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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 ^\ ^