the general case. 
Consider now the equation 

or 23- -= 2 ■=— ^— = 0. 

r-l OPr Oqr r=l OPr 9?r 

As the functions J?, and 4>, are periodic with reapect 
be expanded in series of the form 

*i= 2 a„„^e""'*+-w^= 2 C^,^t 

where m, and fw, are positive or negative integers, and the 
and C^,i», depend only on p,,p,. We have therefore 

80 the equation 2=— ^ 2-5— 5— =0 

f=l OPr OJr r-l OPr 3?r 

becomes ^2__^B™„^ ?( 2^ m,^*) - __2^ C«,.«. ?( I^w,' 
or (since this equation is an identity) 

This equation is valid for all values of pi, p^: and thei 
p, and p, which satisfy the equation 

9p. 3p. 

we must have either 

S«„«,= 0, or m,3*o/ap, + ma3*,/3p,- 
We shall say that a coefficient ^m,, «■, becomes sectdi 
values such that m^ dHt/dp^ + m, dH^ldp^ = 0. 

As fl is a given function, the coefficients B^^^^ are gii 
case of dynamical systems expressed by differential equati 
are considering, no one of these coefficients will vanisl 
secular, and we shall take this case first : so that the equa 

m, 3<I',/3pi + m, d^^/dp, = 
is a consequence of the equation miSSaldpi+tn,dBa/dp,= 
Now let i,, k, be two integers : suppose that we give 1 
such that the equation 

3g, ^ 35. 
kjdpi kidp. 



ie Theorems of Bruns and Po 

can find an infinite number of pairs oi 
is zero: and for each of these sya 
jdpi + -m^dBtldpt is zero, and coosequei 

m, S0t/^ + ma 9't>./9ps 
ing these two equations, we have 
dHJdpr ^ dHJdpi 
3"t'o/3pi ~ 9*o/9pi ' 
(ffo. Oo)/3(Pi. Pi) is zero for all valu 
l^a are commensurable with each other. 
ere are an infinite number of systems 
no is zero : as the Jacobian is a contini 
identically, and consequently Og must 
iry to what was proved in (iii), and 
>n as to the existence of the iotegral < 
liltonian equations possess no one-val 
, provided no one of the coefficients B, 

I of the reatrictionB on the coefficients B, 
to consider the case in which at least • 
vhen it becomes secular. We shall s 
nd (m,', m,') belong to the same claaa ' 
fl^/m,', and that in this case the co 
the same class. 

shew that the result obtained in (v) as 
igrals is true provided that in each of i 
>t' ^Bt,. >m which does not vanish on bo 
coefficient fiB,„«, is zero, but the coel 
ive values such that wti dSJdp, + m, Si 
Hajdpt = 0, and consequently 

relation m, 3<t>o/9pi + m, 3*»/3pi = can 
le equations, it can be inferred from th 
Lhe same as in (v). 

completely defined by the ratio of th< 
sarable number, and let C be the class 
hall say for brevity that this class C 
lis domain, if a set of values of p,, p, < 

Sp, dp. 



166] The Theorems of Brum and Foincar^ 373 

We shall shew that the theorem is still true if in eveiy domaiD S, however 
small, which is contained io D, there are an infinite number of clasaes fur 
which not all the coefficients of the class vanish when they become secular. 

For take any set of values otpi, p,, such that for these value 

dpi dpt 

Suppose that X is commensurable, and that for the class whic 
to this value of \, all the coefficieuta of the class do not vanif 
become secular : the preceding reasoning theu applies to this i 
and so for these values of p, and p, the Jacobian 3(ff,, *«)/9{j 
But, by hypothesis, there exists in every domaiu B, however sn 
contained in J), an infinite number of such sets of values of 
Jacobian cousequently vanishes at all points of D, and therefore 
tion of Ha; so, as before, there exists no one-valued integral disi 

(vii) Deduction of PoinoaTi'a theorem. 
In the four preceding sections, we have considered equations 
dSr^dH dp^^_da 

dt dpr' dt" dqr 
in which H can be expanded in the form 

where the Hessian of H^ with respect to p^ and p^ is not zero 
involve qy and 5,, and B^, jff,, ... are periodic functions of 31,91: 
shewn that no integral of these equations exists which is disti 
equation of energy and is one-valued and regular for all real val 
q^, for values of fi which do not exceed a certain limit, and for va 
p, which form a domaiu D ; provided that in every domain, ho 
contained in D, there are an infinite number of ratios m^jm, for ' 
the corresponding coefficients ■£.■„«, vanish when they become st 
This result can be at once applied to the restricted prob! 
bodies : for we have seen in (i) that the etiuations of motion in 
are of the character specified, and on determining the function 
expansion we find that the last condition is satisfied. Poincan 
thus established. 

This theorem has been extended by Poincard to the general problem ol 
of. Nowi. M&h. da la ilic. CA. i. p. 253 ; it baa also been extended by P(unl( 
(1900), p. 1S99, who has shewn that no int^rals exist which Eire one-value 
in the velocities and involve the coordinates in any way. 



THE GENERAL THEC 

Introduction. 

all now pass to the study of the 
lynaraical systems. For simplii 
isider the motion of a particle 
action of conservative forces, b 
extended to more general dyna 
already been observed (§ 104) tl 
ele with two degrees of freedoi 
edacible to the problem of fin< 
line-element; an account of 
be regarded as falling within th 
}erties are however of no impoi 
eory of geodesies is fully trea 
, we shall only consider thos 
1 interest. 

rincipal results which have been 
167-171), to the stability of a | 
h respect bo small displacemeu 
if a given group of orbits with 
far the orbits preserve their gei 
b time (§§ 177-179). 

Periodic solvtions. 
interest has attached in recen 
modes of motion of dynamici 
of the system is repeated at n 
purely periodic. Such modes o 
periodic solution is also used i 
ite configuration is periodically 
lies, a solution is said to be pei 



'Ae Gfenerai Theory of Orbit- 

Qctions of the time, although the b 
le absolute positioQB at the end ( 



ally the motion of a particle in a 
' the action of cooservative forces, 
its will exist in the ueighbourhooc 
' the particle, namely the orbits ( 
the particle about this equilibrii 
m is unstable, it may happen that 
vibration are imaginary, in which a 
, or that the period of (me of tl 
'hich case these real normal vibrat 
! orbits will evidently however be u 
jrhood of the position of stable equ 

normal variables /or a known perim 
lich define a periodic orbit are a 
le to Poincar^*. 
)f the dynamical system coosiderf 

dt dpr ' dt Zq, 
' does not involve the time t explic 

,(0. 9. = 0.(0, j>. = -^.(0. P.- 

:b define a known periodic orbit of 
enerality if we suppose the coordin 
le lapse of a period the variables { 
1, increases by tv. 

ioDS t can be eliminated : let the res 

Q 

ff,, St, 0i have the period 27r. 
vtem the contact-transformation del 
dW p 9^ 

NouvtUa Uethoda dt la me. Ctl. u. p. 66S 



876 The Gmeral Theory of Orbiti 

The equations of this tranafarmatioD can be written 

p. -ft -9. (P.). 

p.-p.. 

The equations of motion of the dynamical system, 
variabletF, are 

dt BF/ dt 3Q, 
and from the above equations of transformation it is evi( 
solution is now defined by the equations 

Q, = 0, Q, = 0, P, = 0, i'.= 't,(i: 
This form of the equations of the orbit will be called Poi 

169. A criterion for the discovery of periodic orbits. 

We shall now shew that the existence and position of ] 
determined hy a theorem* analogous to those theoren 
position of the roots of an algebraic equation by conside 
the sign of expressions connected with the equation. ^ 
suppose the dynamical problem considered to be thai 
particle of unit mass in a plane under the action of con 
result can be extended to more general systems without 

Let (x, y) be the coordinates of the particle at tir 
fixed rectangular axes in the plane, and let V'{w, y) be 
function, so that the equation of energy is 

H^ + y')+F(<r,y) = A, 
where k is the constant of eneigy. 

The differential equations of motion of the particle J 
fourth order, and their general solution consequently ii 
constants. One of these constants is, however, merely 
to t, which determines the epoch in the orbit, so ther 
distinct orbits. This triple infinity of orbits can be a 
containing a double infinity of orbits, by associating to^ 
which the constant of energy has the same value h : su< 
can be defined analytically by the principle of least a 

* WhitUker, liontkhj Notictt R.A.S. i.ui. (1903), p. 1S6. 
t For the eitensioa to the rastrioted problem of three bodies, e 
Lin. (1003), p. 316. 



ty^eorto^TH \\^ ^^^^'^ ^^^ ^f OrbUs 377 

J. ne mutuf 
the value of twex) jeen two given points (a?o, yo) and (x^, y^ is such as to make 

2i» -Jja* — bression 

L-J f{A-^(^.y)}MW+(dy)'}* 

iipared with other curves joining the given terminal points 



staiioQuyasooir' 



<^'»)"^(^ follows tr*^ 



ample dose"' 
it We) 



mus 
wh( 



vdsf 



no 



81 



Tail 

(» 



Aiy simple closed curve G in the plane of xy \ and let another 
^^id curve C be drawn, enclosing C and differing only slightly fix)m 
lay regard C as defined by an equation of the form 

is the normal distance between the curves C and C (measured out- 
cm 0, and consequently always positive) and 7 is the inclination of 
mal distance to the axis of x. Then if / be the value of the integral 

the integration is taken round the curve (7, and if 7 + S/ denote the 
16 of the same integral when the integration is taken round the curve C 
that the symbol 8 denotes an increment obtained in passing from C to C% 
have 

8/ =j{{dxy + (dy)»}» 8 [h - V(x. y)}* + |{A - V{x, y)}* S [{dxf + (dy)»]*. ' 
But we have 

8 [h- v{x, y)}*=-MA- yi<^. y)}-» (^«^+|^«y) 



dv . 



= - i (A - 1^(«. y))"* ( g- COS 7 + g- sin 7) 8;), 



dy 



and 



_ ^ 



Id, 

Ave 



8 {(da;)' + (dy)'j* = 8p . dy = i<^ l(dx)» + (dy)'}*, 

r 



ib and 



where p is the radius of curvature of the curve C at the point {x, y). 
Thus we have 

8/ =|Kda;)»+(dy)'}*{A - F(a;, y)}-^{^^^li^> - i cos 7 g - 4 sin 7 ^we have 

This equation shews that if the quantity 

^-^-^ - i cos 7 -r * sin 7 15— 

P ' dx ^ dy lantities, we 

is negative at all points of C, then 87 is negative, and so the integr^^* relative 
its value diminished when any curve surrounding C and adjacenbf these resalts 
taken instead of C as the path of integration. ^* 



,)f*. 



878 The General Theory of Orbital 

tea 
Now suppose tbat aoother simple closed curve D can , 
C, and Buch that at all poiots of D the quantity f jo tn^ 



t' 



is positive. Theu, io the same way, it can be shewn tba^ 

diminished when any simple closed curve If, enclosed by D\ 

D, is taken instead of i) as the path of integration, S . tanns oi the 

When, therefore, we consider the aggregate of all simple cti/^. 
situated in the ring-shaped space bounded by C and D — which is ast (r* *• 

contain no singularity of the function Vix, y) — it is clear that X,\ i «« 

which furnishes the least value of / cannot be C or D, and cannot ( tba* "^ * 
with C QT D for any part of ite length. There exist, therefore, ami 
simple closed curves of this aggregate, one or more curves K for wht 
value of J is less than for all other curves of the aggregate. Since /^ norma 
not coincide with C or D along any part of its length, it follows th^ 
curves adjacent to ^ are all members of the aggregate in question, and a 
that the curve K furnishes a stationary value of / as compared with tiUV. orbits 
curves adjacent to it. The curve K is therefore an orbit in the dynamiL|j fun 
system. We have thus arrived at the theorem : If one cloaed carve lujep^' 
enclosed by another closed curve, ami if the quantity v^^ s' 

h-Vix.y) , dV , . dV mo1 

___^_icos7^-i8m7g- .fo: 

negative at all points of the inner curve and positive at all points of the outer y ' 
/urve, then in the ring-shaped apace between the two curves there exists a periodic *^ 
/orbit of the dynamical system, for which the constant of energy is h. As the T 
quantity 

h~y{x.y) . dV . . dV 

•xa be immediately calculated for every point on the curves G and Z>, de> 
ending as it does only on the potential-energy function and the curves 
emselves, this result furnishes a means of detecting the presence of periodic 



70. Lagrange's three particles. 
e shall now consider specially certain periodic solutions of the problem 
e bodies. 
the equations of motion of the problem be taken in the reduced form 
in § 160, and let us first enquire whether these equations admit of a 
- solution in which the mutual distances of the bodies are invariable 
t the motion. 



J 



^^^^^ 169, 170] The General Theory of Orbits 

^^^^ The mutual distances are 



379 



?i , iqi T"^ cos js cos q^ ^ ^ sin 5^, sin q^ ] + p-^^ q^^ 






and ^g2* + ;^-^ cosg,cosg4 ^^ ^^ sin g, sm 94 + , 

(^ 7?Ai4-7nA ^ ^ 2^31)4 ^ ^V (mi 

'^^ it follows that, in the particular solution considered, the quantities 

ji , ^2 , and cos ^8 COS ^4 ^- =^ sin g, sin J4 

must be constant, and hence the functions U, dUjdqi, dJJjdq^ must be constant, 
where U^^mim^rYr^. 

The equations 

^ . dH pi J A • ^^ P2 
9pi /^ ^ dpi fi'' 

shew that jpi and j}, must be permanently zero : while the equations 

dH vJ" dU 



iopleckr' 
il fei 






f 



shew that jpt and pt must be constant. 
Moreover, the equations 



= p, = — ^— s= -i^ + 



0=.p, = - 



aff 






shew that the expressions 

d 



and 






y'-p^-p^' 



cos g, cos 94 2 ' " ' sm g, sin 54 j 



cos g, COS 54 — 



^'-JPs'-p/ . 



sing 



8 sin j4 j 



jDsidered, 
je we have 

oat (§ 46) 



a?4 V"^"- "' ^^^ ^^ 2;,,jp4 

are zero, so we have 

tan 7, cot Qa = cot g, tan q^ = ^? — ^ , 

2/)8i>4 

and therefore p^ + p^a - 42 = ^ 2p,p4, 

or h^^ relative equiIibx.^.M eJBLsily seen to be — n& and 

an equation which shews 

bodies /* and ^i coincjV-* y=*+'7> u^^vb+e, r=waH-<^, 

in other words thp -^PP^^sed to be small quantities : neglecting a constant term, we have 

the motion ofm^ n(i7^-f^)-««(af +617) 



It follows tmii[a 
rest, the par' 



mj' 



^Wj+Wlj 



+«)V(6+,)f *-«. ((«-^+f)V(6+,)'}-*. 



in circular r^^^S *^^ retaining only terms of the second order in the small quantities, we 
xpression for K with which the equations for the vibrations about relative 

One CO 

were discovered by Lagrange in 1772. For references to extensions of these results 

attractlOJ^lgn, of n bodies, cf . Whittaker. BHHih Atsociatian Report, 1809, p. 121. 



I 



The General Theory of Orl 

e to the centre of gravity. This cond 
re in the same straight line. If thej 

in PRO = r^. sin QRO, and two simil 

is the centre of gravity of the particlei 
wtiainf^fiO sinQPa QR 
m^anQRO " sinPQR" PR' 

led with the preceding equation give 

"«. 

• the bodies mu«t he coUinear, or elae 

•uilateral. 

first the collinear cose, let the distan 

i;ravity (measured positively in the sa 

we shall suppose that o,< a, < a,, whi 

e discussion. Since the force acting i 

rcular motion round 0, we have 

n'a, = — mi(at~ a,)~* — m, (a, — o, 
.ngular velocity of the line PQR ; and ; 
-aj)-*+m,(a,-a,)-", n*a,-m,(a,- 
equations we readily find 
+ ky-l] + m,(l+ky(l^-l) + m,{i* 
I the ratio (a, — a,)l(a, — a,), 
ntic equation in k, with real coefficient 
ition is negative when k ia zero, and j 
one positive real root; such a root d« 
itios Oi lOtiOt', and if n is given, the 
letermined. It follows that there are 
•nblem of three bodies, in which the boo 

'-■nt ffinf" "h other; t 

rticles. -^ ("^^ 

'ciaily certain periodic solutions of the 

•n of the problem be taken in the redu 
rsteoq,,re whether these equations ; 
.e mutual distances ofthebiies are k 



170, 171] The Gmeral Theory of Orbits 381 

The conditions relating to the motion of Q and of R reduce to the same 
equation : and hence a motion of the kind indicated is possible, provided n 
and a are connected by this relation. Hence there are an infinite number of 
solutions of the problem of three bodies, in which the triangle formed by the 
bodies remains equilateral and of constant size, and rotates uniformly in the 
plane of the motion : the angular velocity of its rotation can be arbitrarily 
assigned, and the size of the triangle is then determinate. 

The two particular types of motion which have now been found will be 
called Lagramg^s collinear particles and Lagrange's equidistant particles 
respectively*. 

Example, Shew that particular solutions of the problem of three bodies exist, in which 
the bodies are always collinear or always equidistant, although the mutual distances are 
not constant but are periodic functions of the time. 

These are evidently periodic eolutums of the problem, and include Lagrange's particles 
as a limiting case. 

171. Stability of Lagrang^s particles: periodic orbits in the vicinity. 

It has been observed (§ 167) that in the neighbourhood of any configuration of stable 
equilibrium or steady motion there exists in general a family of periodic solutions, namely 
the normal vibrations about the position of equilibrium or steady motion. We shall now 
apply this idea to the case of the Lagrange's-partide solution of the restricted problem of 
three bodies, and thereby obtain certain families of periodic orbits of the planetoid. 

Let iS' and J be the bodies of finite mass, m^ and m^ their masses, their centre of 
gravity, n the angular velocity of SJ, x and y the coordinates of the planetoid P when is 
taken as origin and OJ as axis of x. The equations of motion of the planetoid are (§ 162) 

dx^lK dy^dK du dK dv__dK 
dt^du' dt dv' dt^ dx' dt~ dy' 

where K=^ J (tt'+i;") + w (uy - vx) - mJSP—mJJP, 

Let (a, b) denote the values of (x, y) in the position of relative equilibrium considered, 
so that for the collinear case we have 6=0, and for the equidistant case we have 

a=^(ini'-m^)ll(n%^+fn^j 6=iV3^ where I denotes the distance SJ, so that (§ 46) 

mi+m2=nH\ 

The values of u, v in the position of relative equilibrium are easily seen to be — n& and 
na respectively. 

Write x=a+(, y^zb+rj, u=s-nb+0, v=^na+<f), 

where (, ij, B, <f>, are supposed to be small quantities : neglecting a constant term, we have 

On expanding and retaining only terms of the second order in the small quantities, we 
obtain an expression for K with which the equations for the vibrations about relative 

* They were discovered by Lagrange in 1772. For references to extensions of these results 
to the problem of n bodies, cf. Whittaker, British Atsociation Report, 1899, p. 121. 



The General Theory of Orbit 

brmed : ve shall for definitenesa consider vil 
1 : in this case the expreeaion for K becomes 

motion are 

latioQs in the manner described in Chapter VI 
n ia 2n'/X, where X ia a root of the equation 

f X' given by this equation will be positive proi 
and they will be real provided 4 (}j-i*)<l, 
satisfied provided one of the masses S,JiB aut 
«n tkii condition it aatisjied, there exist two fan 
vicinity of it* equiditlant conjuration 0/ n 
; approximation, 2ir/X, and 2ir/Xg where X,' ant 

X«-»V+CiJ-i')n'=0. 
on leads to the result that the eMinear Lagrangi 
»gjiation, for the periods of normid modes of ' 
itTtlly in the Tteighbourkood of a position of re 
SJ there exists a flxmUy af imstable periodic ori 

tlutt, for one of the modes of normal vibratioi 
listant configuration, the constant of relative e 
relative equilibrium, while for the other mode 1 
of relative equilibrium. 

'erential equation of the normal displace, 
proceed to consider the stability of orbi 
some particular solution of the raotioo 
under the action of forces derived froi 
^, is known ; and consider a solution w 
known solution, and for which the coi 

the normal distance between the two or 
let 8 denote the arc of this orbit froi 
'■ the time at which the particle passE 
i p the radius of curvature of the orl 
on of any point on the adjacent orbit 

lergy of the particle when describing tb 



171, 172] The General Theory of Orbits 883 



and its Lagrangian equations of motion are therefore 

U-(l+-)- = -^ 
\ pj p du 

these equations possess a known integral, namely the integral of energy 

Jti' + J^ f 1 + - j ^'+ Fas A, where A is a constant. 

Writing ^ = t; + A, where A is a small quantity, the first and third of 
these equations become 

p p^ p du 

ft 

it;» + t;A + — +F=A. 
P 



so 



Let V be expanded as a series in the form 

dv_fdv\ f^\ 



where {dV/^\ and (d*F/8u')o are -functions of s, and in particular 
(dV/du\^i^lp. 

Substituting in the two preceding equations, we have 

p 



Eliminating A, we obtain the equation 

'd'V\ . St;*) 



.. ^ (fd'V\ , St;* 



UsO 



or (taking s instead of ^ as the independent variable) 

'^ Idtjdw a/^\ 31 

and tilts %8 tiie differential equation of tiie adjacent orbit 

From this equation we can at once deduce consequences relating to the 
stability of the known orbit. For by Sturm's theorem*, if we have any 
differential equation of the form 

d^u r /.v 

* Cf. Darboux, Th, gin, des Surfaces, Vol. iii. 



The General Theory of Orhi 

1 range of values of t the quantity 1 
□titles a' and b', then an; solution 
the range will be zero again for aot 
- (,) lies between irja and irjh, pi 
comprehend this interval. It folio* 

8 positive at all points of the knowi 
Y adjacent orbit which intersects it 
it will intersect it again infinitely ol 
ailed the coefficient of atability for the 

g'a theorem. 

that the known orbit, with respect 
measured, is a periodic orbit whose p 
uation of an adjacent orbit, it is evid< 
leger, is also the equation of an adja 
lese two equations are in fact congi 
points are separated by one or moi 
djacent to the known orbit let u„, u„ 
ispectively the normal displacements 
iod, at the same place iu the orbit, so 
-1)5). u^^,~4.{8+n^, un+, 
a solution of the equation 

, Un+t Eire three solutions of this linea 
a relation of the form 

e independent of s. 

shew that these constants k and k^ a 
Eicent orbit and of the number n, so 
ir set 

- 1) S), «'^, - ^ <» + mS). u'^ 
near function of the two solutions u„ 

ot BtabUity in 99 173-176, kU powers ot the i 
oing the diSereutiftl equations of th« adjaceu 

9 stability has been studied b; Levi-CiTito, ^n 
IB neglected terms give rise to inBtsbility in oe 
Tst-order terms are considered : this happans w 
is the ohanuiteriatia exponent {} 17S) and T is 



172-174] The General Theory of Orbits 386 

and therefore on adding periods to the argument 8, we have 
But from the equations 

we have Ci M„-h + CjWn+s = * (Cit^+i + C^Un+t) + ii (CiWn + CiUn+i) 

and therefore w',,^^ = ku'^+i + A:i ia'^ , 

which shews that the constants occurring in the linear relation between 
^m+a» y-'m+iy ^'m» are the same as those occurring in the linear relation 
between Wn+a, «n+i, Wn- 

Next, we shall find the value of the constant ki. From the equations 






cb* V da da 

d*Wn d'^+i , ^ dv f dUn dWn- 



wehave ^^,-^ -^^ -^- + -^(^t^. -^-t^n ^^ j =0, 

and hence, on integrating, 

du^_ d^^c ^here c is a constant. 
^ da da V 

Chaoging « to s + S, we have 
and therefore 

dun^i ( 1 dv^^i , , dUf^ 






80 that ki has the value — 1. We thus have the theorem* that if Un» w„+i, Uth-^ 
denote the normul diaplacementa in an orbit adjacent to a known periodic 
orbit in three X)onaecutive revolutionat the ratio k = (i/^+a + y^jun^i haa a conatant 
value, which ia the aamefor all adjacent orbita. 

174. The index of atability. 

The constant ratio Ar = (Wn+a4-Wn)/wn+i» where t^n, t^n+n Un+^^ve the normal 
displacements from a periodic orbit in three consecutive revolutions, is called 
the index of atability of the periodic orbit, for reasons which will now appear. 

* Korteweg, Wiener Sitzungsber. zoni. (1886). 
w. D. 25 



The General Theory of OHnts [ch. xv 

ature of the integral of the difiference-equation 

w«+, - kti»+i + «„ = 
as is well known, on the reality or non-reality of the roots of thp 
equation -X 

X'-k\+l = 0, 
«Qd8 on whether [^1 > 2 or |A;| < 2. 

)8ing first that k is positive and greater than two, we know that 
randent solutions of the difference-equation are of the form 

u = xS,/>(s) and « = X"S^(«), 

md I/X denote the roots of the quadratic (which in this case are 
positive) and ^ (s) and ^ («) are functions of s which have the period 
ing these functions so as to make the solutions u satisfy the 

ves linear differential equations of the second order for the functions 
•), we have two independent particular solutions of the latt«r 
: the general solution is a linear combination of these particular 
and consequently the general equation of the orbits adjacent to 
■n orbit, when k>2, is of the form 

1 and Kt are arbitrary constants, and <f>(a) and "^(s) have the 

irly if i < — 2, the roots X and 1/X are negative, and the general 
of the orbits adjacent to the known orbit is of the form 

M = X, (- X)S ^ (s) +£-,(- X)S t (8), 
and Kj are arbitrary constants, and where ^ and ^ are functions 
1 satisfy the equations 

*(. + «) = -*(.), f(, + S) — + (,). 
suppose that \k\ < 2, so that - 2 < i < 2 r let c" and e"** be the roots 
adratic in X, so that is now real and in fact is cos~' ^k. In the 
r we now End that the general equation of orbits adjacent to the 
■bit ia 

u - A- cos (~ + ^) («) + i" sin (^ + ^) f (»), 

Etnd A are arbiti-ary constants and where if> and ^ are functions of s 
fieriod S. 

these results important consequences relative to the stability of the 
iriodic orbit can be deduced. For if \k\ > 2, it follows from the 



f 



/ 



174, 176] 



The Oenertd Theory of Orbits 



387 



character of the expressions obtained for u that the divergence from the 
periodic orbit (or if ^ and '^ have real zeros, the oscillation about it) becomes 
continually greater as 8 increases; while if |Ar| < 2, the normal displacement 
is represented by circular functions of real arguments, and consequently will 
remain within fixed limits. We thus obtain the theorem that a periodic 
orbit is stable or not, according as the assoda^d index of stability is less or 
greater (in absolute value) than two. 

ExanvpU. Discuss the transitional case in which the index of stability has one of the 
values ± 2 : shewing that the equation of the adjacent orbits is of one of the forms 

where </> and ^ either have the period S or satisfy the equations 

and that the known orbit may be either stable or unstable. (Korteweg.) 

176. Characteristic exponents. 

The stability of types of motion of more general dynamical systems may 
be discussed by the aid of certain constants to which Poincare has given the 
name characteristic exponents*. 

Consider any set of differential equations 

dxi 



dt 



^X, 



(t = l, 2, ...,n), 



where (Zi, Xj, ..., Xn) are functions of {xy^, x^, ..., Xn) and possibly also of t, 
having a period jT in t\ and suppose that a periodic solution of these 
equations is known, defined by the equations 



Xi = (f>i (t) 



(i=l, 2, ...,fi), 



where 0i(< + T) = ^(t) (i= 1, 2, ..., n). 

In order to investigate solutions adjacent to this, we write 

^i = ^i(0+fi (» = 1. 2, ...,n), 

where (fi, fj, ..., fn) are supposed to be small, and are given by the variational 
equations (§112) 

dt jt=i 



"^f-fe 



(i= 1, 2, ..., n). 



As these are linear differential equations, with coefficients periodic in the 
independent variable t, it is known from the general theory of linear 
differential equations that each of the variables {< will be of the form 



* Acta Math, xin. (1S90), p. 1 ; Nouv, M6th, de la M€c, C€l, x. (1892). 



25—2 



888 The General Theory of Orbits 

where the quantities Sn; denote periodic functions of ( with t6 
and the n quantities un are constante, which are called the c> 
exponents of the periodic solution. 

If all the characteristic exponents are purely imaginary, tl 
(fi> £^>t •■•> fn) can evidently be expressed as sums and product 
periodic terms; while this is evidently not the r-ase if the c1 
exponents are not all purely imaginary. Hence the condition for 
the periodic orbit is that all the characteristic exponents mua 
iiru^narif. 

We shall now find the equation which determines the cl 
exponents of a given solution. 

In one of the orbits adjacent to the given periodic orbit, let (j9 
denote the initial values of (fi, f„ .,., f„) and let A + '^j be the 
after the lapse of a period.' As the quantities (^i,^„ ...,^,)are 
functions of (/9i, /3», ... ,ffn), which are zero when (jS,, 0t, ■■.,0„) 
we have by Taylor's theorem (neglecting squares and products of ff 

If Ok is one of the characteristic exponents, one of the adjacen 
be defined by equations of the form 

sothat 0i+>lri = e^'^8a(O) = e't^ffi <t = l. 

and consequently a set of values of j9„ /9j, .... j3n exists for 
equations 

(•-1, 

are satisfied : the quantity o^ must therefore be a root of the equt 

3A + ' *^ iff, ~ 

if, 3+.+ i_,., 3t. 



if. df. 



, Tke characteristic exponents are therefore the roots of this dt 
equation. 



176, 176] The General Theory of Orbits 389 

176. Properties of the characteristic exponents. 

When t is not contained explicitly in the functions (Xj, X^, ..., Xn), it is 
evident that if 

«<=0t(O (t = l, 2, ...,n) 

is a solution of the equations, then 

«^=<^<(< + €) (t = l, 2, ...,n) 

is also a solution, where c is an arbitrary constant. The equations 

fi = ^^(« + e) (i = l,2,...,n) 

therefore define a particular solution of the variational equations; but as 
d(f>i(t-\-€)/d€ is evidently a periodic function of t, it follows that the coeflS- 
cient e^*^ reduces in this case to unity: and hence when t is not contained 
explicitly in the original diffei'ential equations, one of the chara>cteristic 
exponents of every periodic solution is zero. 

Suppose next that the system possesses an integral of the form 

F(xif x^f .... Xn) = Constant 

where JF* is a one-valued function of {xj, x^, .:., Xn) and does not involve t. 
In the notation of the last article, we have 

^ {*< (0) + A + 1<} = i^ {<^i (0) + /94, 

where for brevity F{xi) is written in place of F(xi,X2,..., Xn). DiflFerentiating 
this equation with respect to /3i, we have 

dFd±, dFd±. .M'?±?-o « = 12 n) 

dx,dfii^dx,dPi^-'^dxndpi ^' ^»A...,n;, 

where in dF/dxj, dF/dx^, etc., the quantities (x^ X2, ...y a?») are to be replaced 
by 01 (0), 0j(O), ..., <f>n(0). From these equations it follows that either the 
Jacobian 3(^i, -^j, ..., y^n)l^{fii, At •••! fin) is zero, or else the quantities 
dF/dxj, dF/dx^, ..., dF/dxn are all zero when ^ = 0. If the latter alternative 
is correct, we see that (since the origin of time is arbitrary) the equations 

dF/dx, = 0, dF/dx^ = 0, . . . , dF/dxn = 

must be satisfied at all points of the periodic solution: this is evidently 
a very exceptional case, and the former alternative must be in general the 
true one : but when the Jacobian is zero, the determinantal equation for the 
characteristic exponents is evidently satisfied by the value e*^= 1, Le by 
a = 0: so that one of the characteristic exponents is zero. Thus if the 
differential equaiions possess a one-valued integral, one of ike characteristic 
exponents is zero. 

A comparison of ^ 173, 174 with the theory of characteristic exponents 
shews that in the motion of a particle in a plane under the action of 
conservative forces, the characteristic exponents of any periodic orbit are 



The General Theory of Orbits 

a, — a), where the characteristic exponent a is connected wi 
bility k and the period T by the equation 

it =-2 cosh a?*; 
rbit is stable or unstable according as a is purely im^nary 

imple 1. If the difierential oquatioDs do not involve the time explicit 
valued iDtegrals /*,, ..., Fp which do not involve I, shew that either 
3 exponents are zero, or that all the determinaiits contained in the me 

1151 '-'.^ -- 

v at all points of the periodic solution considered. 
ample 2. If the differential equations form a Hamiltonian syatem, 
teristic exponents of any periodic solution can be arranged in paint, 
1 pair being equal in magnitude but opposite in sign. 

!7. Attractive and repellent regions of afield of force. 
he general character of the motion of a conservative holon 
astrated by a theorem which was given by Hadamard* in 
icity, we shall suppose that the system consists of a pan 
which is free to move on a given smooth surface under fon 
a potential energy function V; a similar result will readil 
for more complex systems. 

et (u, v) be two parameters which specify the position of th 
urface, and let the line-element on the surface be given by 

da" = i'du' + IFSmAv + (Ma? 
i (E, F, 0) are given functions of u and ti. The kinetic e 
:le is 

T'=^[Eu' + 2Fuv + 0v'). 

<he Lagrangian equations of motion are 

dt\.duj 3k du ' (lt\dvj dv dv ' 

1 can be written 

* Joum. dt Math, (a), ul p. S31. 



176, 177] The General Theory of Orbits 391 

We have, by differentiation, 

ou av 

du dv du^ hcdv dv^ 

Substituting for u and v their values from the preceding equations, 
we have 



r.-(«<,-^r. {^ (g)' - ^'^^'^% « (!?)•).*(.,.) 



where 



*c^*)-r^%<*«-^-)-f(^^'-i«'^-i^'^) 



dv 



(i.'^-.i.if-.p]. 



A,^.,.a-n-^J{r^i-o^^) 




dV 

+ 



The quantities occurring in this equation can be expressed in terms of 
defcyrmMion-covariants''^, The principal deformation-co variants connected 
with the surface whose line-element is given by the equation 

d^ = Edu^ + "iFdudv + Gdv* 

are the differential parameters 

where <\> and -^ are arbitrary functions of the variables u and v. 
With this notation, the preceding equation becomes 

* The definition of a deformation-covariant is given in the footnote on page 109. 



892 The Oenierod Theory of Orbitg [ch 

Utilising the eqaation of energy 

Eifi + 2Fuv + G»» = 2 (A - TO. 
and observing that the expression 

a>(tt,^) <t{_dr!dv,-dr/du) 

Eu* + 2 fit* + (?i)» E{d VIdvy - 2F (S V/dv) <9 V/du) + 0(dVlduy 
contains the quantity (uSV/du+vdVjdv) as a factor, we can write 

where \ and fi contain in their denominatorB only the (quantity 

and where ly denotes the expression 

*(3r/a», -dVidu)i{EO-F'); 

we readily find that /p can be expressed in the form 

/r=A,(F)A,(r)-iA(F,A,(7)). 

Consider, on the orbit of the particle, a point at which V has a mini 
value ; at such a point v is zero and V is positive : as A, ( F) is essen' 
positive (since the line-element of the surface is a positive definite fori 
follows that /r > 0, the inequality becoming an equality only when A, ( 
zero, i.e. at an equilibrium-position of the particle. 

As the particle describes any trajectory, the function V will either 
an infinite number of successive maxima and minima (this is the general 
or (in exceptional cases) the function will, after passing some point o 
orbit, vary continually in the same sense. Suppose first that the form 
these alternatives is the true one : then if we divide the given surface 
two regions, in which Ir is positive and negative respectively, it follows 
what has been proved above that the former of these regions contains al 
points of the orbit at which V has a minimum value, i.e. it contains in ge 
an infinite number of distinct parts of the orbit, each of finite length ; wfa 
in the other region, for which /p is negative, the particle cannot remaii 
manently. These two parts of the surface are on this account calle< 
attractive and repellent regions. Each of these regions exists in geuen 
it is easily found that any isolated point of the surface at which Kis a 
mum (i.e. any point where stable equilibrium is possible) is in an attn 
region, and any point at which F is a masiinum is in a repellent region 

It is intereetiag to compare this result witii that which correBpoDdB to it in the i 
of a particle wi^ oue degree of freedom, e.g. a particle which ia &ee to move on a 
under the action of a force which depends only on the position of the particle^ I 
case the particle either ultimatelj travels an indefinite distance in one direct 



177, 178] The Omeral Thewy of Orbits 398 

oscillates about a position of stable equilibrium. The attractive region, in motion with two 
d^;recs of freedom, corresponds to the position of stable equilibrium in motion with one 
degree of freedom. 

Consider next the alternative supposition, namely that after some definite 
instant the variation of Y is always in the same sense. We shall suppose that 
the surface has no infinite sheets and is regular at all points, and that F is 
an everywhere regular function of position on the surface ; so that, since the 
variation of F is always in the same sense, V must tend toward some definite 
finite limit, Y and Y tending to the limit fsero. Considering the equation 

F=-A,(7) + 2(A-F)/^/A,(F) + (Xii + Mv)F, 

we see that if Ai (F) is not very small, X and /i are finite and the last term 
on the right-hand side of the equation is infinitesimal ; and consequently 
either there exist values of f as large as we please for which ly is positive (in 
which case the part of the orbit described in the attractive region is of length 
greater than any assignable quantity) or else Ai(F) tends to zero. But 
Ai (F) can be zero only when dV/du and dY/dv are zero ; if therefore (as is in 
general the case) the surface possesses only a finite number of equilibrium 
positions, the particle will tend to one of these positions, with a velocity 
which tends to zero. A position of equilibrium thus approached asymptotically 
must be a position of unstable equilibrium : for the asymptotic motion re- 
versed is a motion in which the particle, being initially near the equilibrium 
position with a small velocity, does not remain in the neighbourhood of the 
equilibrium position ; and this is inconsistent with the definition of stability. 

Thus finally we obtain Hadamard's theorem, which may be stated as 
follows : If a particle is free to move on a surface which is everywhere regular 
and has no infinite sheets, the potential energy function being regular at aU 
points of the surface ami having only afimte number of maxima and minima 
on it, either the part of the orbit described in the attra^itive region is of length 
greater than any assignable quantity, or else the orbit tends asymptotically to 
one of the positions of unstable equilibrium. 

Example. If all values of t from — oo to + oo are considered, shew that the particle 
must for part of its course be in the attractive region. 

178. Application of the energy integral to the problem of stability. 

A simple criterion for determining the character of a given form of motion 
of a dynamical system is often furnished by the equation of energy of the 
system. Considering the case of a single particle of unit mass which moves 
in a plane under the influence of forces derived from a potential energy 
function F(a?, y), the equation of energy can be written 

i(^ + yO = A-F(^,y). 

Now the branches of the curve F(a?, y) = A separate the plane into regions 
for which (Y(x, y) — h) is respectively positive and negative ; but as (^'^-y*) 



394 The General Theory of Orbits [ch. xv 

is essentially positive, an orbit for which the total energy is h can only exist in 
the regions for which F(a:, y) < h. If then the particle is at any time in the 
interior of a closed branch of the curve F(a?, y) = A, it must always remain 
within this region. The word staMlity is often applied to characterise types 
of motion in which the moving particle is confined to certain limited regions, 
and ill this sense we may say that the motion of the particle in question is 
stable. 

The above method has been used by Hill', Bohlinf, and Darwin J, chiefly 
in connexion with the restricted pcoblem of three bodies. 

179. Application of tntegral-invariarUs to investigations of stahUity, 

' The term stability was applied in a different sense by Poisson to a system which, in 
the lapse of time, returns infinitely often to positions indefinitely near to its original 
position, the intervening oscillations being of any magnitude. It has been shewn by 
Poincar^ that the theory of integral-invariants can be applied to the discussion of Poisson 
stability. 

Considering a system of differential equations 

- , - =^Xy. (j?i, ^2» ••• » ^h) C**— 1> 2, ..,, n), 



for which III'" /^i^2«"^ii 



is an integral-invariant, we regard these equations as defining the trajectory in n dimen- 
sions of a point P whose coordinates are {Xi, x^^ ..., J7n)* ^f the trajectories have no 
branches receding to an infinite distance from the origin, it can be shewn § that if any small 
region R is taken ;n the space, there exist trajectories which traverse R infinitely often : 
and, in fact, the probability that a trajectory issuing from a point of R does not traverse 
this region infinitely often is zero, however small R may be. Poincar<^ has given several 
extensions of this method, and has shewn that under certain conditions it is applicable in 
the restricted problem of three bodies. 



Miscellaneous Examples. 

1. Shew that the motion of a particle in an ellipse under the influence of two fixed 
Newtonian centres of force is stable. (Novikoff.) 

2. A particle of unit mass is free to move in a plane under the action of several centres 
of force which attract it according to the Newtonian law of the inverse square of the 
distance: denoting the resulting potential energy of the particle by V(x, y\ shew that the 
integral 

where the integration is taken over the interior of any periodic orbit for which the constant 
of energy has the value h (the centres of force being excluded from the field of integration 
by small circles of arbitrary infinitesimal radius), is equal to the number of centres of force 
enclosed by the orbit, diminished by two. {Monthly Notices R,A,S. Lxii. p. 186.) 

* Amer, J. Math. x. (1878), p. 75. t Acta Math, x. (1887), p. 109. 

t Acta Math. xxi. (1897), p. 99. 

§ Poinoar^, Acta Math. xiii. (1890), p. 67 ; Nouv, MSth, in. Ch. xxvii. 



7 



178, 179] The General Theory of Orbits 396 

3. Let a fomily of orbits in a plane be defined by a differential eqtiation 

where {Xy y) are the current rectangular coordinates of a point on an orbit of the family ; 
and let dn denote the normal distance from the point (x, y) to some definite adjacent orbit 
of the family. Shew that ^ satisfies the equation 



d^ +^«^=0, 



where 



■-Hmi-t-tty*---'^^- 



and ^ is a variable defined by the equation 

(Sheepshanks Astron. Exam.) 

4. A particle moves under the influence of a repulsive force from a fixed centre : shew 
that the path is always of a hyperbolic character, and never surrounds the centre of force ; 
that the asymptotes do not pass through the centre in the cases when the work, which has 
to be done gainst the force in order to bring the particle to its position from an infinite 
distance, has a finite value; but that when this work is infinitely great, the asymptotes 
pass through the centre, and the duration of the whole motion may be finite. 

(Schouten.) 

5. Shew that in the motion of a particle on a fixed smooth surface under the influence 
of gravity, the curve of separation between the attractive and repellent regions of the 
surface is formed by the apparent horizontal contour of the surface, together with the locus 
of points at which an asymptotic direction is horizontaL 

6. A particle moves freely in space under the influence of two Newtonian centres of 
attraction ; shew that when its constant of energy is negative, it describes a spiral curve 
round the line joining the centres, remaining within a tubular region bounded by two 
ellipsoids of rotation and two hyperboloids of rotation, whose foci are the centres of force : 
and that when the constant of energy is zero or positive, the particle describes a spiral 
path within a region which is bounded by an ellipsoid and two infinite sheets of hyper- 
boloids of the same confocal system. (Bonacini.) 



CHAPTER XVI. 

INTEGRATION BY TRIOONOMETRIO SERIEa 

180. The need for series wkioh converge for aXl vaiuea of the t 
Poincari's series. 

We have already observed (§ 32) that the differential equations of mi 
of a dynamical system can be solved in terms of series of ascending pc 
of the time measured from some fixed epoch ; these series converg 
general for values of t within some definite circle of convergence in 
f-plane, and consequently will not furnish the values of the coordi] 
except for a limited interval of time. By means of the process of conti 
tioD* it would he possible to derive from these series successive sets of i 
power-series, which would converge for vaiuea of the time outside 
interval ; but the process of continuation is too cumbrous to be of i 
use in practice, and the series thus derived give no insight into the gei 
character of the motion, or indication of the remote future of the syi 
The efforts of investigators have therefore been directed to the proble 
expressing the coordinates of a dynamical system by means of expan: 
which converge for all values of the time. One method of achieving 
resultj* is to apply a transformatiou to the j-plane. Assuming that 
motion of the system is always regular (i.e. that there are no coUisioi 
other discontinuities, and that the coordinates are always finite), there 
be no singularities of the system at points on the real axis in the (-plane 
the divergence of the power-series in f — t, after a certain interval of 
must therefore be due to the existence of singularities of the solution ii 
finite part of the f-plane but not on the real axis. Suppose that the si 
larity which is nearest to the real axis is at a distance h from the real 
and let T be a new variable defined by the equation 
, . 2A, 1 + T 

A band which extends to a distance h on either side of the real axis ii 

t-plane evidently corresponds to the interior of the circle |T|al in 

■ Whiltaker, Modem Analyi; § 41. t Due to Poinoari, Acta MUh. iv. (18B4), p. 



^ 



I 



180, 181] Integration by Trigonometric Series 397 

T-plane; the coordinates of the dynamical system are therefore regular 
functions of r at all points in the interior of this circle, and consequently 
they can be expressed as power-series in the variable t, convergent within 
this circle. These series will therefore converge for all real values of t 
between — 1 and 1, i.e. for all real values of t between — oo and + oo . Thus 
these series' are valid for all values of the time, 

181. Trigonometric series. 

The series discussed in the preceding article are all open to the objection 
that they give no evident indication of the nature of the motion of the 
system after the lapse of a great interval of time : they also throw no light 
on the number and character of the distinct types of motion which are 
possible in the problem : and the actual execution of the processes described 
is attended with gi*eat difficulties. Under these circumstances we are led to 
investigate expansions of an altogether different type. 

If in the solution of the problem of the simple pendulum (§ 44) we 
consider the oscillatory type of motion, and replace the elliptic function 
by its expansion as a trigonometric series*, we have 

sini^^?'^ i g*""" ^^ {2s^l)^^(t^t,) 

where d denotes the inclination of the pendulum to the vertical at time t ; 
K ^nd ^ can be regarded as the two arbitrary constants of the solution, and 
/i is a definite constant, while q denotes e'^^'f^, where if' is the complete 
elliptic integral complementary to K. This expansion, each term of which 
is a trigonometric function of t, is valid for all time. Moreover, when the 
constant q is not large, the first few terms of the series give a close approxi- 
mation to the motion for all values of t The circulating type of motion of 
the pendulum can be similarly expressed by a trigonometric series of the 
same general character. 

Turning now to Celestial Mechanics, we find that series of trigonometric 
terms have long been recognised as the most convenient method of expressing 
the coordinates of the members of the solar system ; these series are of the 
type 

2an,,n, n» COS (w^^i + 71^0^ -f ... + W*^*), 

where the summation is taken over positive and negative integer values of 
^1. ^» •••, ^*i and dr is of the form \rt + er ; the quantities a, \, and e being 
constants. Delaunayf shewed in 1860 that the coordinates of the moon can 
be expressed in this way; NewcombJ in 1874 obtained a similar result for 
the coordinates of the planets, and several later writers§ have designed 

* Whittaker, Modem Analysis, § 203. 

t Thiorie du mouvement de la lune. Paris, 1860. X Smithsonian Contrihutiom^ 1874. 

§ e.g. Lindstedt, Tisserand, and Poincar^. 



* 



398 Integration by Trigonometric Series [ch. xvi 

processes for the solution of the general Problem of Three Bodies in this 
form ; these processes are also applicable to other dynamical systems whose 
equations of motion are of a certain type resembling those of the Problem of 
Three Bodies. In the following articles we shall give a method* which is 
applicable to all dynamical systems and leads to solutions in the form of 
trigonometric series : the method consists essentially, as >vill be seen, in the 
repeated application of contact-transformations, which ultimately reduce the 
problem to the equilibrium-problem. 

182. Removal of terms of the first degree frbm the eviergy function. 

Consider then a dynamical system, whose equations of motion are 

dqr_dH dpr__^dH 

dt " dpr' dt " dqr ^ ' ' •••' ^' 

where the energy function H does not involve the time t explicitly. 
The algebraic solution of the 2n simultaneous equations 

g-=:0, 3^=^ (r = l, 2, ..., n) 

will furnish in general one or more sets of values (a,, a^, ..., an> bi*b > &«) 

for the variables (ji, ?j, ..., ?n» Pi» --->Pn)l and each of these sets of values 
will correspond to a form of equilibrium or (if the above equations are those 
of a reduced system) steady motion of the system. 

Let any one of these sets of values (oi, a,, ..., any 6i, 6s> ..., 6«) be 
selected ; we shall shew how to find expansioos which represent the solution 
of the problem when the motion is of a type terminated by this form of 
equilibrium or steady motion. Thus if the system considered were the 
simple pendulum, and the form of equilibrium chosen were that in which 
the pendulum hangs vertically downwards at rest, our aim would be to find 
series which would represent the solution of the pendulum problem when the 
motion is of the oscillatory type. 

Take then new variables (j/, g,', ..., j„', Pi',pa', ...,PnO» defined by the 
equations 

qr^dr + qA Pr^K+Pr (r = 1, 2, ..., n) ; 

the equations of motion become 

d^^dH dp;^_dH 

dt "dp;' dt - dq; ^ ' ' •••' ^' 

and for suflBciently small values of the new variables the function H can be 
expanded as a multiple power series in the form 

H = Hq + Hi + H^ -f- jET, + . . ., 

* Whittaker, Pnyc, Lond, Math, Soe, zxxiy. (1902), p. 206. 



( 



181-183] Integration by Trigonometric Series 399 

where Hj^ denotes terms homogeneous of the A;th degree in the variables 

\?i I ?a I • • • > Jn > Pi* • • • > Pn /• 

Since H,^ does not contain any of the variables, it can be omitted : and the 
fact that the differential equations are satisfied when (9/, 9/, ..., q^, pi\ •••tPn) 
are permanently zero requires that Hi should vanish identically. The 
expansion of H therefore begins with the terms H^, which (suppressing the 
accents of the new variables) can be written in the form 

if, = ^2 (arrqr^ + 2ar,qrqi) + ^b„qrPi + ^2 (CrrPr* + 2c„PrP#), 

where a„ = a«., Cr$ = c„, 

but bn is not necessarily equal to 6^. If the terms ffj, H4, ..., were neglected 
in comparison with H^t the equations would become those of a vibrational 
problem (Chapter VII.). 

183. Determination of the normal coordinates by a contact-trans/ormation. 

We shall now apply a contact-transformation to the system in order to 
express H^ in a simpler form*, — in fact, to obtain variables which correspond 
to normal coordinates for small vibrations of the system. 

Consider the set of 2n equations 

«yr + g^^2(a?i, a-,, ...,^n, yi, ..., yn) = o| 

I (r=l, 2, ...,n) 

— 8Xr + ^ffa{^u ^a, .... ^n, yu .... yn)=OJ 
or — fiyr = an^i + Ctf«^ 

On solving these equations, we obtain for 8 the determinantal equation 
which in § 84 was denoted by /(«) = : we shall suppose that H^ia s, positive 
definite form, and (as in § 84) we shall denote the roots of the equation by 
±isi, ±i8i, ..., ±i8f^; the quantities 81, s^, ..., «n> ^^ stU real, and for 
simplicity we shall suppose no two of them to be equal. 

To each root there will correspond a set of values for the ratios of the 
quantities (xi, x^^ ..., x^, yi, ..., yn)\ let the set which correspond to the root 
is^ be denoted by (^a?i, ^x^^ ..., ^Xn, rVu ..., rVn)^ and let the^set which corre- 
spond to the root - Wr be denoted by (-r^i, -rOJai ...» -r^i -r^i, ..., -ryn^y so 
that we have 

— ^r r^p = Oyif^l + Opara^, + ... + Opnra^n + ftpiryi + .«. + bp^rVn* 
i8r,a!p= bip^i + 6vf^2 + --- + ^np ,^n + Cpi ^y 1 + ... + CpnrVt 



Mr= i,z, ...,n). 

+ ... + bnriPn + Cnyi + ... + CrnVJ 



rn» 



* In obtaining the transformation of this article a method is used which was saggested to the 
aathor by Professor Bromwioh of Qaeen's Ck>llege, Galway, and which furnishes the transforma- 
tion more directly than the method originally devised. 



400 Integration by Trigonometric Series [oh. xvi 

Multiply these equations by ipCp and j^yp respectively, add them, and sum 
with respect to p ; we thus obtain the equation 

n 

K S (^p k}/p - kOJp rVp) "= ff {r, k), 

where 

jy (r, i) = Oil ^ i^i + Oia (riCi fca?a + ia?i ^,) + . . . + 6u (,ii?i ikyi + t^r^ 

so that H(ry k) is symmetrically related to r and k. 
Interchanging r and k, we have 

n 

isk 2 {ycp ^j/p - ,^ptyp) = H (r, i), 
p=i 

n 

and therefore (»,. + «*) ^ (*^p r^p — r^p kVp) = 0. 

p=i 

So, unless «r + «* is zero, we have 

n 

2 (r^i* *yi> - A^^p ,.yp) = 0, 

and consequently if(r, i) is zero: if «r + ** is zero, we have *arp = -,^p» 
jbyp = -y^p, and therefore 

n 

If now we define new variables (g/, g/, ..., qn, p/, ..., pn') by the equations 

Mr = 1, z, ..., n)y 

and if S and A denote any two independent modes of variation, it is evident 

n n 

that the coeflScient of Sj/Api' in ^ (SqiApi — AqiBpi) is 2 {r^^i-kyi-~-^iryi)f 

n 

which is zero when r is not equal to k. Thus 2 {SqiApi — A^jSpj) contains no 
terms except such as (Sj/Ap/ — Aj/Sp/), and the coeflBcient of this term is 

n 

2 (,^/ -,.yi — _^/ ,.y/). Now hitherto the actual values of ^xi, jji have not 

been fixed, as only their ratios are determined from their equations of 
definition; we can therefore choose their values so thai 



n 

2 {r^i ^,yi - ^xi ryi)^l (r = 1, 2, . . ., n), 

•—1 



aad then we shall have 



1=1 * r=l 

SO that (§ 128) the transformation from the variables (ji, Ja* •••>?»> Pi> •••. Pn) 
to the variables (g/, g/, ..., g»', p/, ..., p^) is a contact-transformation. 



183, 184] Integration hy Trigonometric Series 401 

Moreover, if in jET, we substitute for (ji, q^, ,.., q^, pi, ..., p^) in terms of 
(?/, ?»', ..., ?n'. Pi'. ..., P»)» we obtain 

n 

or Hf = %'Si SrqrPr' 

r=l 

Now apply to the variables (g/, gj', ..., gn'» JPi'. •••> Pn) the contact- 
transformation defined by the equations 

*' a^/" ^'=3^' (r- 1,2. ...,«). 

where F = I (p/V + i — - 1 "r?/") , 

r=l \ *r / 

which gives i?2 = i S (p/" + »r'?r"'). 

As all the transformations concerned have been linear, we see that 
H^y H4, ... will be homogeneous polynomials of degrees 3, 4, ... in the new 
variables: and thus, omitting the accents, we have the result that the 
equations of motion of the dynamical system have been brought to the form 

dqr^dH dpr__dH 

dt^dpr' dt~ dqr ^r-1. z, ...,n;, 

where H=^ H^^ H^+ H^^ ..., 

in which H^is a homogeneous polynomial of degree r in the variables, and in 
particular 

J5r,=ii (P.' + V9r'). 

r=l 

It is clear that if we neglect -ff,, H^, ... in comparison with -H,, and 
integrate the equations, the solution obtained will be identical with that 
found in § 84. 

184. Transformation to the trigonometric form of H, 

The system will now be further transformed by applying to it a contact- 
transformation from the variables (ji, g,, ..., qnyPu '-'tPn) to new variables 
{?i', s/i •••» qn't P\y ..., Pn'X defined by the equations 

Where W = J^ [j/ sin- ^^ + 1^ {2.,,/ -^^J , 

so that 

Pr = (2«^/)* sin jp/, q^ = (2g/)* s^^ cosjp/, (r == 1, 2, . . . , n). 

w. D. 26 



t 



I 



V» a 



nr-*. - 



402 Integration by Trigonometric Series [oh. xvi 

The differential equations become 

dgJ dH dpJ dH 
dt dp/' dt a?/ ^^ 1, A...,n;. 

where H = Siqi+8^^'+ ... +«n?n +-^8 + ^4+ ••., 

and now if,, denotes an aggregate of terms which are homogeneous of 
degree ^r in the quantities 5/, and homogeneous of degree r in the 
quantities cosp/, sinjp/. 

Since a product of powers of cosp/, sinp/ can be expressed as a sum of 
sines and cosines of angles of the form (n^pi + ri^pi + . . . + nnPn)^ where 
rij, «,,..., nn have integer or zero values, it follows that H^ can be expressed 
as the sum of a finite number of terms, each of the form 

where mj + ma + . . . + m„ = ^7-, | ^r | ^ 2wy» 

and therefore | ^ I + 1 w^ | + ... + 1 Wn | ^r. 

The function H is thus expressed in the form 

where for each term we have 

and the series is clearly absolutely convergent for all values ofp/, pa', ...,pn, 
provided qi, qj, ,,., qn do not exceed certain limits of magnitude. From the 
absolute convergence it follows that the order of the terms can be rearranged 
in any arbitrary way : we shall suppose them so ordered that all the terms 
involving the same argument n^pi + ... •{-nnPn are collected together, so 
that H takes the form 

^ = ao,o,....o + San„«^...,nn cos (Wi|)/ + . .. + rinPn) 

+ 2^n,.n*....nnSin(niPi'+ ... + TlnPn ), 

where the coefficients a and b are functions of 9/, 9,', ...,qn and the expansion 

of an,.fH....,n» or 6n„»4,....n« in powers of 5/, q^\ ...,qn contains no terms of order 

lower than i{|?^| + |^|+ ... +|wn|}; and where the summations extend 

over all positive and negative integer and zero values of rzi, n,, ..., nn, except 

the combination 

ni = 7ia= ... =nn = 0. 

Moreover, the expansion of ao,o (which will be called the non-periodic part 

of H, the rest of the expansion being called the periodic pan) begins with 
the terms 

«i?/ + ««?«'+ ••• +Snqn; 
and, when q^, q^\ "*,qn are small, these are the most important terms in H, 



/ 



184, 186] hUegration by Trigonometric Series 403 

since they contribute terms independent of g/, q^^ ..., gn' to the diflFerential 
equations. 

For convenience we shall often speak of g/, q^, ,,., qn bs *' small/' in order 
to have a definite idea of the relative importance of the terms which occur. 
It will be understood that g/, q^\ -^^qn are not, however, infinitesimal, and 
in fact are not restricted at all in magnitude except so far as is required to 
ensure the convergence of the various series which are used. 

To avoid unnecessary complexity, we shall ignore the terms 

j 2^»„n,...., nn sin {n^px + . . . + rinPn) 

I in fi^, as they are to be treated in the same way as the terms 

• SOn,. ng..... nn COS {u^pi + . . . + U^Pn), 

I and their presence complicates, but does not in any important respect modify, 

the later developments. 

The form to which the problem has now been brought may therefore be 
stated as follows (suppressing the accents in the variables) : The equations of 
motion are 

dt^dpr' dt ^ dqr (^-1, A...,n), 

where H = ao,o,.... o + ^an..n.,...,n» cos {n^p^ + n^p^ + . . . + n^pn), 

and the coefficients a are*/unctions of q^, q^, ,,,, qn only ; moreover, the periodic 
part of H is small compared with the non-periodic part ao,o, ...o / ^ term which 
has for argument (wiPi+ «2Ps+ ... -^ihiPn) h(^ i^ coefficient an^^n^...,nn ^^ least 
of order i { | ^ | + | w, | + ... + | ^^i | } in the small quantities Ji, gj, . . . , Jn / cL^d 
the expansion ofao^o,..„o begins with the terms (s^qi + s^2+ ... +Snqn)' 

It follows from this that when the variables ji, q^, ..., g» are small they 
vary very slowly, while the variables jpi,j>a, ...,pn vary almost proportionally 
to the time. 

186. Other types of motion which lead to equations of the same form. 

The equations which have now been obtained have been shewn to be 
applicable when the motion is of a type not far removed from a steady motion 
or an equilibrium-configuration, e.g. the oscillatory motion of the simple 
pendulum, or those types of motion of the Problem of Three Bodies which 
have been studied in § 171. But these equations can be shewn to be 
applicable also to motion which is not of this character, and in particular to 
motion such as that of the planets round the sun, or the moon round the 
earth*. 

For let the equations of motion of the Problem of Three Bodies be taken 

* Delaanay, TMoHe de la Lune ; Tisserand, Annalet de VOh$, de Parit, Mimoiret^ zvizi. (18S5). 

26—2 



404 TntegrcUion by Trigonometric Series [ch. xvi 

in the form obtained in § 160 ; and let the con tact- transformation which in 
defined by the equations 

P^-'Wr' ^'^~Wr (^=1.2.3.4) 

be applied to this system, where 



J I ?4" qt qt'i^ 



The new variables can be interpreted in the following way. Suppose that at 
the instant t all the forces acting on the particle /i cease, except a force of 
magnitude mim^jq^ directed to the origin ; and let a be the semi-major axis 
and t the eccentricity of the ellipse described after this instant : then 

?/ = [m^mifia ( 1 - e*)}*, q^ = {r/^mj/Lca}*. 

Further, if the lower limits of the integrals are suitably chosen, pi' + g, is 
the true anomaly of /x in its ellipse, and — ^,' is the mean anomaly. The 
variables q^, q^, p^, p/ stand in a corresponding relation to the particle fi\ 

The equations of motion now take the form 

dqr'^dH dp; aff rr=«12^4V 

dt "ap/' dt " dq; . (r=.l,Ad,4;, 

when the particles m, and m^ are supposed to be of small mass compared with 
TTii, and are describing orbits of a planetary character about nii, it is readily- 
found that H can be expanded iu terms of the new variables in the form 

H = a<».o,o,o + ^'<^,n^.n,,n^ COS (n,p/ + n^p,' 4- Utp^ + n^p^\ 

where the coeflBcients a are functions of (9/, g,', g/, 9/) only, the summation 
extends over positive and negative integer and zero values of w^, n,, Wj, n^^ 
and the coefficient ao,o,o,o is much the most important part of the series. As 
this expansion of H is of the same character as that obtained in § 184, it 
follows that the method of solution given in the following articles is applicable 
either to motion of the planetary type or to motion of the type studied in § 171. 

186. Removal of a periodic term from H. 

We shall now apply to the system another contact-transformation, the 
effect of which will be the removal of one of the periodic terms from H ; this 
will further accentuate the feature already noted, namely that the non-periodic 
part of JET is much more important than the periodic part*. 

* Readers familiar with Celestial Mechanics wiU notice the analogy of this method with that 
of Delaunay's lunar theory : the analysis is different from Delaunay's, but the idea is essentially^ 
the same. 



186, 186] Integration by Trigonometric Series 406 

Let one of the periodic terms in H be selected, say 

^,. «»..., »» cos (ni/j, 4- Ti^pt + . . . + nnjpn). 

Write J3r = aa.a.....o + afh.»4,....n» COS(»iPi + n^p, + ... + WnPn) + -B, 

80 that R denotes the rest of the periodic terms of H ; when we wish to put 
in evidence the arguments of which ani.n„...,n» is a function, we shall write it 

Apply to the system the contact-transformation defined by the equations 

^'^d^- ^'^a^ (r=l,2,...,n), 

where W = q^pi + q^p^ + . . . 4- qnpn +/ (9/, g/, . . . , jn', 0) 

and ^ = WiPi + n^p% + . . . + rinPn \ 

we shall suppose that /is a function, as yet undetermined, of the arguments 
indicated. The problem is now expressed by the equations 

dqr _ dH dp/ _ dH^ / _ i 9 \ 

dt "dpr dt dq; ^r-i. z, ...,n;. 

where 

+ att,.n. tu f ?/ + ni ^ , ... , gn + Tl,» ^ j COS ^ + JB, 

and d and i2 are supposed to be expressed in terms of the new variables by 
means of the equations of transformation 

i>r'=l>r + g^/, ?r = ?/ + rir^ (r = 1, 2, . .. , fl). 

The function / is, as yet, undetermined and at our disposal. It will be 
chosen so as to satisfy the condition that shall identically disappear from 
the expression 

^0,0,..., p l9i *^'^^» •••»?!» • ^'^S^ J 

+ Ctn„fh,....nn f ?/ + »^l g^ » .... ?n' + Hn ^ j COS tf, 

80 that this quantity is a function off/, 9/, ..., q^ alone, say 

Then the equation 

( , ¥ / 3A 

determines dfjdd in terms of g/, g/, ..., q^, a'0,0 oi and costf. 



1 1 



406 Integration by Trigonometric Series [ch. xvi 

Suppose that the solution of this equation for d//d0 is expressed in the 
form of a series of cosines of multiples of (which can be done, for instance, 
by successive approximation), so that 

5^ = Co+ 2 cjc cos kO, 

where Cq, Cj, c,, ... are known functions of y/, ga'» •••> ?n. Gt'o^o,...,o« 

Now a'o,o,...,o is as yet undetermined, and is at our disposal. Impose the 
condition that c© is to be zero; this determines a'o,o,...,o as a function of 
?/> ?a'i ••., Jn'; and, on substituting its value in the series for d/fdO, we have 

dfjd6=^ 2 CfcCosA?^, 

where now Ci, Cj, c,, ... are known functions of y/, g/, ..., g,^'. Integrating 
this equation with respect to dy and for our purpose taking the constant of 
integration to be zero, we have 

/= i ^sinM 
The equations defining the transformation now become 



p/=i>r+ S y^sinA:^ 
?r = ?/+^ 2 CikCosA;^ 



(r=l, 2, ..., n). 



Multiply the first set of these equations by n^, n^, ..., Un respectively, and 
add them: writing 

we have ^ = tf+ 2 yfrii o— , 4- w, r-^, + ... + rin ^— , ) sin M 

Reversing this series, we have 

^ = ^'+ 2 dusinkO', 

where dj, (£,, ... are known functions of y/, q','. •••! ?n- Substituting this 
value of in the equations of transformation, they become 

00 

p^ =2)/ + 2 fSji sin A?^ 

*='^ [• (r=l,2, ...,n), 

?r = ?/ + TV 2 ffk cos A:6^ 

where all the coefficients ^e*, jPt are known functions of j/, q^', ..., gn'. 

Now, before the transformation, the function jB consisted of an aggregate 
of terms of the type 

^ = 2a,n„ tit,. .... m^ cos {rn^pi + . . . + mnPn) ; 



\ 

\ 



186, 187] Integration by Trigonometric Series 407 

when the values which have been found for (qi, q^, ..., q^, pi, ...,jpn) are 
substituted in this expression, and the series is reduced by replacing powers 
and products of trigonometric functions of p/, p^, ..., pn by cosines of sums 
of multiples of pi,Pi\ ..., Pn\ it is clear that 12 will consist of an aggregate 
of terms of the type 

R = %a'm, , m, m» COS (^1^/ + W^Pi +... + TOnPn^ 

where the coefficients a' are known functions of (5/, q^, ..., }»). 

We thus have the result (omitting the accents of the new variables) that 
afier the transformation has been effected, the system is still expressed by a set 
of eqvxitions of the form 

dq^^dH dpr dH / -i 9 \ 

dt^dp/ dt" dq, ^r-i, A...,n; 

where H=^a\^o + 2a «,, m,, .... m„ cos (m^pi + m^p^ + . .. + mnPn)f 

and where the coefficients a' are known functions of q^ q^, ,,,yqn- 

Let us now review the whole effect of the transformation. The differential 
equations of motion have the same general form as before; but from the 
equation 

ao.o....,o + Ctn,, n,, .... n^ COS {^iP\ + Wj^j + ... + n^Pn) = a'o,o,...,o 

we see that one term has been transferred from the periodic part of H to its 
non-periodic part: the periodic part of H is less important, in comparison 
with the non-periodic part, than it was before the transformation was made. 

187. Removal of further periodic terms from H. 

Having now completed the absorption of this periodic term into the non- 
periodic part of Hy we proceed to absorb one of the periodic terms of the new 
expansion of H into the non-periodic part, by a repetition of the same 
process. In this way we can continually enrich the non-periodic part of 
H at the expense of the periodic part, and ultimately, after a number of 
applications of the transformation, the periodic part of H will become so 
insignificant that it may be neglected. Let (a,, Og, ..., o„, /8i, /8„ .... /8n) be 
the variables at which we arrive as a result of the final transformation : then 
the equations of motion are 

dor dH dfir^^dH 

dt 3/9/ dt 8a^ ^ i,A...,«;, 

where H, consisting only of its non-periodic part, is a function of 
(a,, Oj, ..., On) only. We have therefore 



dt 



= 0, ^r=^-jl^dt (r=l,2, ...,7i). 



408 IntegrcUion by Trigonometric Series [oh. xvi 

which shews that the quantities a are constants, and the quantities fi are of 
the fonn 

/9r = M + fr, where Mr = -^ (r = l, 2, ..., n); 

the quantities e^ are arbitrary constants, and the part of fMr independent of 
(ai,aa, ..., On) is -«r. 

188. Reversion to the original coordinates. 

Having now solved the equations of motion in their final form, it remains 
only to express the original coordinates of the dynamical system in terms of 
the ultimate coordinates («!, Og, ..., On, A. ...i fin)- Remembering that the 
result of performing any number of contact-transformations in succession is a 
contact- transformation, it is easily seen that the variables (qi, q^, ..., 9n> 
P\9 •••»/>n) used at the end of § 184 can be expressed in terms of («!, a,, ..., Oni 
A» •••, fin) by equations of the form 

Jr = Or + Sm^^i^,, „ rnn COS (w^p^ + m,jp, + . . . + m^Pn)) 

or 

?r =/r(ai, O,, .... an) + ^tO^,, «,. .... mn COS {m^fii -h mj/Sa + . . . + Wn/8«)1 
Pr = /8r + 2,Am,, in, nm sin {ttI^P^ -h Wj/8a + . . . + mn/9n) J 

(r=l, 2, ....n), 

where the coefficients a and 6 are functions of (ai, a,, ..., ce„). 

From this it follows that the variables (g,, q^, ..., gn»Pi» •••! i>n) of § 182, 
in terms of which the configuration of the dynamical system was originally 
expressed, are obtained in the form of trigonometric series, proceeding in 
sines and cosines of sums of multiples of the n angles /9i, /S^, ..., ^8^. These 
angles are linear functions of the time, of the form ^^t + 6,. ; the quantities 
€^ are n of the 2n arbitrary constants of the solution, while the quantities 
/i^ are of the form 

the coefficients c being independent of the constants of integration. The 
coefficients in the trigonometric series are functions of the arbitrary constants 
{a^,a^, ..., On) only. 

The expansions thus obtain^ represent a family of solutions of the 
dynamical system^ the limiting member of the family being the position of 
equilibrium or steady motion which was our staHing-point 

Evidently also, by applying the integration-process of §§ 186 — 188 to the 
equations of motion found in § 185, we obtain a solution of the Problem of 
Three Bodies^ when the motion is of the planetary type, in terms of trigono- 
metric series of the kind above specified. 



187, 188] Integration by Trigonometric Series 409 

For the further development of the theory of the present chapter, in connexion with 
the Problem of Three Bodies, reference may be made to treatises on Celestial Mechanics : 
in particular, the second volume of Poincar^'s NouveUes MUhodes de la M4caniquB Celeste 
contains an account of several methods of deriving expansions, with a discussion of the 
convergence of the series obtained. 



Miscellaneous Examples. 

1. Let <^ denote any function of the variables qi, Qn —y ^mpu "*} Pn of a dynamical 
system which possesses an integral of energy 

^(?i» ?2> — » 9n,Pi^ ...,jpO = Constant; 

let a|, O), ..., a^, 6^, ..., b^ be the values of ^|, ^j) •••) ?»> Pi9 •••> Pn respectively at the 
instant t^t^; and let {/, g} denote the value of the Poisson-bracket (/, g) when the quan- 
tities ^1, ^2) •••» ?n> PiJ •••) pn occurring in it are replaced respectively by Oj, a,, ..., a», 
^i> •••> ^n* 
Shew that 

^tei> 9ii •••» ^n^Pu ••-»i'»)=0(«i» «2> •••> «»> ^i> •••> ^»)+(<-^o){0» ^} 

2. Shew that the dynamical system whose equations of motion are 

di~^' dt" dg' 
where £r=i^+___, 

possesses a family of solutions represented by the expansion (retaining only terms of order 
less than a') 

wh« 0=^fk+^t+t, 

and a and c are arbitrary constants. 






i 



^8 



>■■ 



s 






•J 



INDEX OF TEEMS EMPLOYED. 

(The numbers refer to the pages, where the term occurs for the first time 

in the book or is defined.) 



<^. 



Abaolate integral-invariants, 266 

Acceleration, 14 

Action and Beaction, Law of, 29 

Action, Integral of, 243 

Adjoint system, 281 

Admission of a contact-transformation by a 

dynamical system, 808 
Angles, Eulerian, 9 
Angular momentum, 58 
Angular velocity, 14 
Anomaly, 88 
Apex of a top, 151 
Aphelion, 84 
Apocentre, 84 
Appell's equations, 258 
Apse, 84 

Arc-coordinates, 19 

Attractive regions of a field of force, 892 
Axes, principal, 122 
Axis, instantaneous, 2 
Azimuth, 19 

Bernoulli's principle, 182 

Bertrand*s theorem on determination of forces, 

319 
Bertrand's theorem on impulses, 255 



odf^y^^ Bilinear covariant, 286 

iilomei^.'>.-oblem of Three, 827 






'*^. 



«> 



„ inerur representation of the Last 



^'^ojtaental ellipsoid, i. 
b1^^^ ^tum, 47 
Br''' 9^ angular, 58 te's, 287 



o/i. 



^' ^^ corresponding to 288 

Br. -^^'^6. integral of, 58 

^ ^^W,^pulkve, 47 
Can iti ( 44 

I 86 



^ion, 259 



Qe^tUti 



Gentr 



'^ 



Of 



>> 



»» 



fd, 293 

189 
H 270 



»» 



tf 



ft 



»f 



Centrifugal forces, 41 
Characteristic exponents, 388 
Christoffers symbol, 39 
Classical integrals, 346 
Coefficient of friction, 223 
„ stability, 884 
CoUinear Lagrange's particles, 381 
Collision, 230 

Components of a vector, 13 
Conjugate determinants, 289 

,, points on a trajectory, 248 
Conservation of energy, 61 

I, „ momentum, 58 

„ „ angular momentum, 59 

Conservative fields of force, 37 
Constraint, Gauss*, 250 
Contact-transformations, 282 

homogeneous, 290 
infinitesimal, 291 
Coordinates of a dynamical system, 32 
elliptic, 95 

ignorable or cyclic, 53 
ignoration of, 55 
norinal or principal, 177 
quasi-, 41 
Cotes' spirals, 81 
Covariant, bilinear, 285 

„ deformation-, 109 

Curvature, least, 250 
Cyclic coordinates, 53 

Deformation-oovariant, 109 
Degrees of freedom, 33 
Density, 115 
Differential form, 285 

,, parameters, 109 

Displacement of a body, 1 
„ possible, 33 

Dissipation function, 226 
Dissipative systems, 222 



»» 



»t 



»t 



it 



tf 



} 



412 Index 



/ 



t» 



It 



11 



t» 



»> 



>> 



t> 



» 



»i 



ft 



f> 



»i 



*> 



Distanoe, mean, 86 
Divisors, elementary, 179 

Eccentrio anomaly, 88 
Elementary divisors, 179 
Elimination of the nodes, 829 
Ellipsoid, momental, 122 
of inertia, 122 
of gyration, 122 
Elliptic coordinates, 95 
Energy, integral of, 61 
kinetic, 35 
potential, 37 
total, 62 
Equations, Appell's, 263 

first Pfaff's system of, 296 

Hamilton's, 258 

Hamilton-Jacobi, 303 

Jacobi's, 329 

LagraDgian, 37 

Lagrangian in quasi-coordinates, 

41 
Lagrangian with undetermined 

multipliers, 211 
variational, 262 
Equidistant Lagrange's particles, 881 
Equilibrium configuration, 173 
Equimomental, 115 
Eulerian angles, 9 
Exponents, characteristicf, 388 
Expressions, Pfaff's, 285 

„ Lagrange's bracket-, 287 

„ Poisson's bracket-, 288 

Extended point-transformations, 282 
External forces, 36 

Field of force, 29 

conservative, 37 
parallel, 91 
First Pfa£F's system, 296 
Fixity, 26 

Fixture, sudden, 165 
Flux of a vector, 13 
Focus, kiuetic, 248 
Forces, 29 

central, 76 
centrifugal, 41 

external and molecular, 31, 36 
„ reversed, 47 
Form, differential, 285 
Frame of reference, 26 
Freedom, degrees of, 33 
Friction, 223 

„ coefficient of, 223 
Function, dissipation, 226 



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Function, Jacobi's, 880 
Function-group, 310 

Gauss' principle, 250 
Gravity, 27 
Group, Function-, 310 
Group property, 283 
Gyration, ellipsoid of, 122 

„ radius of, 116 
Gyroscopic terms, 191 

Hamilton's equations, 258 

principle, 242, 245 

theorem, 78 
Hamilton-Jacobi equation, 303 
Helmholtz's reciprocal theorem, 293 
Herpolhode, 150 
Hertz's principle, 250 
Holonomic systems, SS^ai^^-^* 
Homogeneous contact-transformations, 290 

Ignorable coordinates, 53 
Ignoration of coordinates, 55 
Impact, 230 
Impulsive motion, 47 

„ „ Lagrangian equations of, 

49 
Index of stability, 385 
Inelastic bodies, 230 
Inertia, ellipsoid of, 122 

,, moments and products of, 115 
Infinitesimal contact-transformations, 291 
Initial motions, 44 

Instantaneous centre and axis of rotation, 2 
Integral of angular momentum, 59 

classical, 346 

of energy, 61 

Jacobian, 342 

of momentum, 58 

of a system of equations, ^2 
Integral-invariants, 261 

„ „ absolute and relative, 265 

Invariable line and plane, 142, 334 
Invariant relations, 314 
Invariants, integral-, 2f^^ 
Inverse of a transfo* 
Involution, involu' 
Isoperimetrical 

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413 



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Joakovsky's theorem, 107 

KinematicB, 1 
Kinetic energy, 85 

„ foons, 248 

„ potential, 88 
Kineto-statics, 37 
Klein's parameters, 11 
Koenigs and Lie*8 theorem, 269 
Kovalevski's top, 160 

Lagrange's braoket-ezpressions, 287 
equations of motion, 87 

with andetermined mal- 

tipliers, 211 
of impulsive motion, 49 
for quasi-coordinates, 48 
particles, 881 
Lagrangian function, 88 
Lambert's theorem, 90 
Larmor-Boltzmann representation of the Ijast 

Multiplier, 272 
Last Multiplier, 270 
Law, Newtonian, 85 
Least Action, 243 

„ curvature, 250 
Levi-Civita's theorem, 318 
Levy's theorem, 818 
Lie and Koenigs' theorem, 269 
Line and plane, invariable, 142, 884 
Liouville's theorem, 274, 811 

«f ^7P®> systems of, 66 

Localised vectors, 15 

Mass, 28 

Mathieu transformations, 290 
Mean anomaly, 88 
distance, 86 
motion, 86 
Meridian plane, 18 
Model, 46 

Molecular forces, 31 
Moment of a force, 29 
„ „ inertia, 116 
Momental ellipsoid, 122 
Momentum, 47 

angplar, 58 

conresponding to coordinate, 58 
integral of, 58 
Motion, impul kve, 47 
initi [ 44 
86 

,d, 293 
189 
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Natural dynamical systems, 56 
Newtonian law, 85 

Newton's theorem on revolving orbits, 82 
Node, 337 

Nodes, elimination of, 329 
Non-holonomic systems, 83 
Non-natural systems, 56 
Normal coordinates, 177 
form, 876 



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vibrations, 182, 191 



Orbit, 77 

„ periodic, 874 
Order of an integral-invariant, 262 

„ „ a system of equations, 51 
Oscillation, centre of, 180 



Parallel fields of force, 91 
Parameters, differential, 109 
„ Klein's, 11 

„ symmetrical, 8 

Particles, 27 

„ Lagrange^s, 881 
Pattern, 46 
Pendulum, simple, 71 

„ spherical, 102 

Perfect roughness, 81 
Pericentre, 84 
Perihelion, 84 

,, -constant, 85 

Periodic solutions or orbits, 874 
„ time, 86 

„ and non-periodic parts of Hamiltouian 
function, 402 
Pfaff's expression, 285 

,, system of equations, 296 
Pitch of a screw, 5 
Plane, invariable line and, 142, 834 
Planetoid, 341 

Poinsot's representation, 148 
Point-transformations, 282 
Poisson's bracket-expressions, 288 

„ theorem, 308 
Polhode, 150 

Possible displacements, 38 
Potential energy, 37 
kinetic, 88 
Schering's, 43 
Principal axes and moments of inertia, 122 

„ coordinates, 177 
Principle, HamUton's, 242, 245 
of Least Action, 243 
„ „ Curvature, 250 
„ superposition of vibrations, 182 
Problem of Three Bodies, 827 



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414 



Index 



Problem of Three Bodies, in a plane, 339 
„ ,, ,, restricted, 341 

,, two centres of gravitation, 95 

Product of inertia, 115 



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Quadratures, problems soluble by, 53 
Quasi-coordinates, 41 

Badius of gyration, 116 
Bayleigh's dissipation function, 226 
Reciprocal theorem, Helmholtz's, 293 
Relations, invariant, 314 
Relative velocity, 14 

„ integral-invariants, 265 
Repellent regions of field of force, 392 
Restricted Problem of Three Bodies, 341 
Resultant of vectors, 13 
Reversed forces, 47 

„ motion, 293 
Revolving orbits, 82 
Rigid body, 1, 31 
Rotation about a line or point, 1 
instantaneous axis of, 2 
,, centre of, 3 

Roughness, perfect, 31 



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Schering's potential function, 43 

Screw, 5 

Similarity in dynamical systems, 46 

Sleeping top, 201 

Smoothness, 31 

Solubility by quadratures, 53 

Solution, periodic, 374 

Spherical pendulum, 102 

,, top, 155 
Spirals, Cotes', 81 
Stability of equilibrium, 182 

,, steady motion, 189 

„ orbits, 384, 394 

coefiicient of, 384 

index of, 385 
Steady motion, 159, 189 
Sub-groups, 290 
Sudden fixture, 165 
Superposition of vibrations, 182 



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Suspension, centre of, 130 
Sylvester's theorem, 179 
Symbol, Christoffers, 89 

„ of a transformation, 292 
Symmetrical parameters, 8 
System, adjoint, 281 

dissipative, 222 

involution-, 310 

isoperimetrical, 261 

PfaflPs, 296 



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Thomson's theorem, 256 
Three Bodies, Problem of, 327 
„ n M t> in ft plane, 339 

restricted, 341 



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Time, 27 

„ periodic, 86 
Top, 151 

Kowalevski's, 160 
spherical, 155 
sleeping, 201 
Trajectory, 77, 241 
Transformations, contact-, 282 

Mathieu, 290 
point-, 282 
Translation of a body, 1 
True anomaly, 88 
Two centres of gravitation, 95 
Type, Liouville's, 66 

Variational equations, 262 
Vectors, 13 

,, localised, 15 
Velocity, 14 

angular, 14 
relative, 14 
,, corresponding to a coordinate, 32 
Vertex of a top, 151 
Vibrations about equilibrium, 173 

„ steady motion, 189 
normal, 182, 191 
of dissipative systems, 228 
,, non-holonomic systems, 217 



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