Google
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at |http: //books .google .com/I
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
ft
t»
«f
ti
>»
11
»»
If
»»
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
Jacobi's eqr
ft
»»
If
ff
ff
ft
ff
f»
Jacob'
Jaco*
Index
413
«H
fi
>>
i»
if
ft
»
t>
it
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
270
t«
i»
ft
«*
ft
It
»
mer
re'
\ .. 8f
fultipliei
ft
f
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
ff
ff
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
ft
ft
ft
ft
ft
414
Index
Problem of Three Bodies, in a plane, 339
„ ,, ,, restricted, 341
,, two centres of gravitation, 95
Product of inertia, 115
»*
>f
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
»»
i»
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
a
»»
f f
>f
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
»»
»»
»»
»»
Thomson's theorem, 256
Three Bodies, Problem of, 327
„ n M t> in ft plane, 339
restricted, 341
It
It
It
11
ft
ft
It It
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
It
It
tt
It
It
»t
Work, 30
CAMBRIDOB : PBINTBD BY J. AND C. F. CLAY, AT THK UNIVKB8ITY''®*^y',^eP^*'
-, Las
\)S»^
i-^*^.^
V
^
.w
A^**
v^-t^ft
\>V»^
SJ»^
f
^\
^