on a general method in dynamics - william rowan hamilton

8/11/2019 On a General Method in Dynamics - William Rowan Hamilton

1/65

ON A GENERAL METHOD IN DYNAMICS

By

William Rowan Hamilton

(Philosophical Transactions of the Royal Society, part II for 1834, pp. 247308.)

Edited by David R. Wilkins

2000


2/65

NOTE ON THE TEXT

This edition is based on the original publication in the Philosophical Transactions of the Royal Society , part II for 1834.

The following errors in the original published text have been corrected:a term w(n ) in the last summand on the right hand side of equation (S 5 .) has beencorrected to w(n 1) ;a minus sign ( ) missing from equation (K 6 .) has been inserted.The paper On a General Method in Dynamics has also been republished in The Mathe-

matical Papers of Sir William Rowan Hamilton, Volume II: Dynamics , edited for the Royal

Irish Academy by A. W. Conway and A. J. McConnell, and published by Cambridge Univer-sity Press in 1940.

David R. WilkinsDublin, February 2000

i


3/65

On a General Method in Dynamics; by which the Study of the Motions of all free Systems of attracting or repelling Points is reduced to the Search and Dif- ferentiation of one central Relation, or characteristic Function. By WilliamRowan Hamilton , Member of several scientic Societies in the British Do-minions, and of the American Academy of Arts and Sciences, Andrews Profes-sor of Astronomy in the University of Dublin, and Royal Astronomer of Ireland.Communicated by Captain Beaufort , R.N. F.R.S.

Received April 1,Read April 10, 1834.

[Philosophical Transactions of the Royal Society , part II for 1834, pp. 247308.]

Introductory Remarks.

The theoretical development of the laws of motion of bodies is a problem of such interestand importance, that it has engaged the attention of all the most eminent mathematicians,since the invention of dynamics as a mathematical science by Galileo , and especially sincethe wonderful extension which was given to that science by Newton . Among the successorsof those illustrious men, Lagrange has perhaps done more than any other analyst, to giveextent and harmony to such deductive researches, by showing that the most varied conse-quences respecting the motions of systems of bodies may be derived from one radical formula;the beauty of the method so suiting the dignity of the results, as to make of his great worka kind of scientic poem. But the science of force, or of power acting by law in space andtime, has undergone already another revolution, and has become already more dynamic, byhaving almost dismissed the conceptions of solidity and cohesion, and those other materialties, or geometrically imaginably conditions, which Lagrange so happily reasoned on, andby tending more and more to resolve all connexions and actions of bodies into attractionsand repulsions of points: and while the science is advancing thus in one direction by theimprovement of physical views, it may advance in another direction also by the inventionof mathematical methods. And the method proposed in the present essay, for the deduc-tive study of the motions of attracting or repelling systems, will perhaps be received withindulgence, as an attempt to assist in carrying forward so high an inquiry.

In the methods commonly employed, the determination of the motion of a free point inspace, under the inuence of accelerating forces, depends on the integration of three equationsin ordinary differentials of the second order; and the determination of the motions of a systemof free points, attracting or repelling one another, depends on the integration of a system of such equations, in number threefold the number of the attracting or repelling points, unless wepreviously diminish by unity this latter number, by considering only relative motions. Thus, inthe solar system, when we consider only the mutual attractions of the sun and the ten knownplanets, the determination of the motions of the latter about the former is reduced, by theusual methods, to the integration of a system of thirty ordinary differential equations of the

1


4/65

second order, between the coordinates and the time; or, by a transformation of Lagrange , tothe integration of a system of sixty ordinary differential equations of the rst order, betweenthe time and the elliptic elements: by which integrations, the thirty varying coordinates, orthe sixty varying elements, are to be found as functions of the time. In the method of thepresent essay, this problem is reduced to the search and differentiation of a single function,which satises two partial differential equations of the rst order and of the second degree:and every other dynamical problem, respecting the motions of any system, however numerous,of attracting or repelling points, (even if we suppose those points restricted by any conditionsof connexion consistent with the law of living force,) is reduced, in like manner, to the studyof one central function, of which the form marks out and characterizes the properties of themoving system, and is to be determined by a pair of partial differential equations of therst order, combined with some simple considerations. The difficulty is therefore at leasttransferred from the integration of many equations of one class to the integration of two of another: and even if it should be thought that no practical facility is gained, yet an intellectualpleasure may result from the reduction of the most complex and, probably, of all researches

respecting the forces and motions of body, to the study of one characteristic function,* theunfolding of one central relation.The present essay does not pretend to treat fully of this extensive subject,a task which

may require the labours of many years and many minds; but only to suggest the thoughtand propose the path to others. Although, therefore, the method may be used in the mostvaried dynamical researches, it is at present only applied to the orbits and perturbations of asystem with any laws of attraction or repulsion, and with one predominant mass or centre of predominant energy; and only so far, even in this one research, as appears sufficient to makethe principle itself understood. It may be mentioned here, that this dynamical principle isonly another form of that idea which has already been applied to optics in the Theory of systems of rays , and that an intention of applying it to the motion of systems of bodieswas announced at the publication of that theory. And besides the idea itself, the mannerof calculation also, which has been thus exemplied in the sciences of optics and dynamics,seems not conned to those two sciences, but capable of other applications; and the peculiarcombination which it involves, of the principles of variations with those of partial differentials,for the determination and use of an important class of integrals, may constitute, when it shallbe matured by the future labours of mathematicians, a separate branch of analysis.

WILLIAM R. HAMILTON.

Observatory, Dublin, March 1834.

* Lagrange and, after him, Laplace and others, have employed a single function toexpress the different forces of a system, and so to form in an elegant manner the differentialequations of its motion. By this conception, great simplicity has been given to the statementof the problem of dynamics; but the solution of that problem, or the expression of the mo-tions themselves, and of their integrals, depends on a very different and hitherto unimaginedfunction, as it is the purpose of this essay to show.

Transactions of the Royal Irish Academy, Vol. xv , page 80. A notice of this dynamicalprinciple was also lately given in an article On a general Method of expressing the Paths of Light and of the Planets, published in the Dublin University Review for October 1833.

2


5/65

Integration of the Equations of Motion of a System, characteristic Function of such Motion,and Law of varying Action.

1. The known differential equations of motion of a system of free points, repelling orattracting one another according to any functions of their distances, and not disturbed byany foreign force, may be comprised in the following formula:

.m (x x + y y + z z ) = U. (1.)

In this formula the sign of summation extends to all the points of the system; m is, for anyone such point, the constant called its mass; x , y , z , are its component accelerations, orthe second differential coefficients of its rectangular coordinates x, y, z, taken with respectto the time; x , y, z , are any arbitrary innitesimal displacements which the point canbe imagined to receive in the same three rectangular directions; and U is the innitesimalvariation corresponding, of a function U of the masses and mutual distances of the several

points of the system, of which the form depends on the laws of their mutual actions, by theequation

U = .mm f (r ), (2.)

r being the distance between any two points m , m , and the function f (r ) being such that thederivative or differential coefficient f (r ) expresses the law of their repulsion, being negativein the case of attraction. The function which has been here called U may be named the force-function of a system: it is of great utility in theoretical mechanics, into which it wasintroduced by Lagrange , and it furnishes the following elegant forms for the differentialequations of motion, included in the formula (1.):

m 1x 1 = U x 1

; m2x 2 = U x 2

; . . . m n x n = U x n

;

m 1y1 = U y1

; m2y2 = U y2

; . . . m n yn = U yn

;

m 1z1 = U z1

; m2z2 = U z2

; . . . m n zn = U zn

;

(3.)

the second members of these equations being the partial differential coefficients of the rstorder of the function U . But notwithstanding the elegance and simplicity of this knownmanner of stating the principal problem of dynamics, the difficulty of solving that problem,or even of expressing its solution, has hitherto appeared insuperable; so that only sevenintermediate integrals, or integrals of the rst order, with as many arbitrary constants, havehitherto been found for these general equations of motion of a system of n points, insteadof 3n intermediate and 3 n nal integrals, involving ultimately 6 n constants; nor has anyintegral been found which does not need to be integrated again. No general solution has beenobtained assigning (as a complete solution ought to do) 3 n relations between the n massesm 1 , m 2 , . . . m n , the 3n varying coordinates x1 , y1 , z1 , . . . x n , yn , zn , the varying time t , and the6n initial data of the problem, namely, the initial coordinates a1 , b1 , c1 , . . . a n , bn , cn , and theirinitial rates of increase a1 , b1 , c1 , . . . a n , bn , cn ; the quantities called here initial being those

3


6/65

which correspond to the arbitrary origin of time. It is, however, possible (as we shall see)to express these long-sought relations by the partial differential coefficients of a new centralor radical function, to the search and employment of which the difficulty of mathematicaldynamics becomes henceforth reduced.

2. If we put for abridgement

T = 12 .m (x2 + y 2 + z 2), (4.)

so that 2 T denotes, as in the Mecanique Analytique, the whole living force of the system;(x , y , z , being here, according to the analogy of our foregoing notation, the rectangularcomponents of velocity of the point m , or the rst differential coefficients of its coordinatestaken with respect to the time;) an easy and well known combination of the differential equa-tions of motion, obtained by changing in the formula (1.) the variations to the differentialsof the coordinates, may be expressed in the following manner,

dT = dU, (5.)

and gives, by integration, the celebrated law of living force, under the form

T = U + H. (6.)

In this expression, which is one of the seven known integrals already mentioned, thequantity H is independent of the time, and does not alter in the passage of the points of thesystem from one set of positions to another. We have, for example, an initial equation of thesame form, corresponding to the origin of time, which may be written thus,

T 0 = U 0 + H. (7.)

The quantity H may, however, receive any arbitrary increment whatever, when we passin thought from a system moving in one way, to the same system moving in another, withthe same dynamical relations between the accelerations and positions of its points, but withdifferent initial data; but the increment of H , thus obtained, is evidently connected with theanalogous increments of the functions T and U , by the relation

T = U + H, (8.)

which, for the case of innitesimal variations, may be conveniently be written thus,

T = U + H ; (9.)

and this last relation, when multiplied by dt , and integrated, conducts to an important result.For it thus becomes, by (4.) and (1.),

.m (dx.x + dy . y + dz . z ) = .m (dx . x + dy . y + dz . z ) + H.dt , (10.)4


7/65

that is, by the principles of the calculus of variations,

V = .m (x x + y y + z z ) .m (a a + b b + c c) + t H, (A.)if we denote by V the integral

V = .m (x dx + y dy + z dz ) = t

02T dt, (B.)

namely, the accumulated living force, called often the action of the system, from its initial toits nal position.

If, then, we consider (as it is easy to see that we may) the action V as a function of the initial and nal coordinates, and of the quantity H , we shall have, by (A.), the followinggroups of equations; rst, the group,

V x 1

= m 1 x 1 ; V x 2= m 2x2 ; . . . V x n

= m n x n ;

V y1

= m 1 y1 ; V

y2= m 2y2 ; . . .

V yn

= m n yn ;

V z1

= m 1 z1 ; V

z2= m 2z2 ; . . .

V zn

= m n zn .

(C .)

Secondly, the group,

V

a 1=

m 1a 1 ;

V

a 2=

m 2 a 2 ; . . .

V

a n=

m n a n ;

V b1

= m 1b1 ; V

b2= m 2 b2 ; . . .

V bn

= m n bn ;V c1

= m 1c1 ; V

c2= m 2 c2 ; . . .

V cn

= m n cn ;(D .)

and nally, the equation,V H

= t. (E .)

So that if this function V were known, it would only remain to eliminate H between the 3 n +1equations (C.) and (E.), in order to obtain all the 3 n intermediate integrals, or between (D.)and (E.) to obtain all the 3 n nal integrals of the differential equations of motion; that is,ultimately, to obtain the 3 n sought relations between the 3 n varying coordinates and the time,involving also the masses and the 6 n initial data above mentioned; the discovery of whichrelations would be (as we have said) the general solution of the general problem of dynamics.We have, therefore, at least reduced that general problem to the search and differentiation of a single function V , which we shall call on this account the characteristic function of motion of a system; and the equation (A.), expressing the fundamental law of its variation,we shall call the equation of the characteristic function , or the law of varying action .

5


8/65

3. To show more clearly that the action or accumulated living force of a system, or inother words, the integral of the product of the living force by the element of the time, maybe regarded as a function of the 6 n + 1 quantities already mentioned, namely, of the initialand nal coordinates, and of the quantity H , we may observe, that whatever depends onthe manner and time of motion of the system may be considered as such a function; becausethe initial form of the law of living force, when combined with the 3 n known or unknownrelations between the time, the initial data, and the varying coordinates, will always furnish3n + 1 relations, known or unknown, to connect the time and the initial components of velocities with the initial and nal coordinates, and with H . Yet from not having formedthe conception of the action as a function of this kind, the consequences that have beenhere deduced from the formula (A.) for the variation of that denite integral appear to haveescaped the notice of Lagrange , and of the other illustrious analysts who have written ontheoretical mechanics; although they were in possession of a formula for the variation of thisintegral not greatly differing from ours. For although Lagrange and others, in treating of the motion of a system, have shown that the variation of this denite integral vanishes when

the extreme coordinates and the constant H are given, they appear to have deduced fromthis result only the well known law of least action ; namely, that if the points or bodies of a system be imagined to move from a given set of initial to a given set of nal positions,not as they do nor even as they could move consistently with the general dynamical laws ordifferential equations of motion, but so as not to violate any supposed geometrical connexions,nor that one dynamical relation between velocities and congurations which constitutes thelaw of living force; and if, besides, this geometrically imaginable, but dynamically impossiblemotion, be made to differ innitely little from the actual manner of motion of the system,between the given extreme positions; then the varied value of the denite integral calledaction, or the accumulated living force of the system in the motion thus imagined, will differinnitely less from the actual value of that integral. But when this well known law of least,or as it might be better called, of stationary action , is applied to the determination of theactual motion of the system, it serves only to form, by the rules of the calculus of variations,the differential equations of motion of the second order, which can always be otherwise found.It seems, therefore, to be with reason that Lagrange , Laplace , and Poisson have spokenlightly of the utility of this principle in the present state of dynamics. A different estimate,perhaps, will be formed of that other principle which has been introduced in the presentpaper, under the name of the law of varying action , in which we pass from an actual motionto another motion dynamically possible, by varying the extreme positions of the system, and(in general) the quantity H , and which serves to express, by means of a single function, notthe mere differential equations of motion, but their intermediate and their nal integrals.

Verication of the foregoing Integrals.

4. A verication, which ought not to be neglected, and at the same time an illustrationof this new principle, may be obtained by deducing the known differential equations of motionfrom our system of intermediate integrals, and by showing the consistence of these again withour nal integral system. As preliminary to such verication, it is useful to observe that thenal equation (6.) of living force, when combined with the system (C.), takes this new form,

6


9/65


10/65


11/65

and nally, for the last equation,ddt

V H

= 1 . (K.)

By combining the equations (C.) with their differentials (H.), and with the relation (F.),we deduced, in the foregoing number, the known equations of motion (3.); and we are now toshow the consistence of the same intermediate integrals (C.) with the group of differentials(I.) which have been obtained from the nal integrals.

The rst equation of the group (I.) may be developed thus:

0 = x 1 2V

a 1 x 1+ x2

2V a 1 x 2

+ + xn 2V

a 1 x n

+ y1 2V

a 1 y1+ y2

2V a 1 y2

+ + yn 2V

a 1 yn

+ z1 2V

a 1 z1+ z2

2V a 1 z2

+ + zn 2V

a 1 zn

(14.)

and the others may be similarly developed. In order, therefore, to show that they are satisedby the group (C.), it is sufficient to prove that the following equations are true,

0 = a i

. 12m

V x

2

+V y

2

+V z

2

,

0 = bi

. 12m

V x

2

+V y

2

+V z

2

,

0 = c i

. 12m

V x

2

+V y

2

+V z

2

,

(L .)

the integer i receiving any value from 1 to n inclusive; which may be shown at once, andthe required verication thereby be obtained, if we merely take the variation of the relation(F.) with respect to the initial coordinates, as in the former verication we took its variationwith respect to the nal coordinates, and so obtained results which agreed with the knownequations of motion, and which may be thus collected,

x i

. 12m

V x

2

+V y

2

+V z

2

= U x i

;

y i . 12m

V

x

2

+V

y

2

+V

z

2

= U

y i ;

z i

. 12m

V x

2

+V y

2

+V z

2

= U z i

.

(M .)

The same relation (F.), by being varied with respect to the quantity H , conducts to theexpression

H

. 12m

V x

2

+V y

2

+V z

2

= 1; (N .)

9


12/65


13/65

In this manner, therefore, we can deduce from the properties of our characteristic functionthe six other known integrals above mentioned, in addition to that seventh which containsthe law of living force, and which assisted in the discovery of our method.

Introduction of relative or polar Coordinates, or other marks of position of a System.

7. The property of our characteristic function, by which it depends only on the internalor mutual relations between the positions initial and nal of the points of an attracting orrepelling system, suggests an advantage in employing internal or relative coordinates; andfrom the analogy of other applications of algebraical methods to researches of a geometricalkind, it may be expected that polar and other marks of position will also often be founduseful. Supposing, therefore, that the 3 n nal coordinates x1 y1 z1 . . . xn yn zn have beenexpressed as functions of 3 n other variables 1 2 . . . 3n , and that the 3 n initial coordinateshave in like manner been expressed as functions of 3 n similar quantities, which we shall calle1 e2 . . . e3n , we shall proceed to assign a general method for introducing these new marks of

position into the expressions of our fundamental relations.For this purpose we have only to transform the law of varying action, or the fundamentalformula (A.), by transforming the two sums,

.m (x x + y y + z z ), and .m (a a + b b + c c),

which it involves, and which are respectively equivalent to the following more developedexpressions,

.m (x x + y y + z z ) = m 1(x 1 x 1 + y1 y1 + z1 z1)+ m 2(x 2 x 2 + y2 y2 + z2 z2)

+ &c. + m n (x n x n + yn yn + zn zn );

(17.)

.m (a a + b b + c c) = m 1(a 1 a 1 + b1 b1 + c1 c1)+ m 2(a 2 a 2 + b2 b2 + c2 c2)

+ &c. + m n (a n a n + bn bn + cn cn ).(18.)

Now x i being by supposition a function of the 3 n new marks of position 1 . . . 3n , itsvariation x i , and its differential coefficient x i may be thus expressed:

x i = x i1

1 + x i2

2 + + x i3n

3n ; (19.)

x i = x i1

1 + x i2

2 + + x i3n

3n ; (20.)

and similarly for yi and zi . If, then, we consider x i as a function, by (20.), of 1 . . . 3n ,involving also in general 1 . . . 3n , and if we take its partial differential coefficients of therst order with respect to 1 . . . 3n , we nd the relations,

x i1

= x i1

; x i

2=

x i2

; . . . x i

3n=

x i3n

; (21.)

11


14/65

and therefore we obtain these new expressions for the variations x i , yi , zi ,

x i = x i1

1 + x i2

2 + + x i3n

3n ,

y i = yi1

1 + yi2

2 + + yi3n

3n ,

z i = zi1

1 + zi2

2 + + zi3n

3n .

(22.)

Substituting these expressions (22.) for the variations in the sum (17.), we easily trans-form it into the following,

.m (x x + y y + z z ) = .m xx1

+ y y1

+ z z1

. 1

+ .m x x2+ y y2

+ z z2. 2

+ &c.+ .m x x3n

+ y y3n

+ z z3n

. 3n

= T 1

1 + T 2

2 + + T 3n

3n ;

(23.)

T being the same quantity as before, namely, the half of the nal living force of system, butbeing now considered as a function of 1 . . . 3n , involving also the masses, and in general1 . . . 3n , and obtained by substituting for the quantities x y z their values of the form

(20.) in the equation of denition

T = 12 .m (x2 + y 2 + z 2). (4.)

In like manner we nd this transformation for the sum (18.),

.m (a a + b b + c c) = T 0e1

e1 + T 0e2

e2 + + T 0e3n

e3n . (24.)

The law of varying action, or the formula (A.), becomes therefore, when expressed bythe present more general coordinates or marks of position,

V = .T

.T e

e + t H ; (Q.)

and instead of the groups (C.) and (D.), into which, along with the equation (E.), this lawresolved itself before, it gives now these other groups,

V 1

= T 1

; V

2=

T 2

; V

3n=

T 3n

; (R.)

12


15/65

andV e1

= T 0e1

; V

e2=

T 0e2

; V

e3n=

T 0e3n

. (S.)

The quantities e1 e2 . . . e3n , and e1 e2 . . . e3n , are now the initial data respecting themanner of motion of the system; and the 3 n nal integrals, connecting these 6 n initial data,and the n masses, with the time t, and with the 3 n nal or varying quantities 1 2 . . . 3n ,which mark the varying positions of the n moving points of the system, are now to beobtained by eliminating the auxiliary constant H between the 3 n +1 equations (S.) and (E.);while the 3n intermediate integrals, or integrals of the rst order, which connect the samevarying marks of position and their rst differential coefficients with the time, the masses,and the initial marks of position, are the result of elimination of the same auxiliary constantH between the equations (R.) and (E.). Our fundamental formula, and intermediate andnal integrals, can therefore be very simply expressed with any new sets of coordinates; andthe partial differential equations (F.) (G.), which our characteristic function V must satisfy,and which are, as we have said, essential in the theory of that function, can also easily be

expressed with any such transformed coordinates, by merely combining the nal and initialexpressions of the law of living force,

T = U + H, (6.)

T 0 = U 0 + H, (7.)

with the new groups (R.) and (S.). For this purpose we must now consider the function U ,of the masses and mutual distances of the several points of the system, as depending onthe new marks of position 1 2 . . . 3n ; and the analogous function U 0 , as depending simi-larly on the initial quantities e1 e2 . . . e3n ; we must also suppose that T is expressed (as it

may) as a function of its own coefficients, T 1 ,

T 2 , . . .

T 3n , which will always be, with re-

spect to these, homogeneous of the second dimension, and may also involve explicitly thequantities 1 2 . . . 3n ; and that T 0 is expressed as a similar function of its coefficientsT 0e1

, T 0e2

, . . . T 0e3n

; so that

T = F T 1

, T 2

, . . . T 3n

,

T 0 = F T 0e1

, T 0e2

, . . . T 0e3n

;(25.)

and that then these coefficients of T and T 0 are changed to their values (R.) and (S.), so asto give, instead of (F.) and (G.), two other transformed equations, namely,

F V 1

, V 2

, . . . V 3n

= U + H, (T .)

and, on account of the homogeneity and dimension of T 0 ,

F V e1

, V e2

, . . . V e3n

= U 0 + H. (U.)

13


16/65

8. Nor is there any difficulty in deducing analogous transformations for the known dif-ferential equations of motion of the second order, of any system of free points, by taking thevariation of the new form (T.) of the law of living force, and by attending to the dynamicalmeanings of the coefficients of our characteristic function. For if we observe that the nalliving force 2T , when considered as a function of 1 2 . . . 3n , and of 1 2 . . . 3n , is neces-sarily homogeneous of the second dimension with respect to the latter set of variables, andmust therefore satisfy the condition

2T = 1T 1

+ 2T 2

+ + 3nT

3n, (26.)

we shall perceive that its total variation,

T = T 1

1 + T 2

2 + + T 3n

3n

+ T 1

1 + T 22 + + T 3n 3n ,

(27.)

may be put under the form

T = 1 T 1

+ 2 T 2

+ + 3n T 3n

T 1

1 T 2

2 T 3n

3n

= . T

.

T

= . V

T

,

(28.)

and therefore that the total variation of the new partial differential equation (T.) may bethus written,

. V

T

= .U

+ H : (V.)

in which, if we observe that = ddt

, and that the quantities of the form are the only ones

which vary with the time, we shall see that

. V

= ddt

V

. + ddt

V e

. e + ddt

V H

. H, (29.)

because the identical equation dV = dV gives, when developed,

V

. d + V e

. de + V H

. dH = dV

. + dV e

. e + dV H

. H. (30.)

14


17/65

Decomposing, therefore, the expression (V.), for the variation of half the living force, intoas many separate equations as it contains independent variations, we obtain, not only theequation

ddt

V H

= 1 , (K.)

which had already presented itself, and the group

ddt

V e1

= 0 , d

dtV e2

= 0 , d

dtV

e3n= 0 , (W .)

which might have been at once obtained by differentiation from the nal integrals (S.), butalso a group of 3n other equations of the form

ddt

V

T

= U

, (X.)

which give, by the intermediate integrals (R.),

ddt

T

T

= U

: (Y.)

that is, more fully,ddt

T 1

T 1

= U 1

;

ddt

T

2

T 2

= U 2

;

ddt

T 3n

T 3n

= U 3n

.

(Z.)

These last transformations of the differential equations of motion of the second order,of an attracting or repelling system, coincide in all respects (a slight difference of notationexcepted,) with the elegant canonical forms in the Mecanique Analytique of Lagrange ; butit seemed worth while to deduce them here anew, from the properties of our characteristicfunction. And if we were to suppose (as it has often been thought convenient and evennecessary to do,) that the n points of a system are not entirely free, nor subject only totheir own mutual attractions or repulsions, but connected by any geometrical conditions, andinuenced by any foreign agencies, consistent with the law of conservation of living force;so that the number of independent marks of position would be now less numerous, and theforce-function U less simple than before; it might still be proved, by a reasoning very similarto the foregoing, that on these suppositions also (which however, the dynamical spirit istending more and more to exclude,) the accumulated living force or action V of the systemis a characteristic motion-function of the kind already explained; having the same law andformula of variation, which are susceptible of the same transformations; obliged to satisfy inthe same way a nal and an initial relation between its partial differential coefficients of the

15


18/65

rst order; conducting, by the variation of one of these two relations, to the same canonicalforms assigned by Lagrange for the differential equations of motion; and furnishing, on thesame principles as before, their intermediate and their nal integrals. To those imaginablecases, indeed, in which the law of living force no longer holds, our method also would notapply; but it appears to be the growing conviction of the persons who have meditated themost profoundly on the mathematical dynamics of the universe, that these are cases suggestedby insufficient views of the mutual actions of body.

9. It results from the foregoing remarks, that in order to apply our method of thecharacteristic function to any problem of dynamics respecting any moving system, the knownlaw of living force is to be combined with our law of varying action; and that the generalexpression of this latter law is to be obtained in the following manner. We are rst to expressthe quantity T , namely, the half of the living force of the system, as a function (which willalways be homogeneous of the second dimension,) of the differential coefficients or rates of increase 1 , 2 &c., of any rectangular coordinates, or other marks of position of the system:

we are next to take the variation of this homogeneous function with respect to those rates of increase, and to change the variations of those rates 1 , 2 , &c., to the variations 1 , 2 ,&c., of the marks of position themselves; and then to subtract the initial from the nal valueof the result, and to equate the remainder to V t H . A slight consideration will showthat this general rule or process for obtaining the variation of the characteristic function V ,is applicable even when the marks of position 1 , 2 , &c. are not all independent of eachother; which will happen when they have been made, from any motive of convenience, morenumerous than the rectangular coordinates of the several points of the system. For if wesuppose that the 3 n rectangular coordinates x 1 y1 z1 . . . x n yn zn have been expressed by anytransformation as functions of 3 n + k other marks of position, 1 2 . . . 3n + k , which musttherefore be connected by k equations of condition,

0 = 1(1 , 2 , . . . 3n + k ),0 = 2(1 , 2 , . . . 3n + k ),

0 = k (1 , 2 , . . . 3n + k ),

(31.)

giving k of the new marks of position as functions of the remaining 3 n ,

3n +1 = 1(1 , 2 , . . . 3n ),

3n +2 = 2(1 , 2 , . . . 3n ),

3n + k = k (1 , 2 , . . . 3n ),

(32.)

the expressionT = 12 .m (x

2 + y 2 + z 2), (4.)

will become, by the introduction of these new variables, a homogeneous function of thesecond dimension of the 3 n + k rates of increase 1 , 2 , . . . 3n + k , involving also in general

16


19/65


20/65

which give, by (32.),T 1

1 + T 2

2 + + T 3n

3n =T 1

1 +T 2

2 + + T

3n + k3n + k ; (38.)

we may therefore put the expression (Q.) under the following more general form,

V = .T

.T 0e

e + t H, (A1 .)

the coefficientsT

being formed by treating all the 3 n + k quantities 1 , 2 , . . . , 3n + k , as

independent; which was the extension above announced, of the rule for forming the variationof the characteristic function V .

We cannot, however, immediately decompose this new expression (A 1 .) for V , as wedid the expression (Q.), by treating all the variations , e, as independent; but we maydecompose it so, if we previously combine it with the nal equations of condition (31.), andwith the analogous initial equations of condition, namely,

0 = 1(e1 , e2 , . . . e 3n + k ),0 = 2(e1 , e2 , . . . e 3n + k ),

0 = k (e1 , e2 , . . . e 3n + k ),

(39.)

which we may do by adding the variations of the connecting functions 1 , . . . , k , 1 , . . . kmultiplied respectively by the factors to be determined, 1 , . . . k , 1 , . . . k . In this mannerthe law of varying action takes this new form,

V = .T

.

T 0

ee + t H + . + . ; (B1 .)

and decomposes itself into 6 n + 2 k + 1 separate expressions, for the partial differential coef-cients of the rst order of the characteristic function V , namely, into the following,

V 1

=T 1

+ 111

+ 221

+ + kk1

,

V 2

=T 2

+ 112

+ 222

+ + kk2

,

V

3n + k

= T

3n + k

+ 11

3n + k

+ + kk

3n + k

,

(C 1 .)

andV e1

= T e1

+ 1 1e1

+ 2 2e1

+ + k ke1

,

V e2

= T e2

+ 1 1e2

+ 2 2e2

+ + k ke2

,

V e3n + k

= T

e3n + k+ 1

1e3n + k

+ + k k

e3n + k,

(D1 .)

18


21/65

besides the old equation (E.). The analogous introduction of multipliers in the canonicalforms of Lagrange , for the differential equations of motion of the second order, by which a

sum such as .

is added to U

in the second member of the formula (Y.), is also easily

justied on the principles of the present essay.

Separation of the relative motion of a system from the motion of its centre of gravity; char-acteristic function for such motion, and law of its variation.

10. As an example of the foregoing transformations, and at the same time as an importantapplication, we shall now introduce relative coordinates, x y z , referred to an internal originx y z ; that is, we shall put

x i = x i + x , yi = y i + y , zi = z i + z , (40.)

and in like mannera i = a i + a , bi = b i + b , ci = c i + c ; (41.)

together with the differentiated expressions

x i = x i + x , yi = y i + y , zi = z i + z , (42.)

anda i = a i + a , bi = b i + b , ci = c i + c . (43.)

Introducing the expressions (42.) for the rectangular components of velocity, we nd that thevalue given by (4.) for the living force 2 T decomposes itself into the three following parts,

2T = .m (x 2 + y 2 + z 2)= .m (x 2 + y 2 + z 2 ) + 2( x .mx + y .my + z .mz )

+ ( x 2 + y 2 + z 2 ) m ;(44.)

if then we establish, as we may, the three equations of condition,

.mx = 0 , .my = 0 , .mz = 0 , (45.)

which give by (40.),x = .mx

m , y = .my

m , z = .mz

m , (46.)

so that x y z are now the coordinates of the point which is called the centre of gravity of the system, we may reduce the function T to the form

T = T + T , (47.)

in whichT = 12 .m (x

2 + y 2 + z 2 ), (48.)

19


22/65

andT = 12 (x

2 + y 2 + z 2 ) m. (49.)

By this known decomposition, the whole living force 2 T of the system is resolved intothe two parts 2 T and 2T , of which the former, 2 T , may be called the relative living force ,

being that which results solely from the relative velocities of the points of the system, in theirmotions about their common centre of gravity x y z ; while the latter part, 2 T , resultsonly from the absolute motion of that centre of gravity in space, and is the same as if all themasses of the system were united in that common centre. At the same time, the law of livingforce, T = U + H , (6.), resolves itself by the law of motion of the centre of gravity into thetwo following separate equations,

T = U + H , (50.)

andT = H ; (51.)

H and H being two new constants independent of the time t, and such that their sum

H + H = H. (52.)

And we may in like manner decompose the action, or accumulated living force V , which

is equal to the denite integral t

02T dt , into the two following analogous parts,

V = V + V , (E1 .)

determined by the two equations,

V =

t

02T dt, (F1 .)

and

V = t

02T dt. (G1 .)

The last equation gives by (51.),V = 2H t ; (53.)

a result which, by the law of motion of the centre of gravity, may be thus expressed,

V = (x a )2 + ( y b )2 + ( z c )2 . 2H m : (H1 .)a b c being the initial coordinates of the centre of gravity, so thata =

.ma m

, b = .mb

m , c =

.mc m

. (54.)

And for the variation V of the whole function V , the rule of the last number gives

V = .m (x x a a + y y b b + z z c c )+ ( x x a a + y y b b + z z c c ) m+ t H + 1 .mx + 2 .my + 3 .mz+ 1 .ma + 2 .mb + 3 .mc ;

(I1 .)

20


23/65

while the variation of the part V , determined by the equation (H 1 .), is easily shown to beequivalent to the part

V = ( x x a a + y y b b + z z c c ) m + t H ; (K1 .)the variation of the other part V may therefore be thus expressed,

V = .m (x x a a + y y b b + z z c c )+ t H + 1 .mx + 2 .my + 3 .mz+ 1 .ma + 2 .mb + 3 .mc :

(L1 .)

and it resolves itself into the following separate expressions, in which the part V is consideredas a function of the 6 n + 1 quantities x i y i z i a i b i c i H , of which, however, only 6 n 5are really independent:rst group,

V x 1

= m 1x 1 + 1m 1 ; V x n = m n x n + 1m n ;V y 1

= m 1y 1 + 2m 1 ; V

y n= m n y n + 2m n ;

V z 1

= m 1z 1 + 3m 1 ; V

z n= m n z n + 3m n ;

(M1 .)

second group,

V a 1

= m 1 a 1 + 1m 1 ; V

a n= m n a n + 1m n ;

V b 1

= m 1 b 1 + 2m 1 ; V

b n= m n b n + 2m n ;

V c 1

= m 1 c 1 + 3m 1 ; V

c n= m n c n + 3m n ;

(N1 .)

and nally,V H

= t. (O1 .)

With respect to the six multipliers 1 2 3 1 2 3 which were introduced by the 3nal equations of condition (45.), and by the 3 analogous initial equations of condition,

.ma = 0 , .mb = 0 , .mc = 0; (55 .)

we have, by differentiating these conditions,

.mx = 0 , .my = 0 , .mz = 0 , (56.)

and .ma = 0 , .mb = 0 , .mc = 0; (57 .)

21


24/65


25/65

and the groups (M 1 .) and (N 1 .) will reduce themselves to the two following:V x 1

= m 1x 1 ; V

x 2= m 2x 2 ;

V x n

= m n x n ;

V

y 1= m 1y 1 ;

V

y 2= m 2y 2 ;

V

y n= m n y n ;

V z 1

= m 1z 1 ; V

z 2= m 2z 2 ;

V z n

= m n z n ;

(Q 1 .)

andV a 1

= m 1 a 1 ; V

a 2= m 2a 2 ;

V a n

= m n a n ;V b 1

= m 1 b 1 ; V

b 2= m 2b 2 ;

V b n

= m n b n ;V c 1

= m 1 c 1 ; V

c 2= m 2c 2 ;

V c n

= m n c n ;(R 1 .)

analogous in all respects to the groups (C.) and (D.). We nd, therefore, for the relativemotion of a system about its own centre of gravity, equations of the same form as thosewhich we had obtained before for the absolute motion of the same system of points in space,And we see that in investigating such relative motion only, it is useful to conne ourselves tothe part V of our whole characteristic function, that is, to the relative action of the system,or accumulated living force of the motion about the centre of gravity; and to consider thispart as the characteristic function of such relative motion, in a sense analogous to that whichhas been already explained.

This relative action, or part V , may, however, be otherwise expressed, and even in aninnite variety of ways, on account of the six equations of condition which connect the 6 n

centrobaric coordinates; and every different preparation of its form will give a different setof values for the six multipliers 1 2 3 1 2 3 . For example, we might eliminate, by aprevious preparation, the six centrobaric coordinates of the point m n from the expression of V , so as to make this expression involve only the centrobaric coordinates of the other n 1points of the system, and then we should have

V x n

= 0 , V

y n= 0 ,

V z n

= 0 , V

a n= 0 ,

V b n

= 0 , V

c n= 0 , (S1 .)

and therefore, by the six last equations of the groups (M 1 .) and (N 1 .), the multipliers wouldtake the values

1 =

x n , 2 =

y n , 3 =

z n , 1 = a n , 2 = b n , 3 = c n , (65.)

and would reduce, by (60.) and (61.), the preceding 6 n 6 equations of the same groups(M1 .) and (N 1 .), to the formsV x 1

= m 1 1 , V

x 2= m 2 2 ,

V x n 1

= m n 1 n 1 ,

V y 1

= m 11 , V

y 2= m 2 2 ,

V y n 1

= m n 1 n 1 ,

V z 1

= m 1 1 , V

z 2= m 2 2 ,

V z n 1

= m n 1 n 1 ,

(T 1 .)

23


26/65

andV a 1

= m 1 1 , V

a 2= m 2 2 ,

V a n 1

= m n 1 n 1 ,V

b 1=

m 1 1 ,

V

b 2=

m 2 2 ,

V

b n 1=

m n 1 n 1 ,

V c 1

= m 1 1 , V

c 2= m 2 2 ,

V c n 1

= m n 1 n 1 .(U1 .)

12. We might also express the relative action V , not as a function of the centrobaric,but of some other internal coordinates, or marks of relative position. We might, for instance,express it and its variation as functions of the 6 n 6 independent internal coordinates already mentioned, and of their variations, dening these without any reference to thecentre of gravity, by the equations

i = x i xn , i = yi yn , i = zi zn , i = a i a n , i = bi bn , i = ci cn .

(66.)

For all such transformations of V it is easy to establish a rule or law, which may be calledthe law of varying relative action (exactly analogous to the rule (B 1 .)), namely, the following:

V = .T

.T 0e

e + t H + . + . ; (V1 .)

which implies that we are to express the half T of the relative living force of the system asa function of the rates of increase of any marks of relative position; and after taking itsvariation with respect to these rates, to change their variations to the variations of the marksof position themselves; then to subtract the initial from the nal value of the result, and

to add the variations of the nal and initial functions , which enter into the equationsof condition, if any, of the form = 0, = 0, (connecting the nal and initial marks of relative position,) multiplied respectively by undetermined factors ; and lastly, to equatethe whole result to V t H , H being the quantity independent of the time in the equation(50.) of relative living force, and V being the relative action, of which we desired to expressthe variation. It is not necessary to dwell here on the demonstration of this new rule (V 1 .),which may easily be deduced from the principles already laid down; or by the calculus of variations from the law of relative living force, combined with the differential equations of second order of relative motion.

But to give an example of its application, let us resume the problem already mentioned,namely to express V by means of the 6n

5 independent variations i i i i i i

H . For this purpose we shall employ a known transformation of the relative living force2T , multiplied by the sum of the masses of the system, namely the following:

2T m = .m i m k {(x i x k )2 + ( yi yk )2 + ( zi zk )2}: (67.)the sign of summation extending, in the second member, to all the combinations of pointstwo by two, which can be formed without repetition. This transformation gives, by (66.),

2T m = m n .m ( 2 + 2 + 2)+ .m i m k {( i k )2 + ( i k )2 + ( i k )2};

(68.)

24


27/65

the sign of summation extending only to the rst n 1 points of the system. Applying,therefore, our general rule or law of varying relative action, and observing that the 6 n 6internal coordinates are independent, we nd the following new expression:V = t H +

m n

m . .m (

+

+

)

+ 1 m

. .m i m k {( i k )( i k ) + ( i k )(i k )+ ( i k )( i k )}

1 m

. .m i m k {( i k )( i k ) + ( i k )( i k )+ ( i k )( i k )}:

(W 1 .)

which gives, besides the equation (O 1 .), the following groups:

V i

= m i m

. .m ( i ) = m i i m m ,V i

= m i m

. .m (i ) = m i i m

m,

V i

= m i m

. .m ( i ) = m i i m

m,

(X1 .)

andV i

= m i m

. .m ( i ) = m i i m

m,

V i

= m i m

. .m ( i ) = m i i m

m,

V i

= m i m

. .m ( i ) = m i i m

m;

(Y 1 .)

results which may be thus summed up:

V = t H + .m ( + + )

1 m

( m . m + m . m + m . m )

+ 1 m

( m . m + m . m + m . m ),

(Z1 .)

and might have been otherwise deduced by our rule, from this other known transformationof T ,

T = 12 .m ( 2 + 2 + 2)

( m )2 + ( m )2 + ( m )2

2 m . (69.)

And to obtain, with any set of internal or relative marks of position, the two partial dif-ferential equations which the characteristic function V of relative motion must satisfy, and

25


28/65

which offer (as we shall nd) the chief means of discovering its form, namely, the equationsanalogous to those marked (F.) and (G.), we have only to eliminate the rates of increase of the marks of position of the system, which determine the nal and initial components of therelative velocities of its points, by the law of varying relative action, from the nal and initialexpressions of the law of relative living force; namely, from the following equations:

T = U + H , (50.)

andT 0 = U 0 + H . (70.)

The law of areas, or the property respecting rotation which was expressed by the partialdifferential equations (P.), will also always admit of being expressed in relative coordinates,and will assist in discovering the form of the characteristic function V ; by showing that thisfunction involves only such internal coordinates (in number 6 n 9) as do not alter by anycommon rotation of all points nal and initial, round the centre of gravity, or round anyother internal origin; that origin being treated as xed, and the quantity H as constant, indetermining the effects of this rotation. The general problem of dynamics, respecting themotions of a free system of n points attracting or repelling one another, is therefore reduced,in the last analysis, by the method of the present essay, to the research and differentiationof a function V , depending on 6 n 9 internal or relative coordinates, and on the quantityH , and satisfying a pair of partial differential equations of the rst order and second degree;in integrating which equations, we are to observe, that at the assumed origin of the motion,namely at the moment when t = 0, the nal or variable coordinates are equal to their initial

values, and the partial differential coefficient V H

vanishes; and, that at a moment innitely

little distant, the differential alterations of the coordinates have ratios connected with the

other partial differential coefficients of the characteristic function V , by the law of varyingrelative action. It may be here observed, that, although the consideration of the point, calledusually the centre of gravity, is very simply suggested by the process of the tenth number,yet this internal centre is even more simply indicated by our early corollaries from the lawof varying action; which show that the components of relative nal velocities, in any systemof attracting or repelling points, may be expressed by the differences of quantities of the

form 1m

V x

, 1m

V y

, 1m

V z

: and that therefore in calculating these relative velocities, it is

advantageous to introduce the nal sums mx , my , mz , and, for an analogous reason,the initial sums ma , mb , mc , among the marks of the extreme positions of the system,in the expression of the characteristic function V ; because, in differentiating that expressionfor the calculation of relative velocities, those sums may be treated as constant.

On Systems of two Points, in general; Characteristic Function of the motion of any Binary System.

13. To illustrate the foregoing principles, which extend to any free system of points,however numerous, attracting or repelling one another, let us now consider, in particular, asystem of two such points. For such a system, the known force-function U becomes, by (2.)

U = m 1m 2 f (r ), (71.)

26


29/65


30/65

as a function of those thirteen quantities x1 y1 z1 x2 y2 z2 a1 b1 c1 a2 b2 c2 H : and mightthen calculate the variation of this function,

V = V x 1

x 1 + V y1

y1 + V z1

z1 + V x 2

x 2 + V y2

y2 + V z2

z2

+ V a 1

a 1 + V b1

b1 + V c1

c1 + V a 2

a 2 + V b2

b2 + V c2

c2

+ V H

H.

(B2 .)

But the essence of our method consists in forming previously the expression of this variation by our law of varying action , namely,

V = m 1(x 1 x 1 a 1 a 1 + y1 y1 b1 b1 + z1 z1 c1 c1)+ m 2(x 2 x 2

a 2 a 2 + y2 y2

b2 b2 + z2 z2

c2 c2)

+ t H ;(C 2 .)

and in considering V as a characteristic function of the motion , from the form of which maybe deduced all the intermediate and all the nal integrals of the known differential equations,by resolving the expression (C 2 .) into the following separate groups, (included in (C.) and(D.),)

V x 1

= m 1x 1 , V

y1= m 1y1 ,

V z1

= m 1z1 ,

V x 2

= m 2x 2 , V

y2= m 2y2 ,

V z2

= m 2z2 ;(D2 .)

andV a 1

= m 1a 1 , V

b1= m 1 b1 ,

V c1

= m 1 c1 ,V a 2

= m 2a 2 , V

b2= m 2 b2 ,

V c2

= m 2 c2 ;(E 2 .)

besides this other equation, which had occurred before,

V H

= t. (E .)

By this new method, the difficulty of integrating the six known equations of motion of the second order (74.) is reduced to the search and differentiation of a single function V ; andto nd the form of this function, we are to employ the following pair of partial differentialequations of the rst order:

12m 1

V x 1

2

+V y1

2

+V z1

2

+ 12m 2

V x 2

2

+V y2

2

+V z2

2

= m 1m 2 f (r ) + H,

(F 2 .)

28


31/65

12m 1

V a 1

2

+V b1

2

+V c1

2

+ 12m 2

V a 2

2

+V b2

2

+V c2

2

= m 1m 2 f (r 0) + H,

(G 2 .)

combined with some simple considerations. And it easily results from the principles alreadylaid down, that the integral of this pair of equations, adapted to the present question, is

V = (x a )2 + ( y b )2 + ( z c )2 . 2H (m 1 + m 2)+

m1m 2m 1 + m 2

h + r

r 0 dr ;

(H2 .)

in which x y z a b c denote the coordinates, nal and initial, of the centre of gravityof the system,

x = m1x 1 + m 2x 2

m 1 + m 2, y =

m1y1 + m 2y2m 1 + m 2

, z = m1z1 + m 2z2

m 1 + m 2,

a = m1a 1 + m 2a 2m 1 + m 2

, b = m1b1 + m 2b2m 1 + m 2

, c = m1c1 + m 2c2m 1 + m 2

,(78.)

and is the angle between the nal and initial distances r , r0 : we have also put for abridge-ment

= 2(m 1 + m 2) f (r ) + H m 1m 2 h2r 2 , (79.)the upper or the lower sign to be used, according as the distance r is increasing or decreasing,and have introduced three auxiliary quantities h , H , H , to be determined by this condition,

0 = + r

r 0

h dr, (I

2.)

combined with the two following,

m 1m 2m 1 + m 2

r

r 0

H

dr = (x a )2 + ( y b )2 + ( z c )2 . m 1 + m 2

2H ,

H + H = H ;(K 2 .)

which auxiliary quantities, although in one view they are functions of the twelve extremecoordinates, are yet to be treated as constant in calulating the three denite integrals, orlimits of sums of numerous small elements,

r

r 0 dr,

r

r 0

h

dr, r

r 0

H

dr.

The form (H 2 .), for the characteristic function of a binary system , may be regarded asa central or radical relation, which includes the whole theory of the motion of such a system;so that all the details of this motion may be deduced from it by the application of our generalmethod. But because the theory of binary systems has been brought to great perfectionalready, by the labours of former writers, it may suffice to give briey here a few instancesof such deduction.

29


32/65

14. The form (H 2 .), for the characteristic function of a binary system, involves explicitly,when is changed to its value (79.), the twelve quantities x y z a b c r r 0 h H H ,(besides the masses m 1 m2 which are always considered as given;) its variation may thereforebe thus expressed:

V = V x

x + V y

y + V z

z + V a

a + V b

b + V c

c

+ V r

r + V r 0

r 0 + V

+ V H

H + V H

H .(L2 .)

In this expression, if we put for abridgement

= 2H (m 1 + m 2 )(x a )2 + ( y b )2 + ( z c )2 , (80.)we shall have

V x

= (x a ), V

y= (y b ),

V z

= (z c ),V a

= (a x ), V

b= (b y ),

V c

= (c z );(M2 .)

and if we put

0 = 2(m 1 + m 2) f (r 0) + H m 1m 2 h2r 20 , (81.)the sign of the radical being determined by the same rule as that of , we shall have

V r

= m1 m 2m 1 + m 2

, V

r 0= m 1m 2 0

m 1 + m 2,

V

= m 1m 2hm 1 + m 2

; (N2 .)

besides, by the equations of condition (I 2 .), (K 2 .), we have

V h

= 0 , (O2 .)

and V H

= V H

= r

r 0

dr

, H + H = H. (P 2 .)

The expression (L 2 .) may therefore be thus transformed:

V = {(x a )(x a ) + ( y b )(y b ) + ( z c )(z c )}+

m1m 2m 1 + m 2

( r 0 r 0 + h ) + r

r 0

r

. H ;(Q 2 .)

30


33/65


34/65

three conditions (I 2 .) (T 2 .) (U2 .). Thus, if we observe that the distances r , r0 , and theincluded angle , depend only on relative coordinates, which may be thus denoted,

x 1 x2 = , y1 y2 = , z1 z2 = ,a

1 a

2 = , b

1 b

2 = , c

1 c

2 = ,

(82.)

we obtain by easy combinations the three following intermediate integrals for the centre of gravity of the system:

x t = x a , y t = y b , z t = z c , (83.)and the three following nal integrals,

a t = x a , b t = y b , c t = z c , (84.)expressing the well-known law of the rectilinear and uniform motion of that centre. Weobtain also the three following intermediate integrals for the relative motion of one point of the system about the other:

= r

+ h

,

= r

+ h

,

= r

+ h

,

(85.)

and the three following nal integrals,

= 0r 0 h

,

= 0r 0 h

,

= 0r 0 h

;

(86.)

in which the auxiliary quantities h , H are to be determined by (I 2 .), (T 2 .), and in which the

dependence of r , r0 , , on , , , , , , is expressed by the following equations:

r = 2 + 2 + 2 , r 0 = 2 + 2 + 2 ,rr 0 cos = + + . (87.)If then we put, for abridgement,

A = r

+ h

r 2 tan, B =

hrr 0 sin

, C = 0r 0

+ h

r 20 tan, (88.)

32


35/65

we shall have these three intermediate integrals,

= A B, = A B, = A B , (89.)and these three nal integrals,

= B C, = B C, = B C, (90.)of the equations of relative motion. These integrals give,

= = B ( ), = = B ( ), = = B ( ),

(91.)

and ( ) + ( ) + ( ) = 0; (92 .)

they contain therefore the known law of equable description of areas, and the law of a planerelative orbit. If we take for simplicity this plane for the plane , the quantities will vanish; and we may put,

= r cos , = r sin , = 0 , = r 0 cos 0 , = r 0 sin 0 , = 0 ,

(93.)

and

= r cos r sin , = r sin + r cos , = 0 , = r 0 cos0 0r 0 sin 0 , = r 0 sin 0 + 0r 0 cos 0 , = 0 , (94.)

the angles 0 being counted from some xed line in the plane, and being such that theirdifference

0 = . (95.)These values give

= r 2 , = r 20 0 , = rr 0 sin , (96.)

and therefore, by (88.) and (91.),r 2 = r 20 0 = h ; (97.)

the quantity 12 h is therefore the constant areal velocity in the relative motion of the system;a result which is easily seen to be independent of the directions of the three rectangularcoordinates. The same values (93.), (94.), give

cos + sin = r, cos + sin = r , cos + sin = r 0 cos , cos 0 + sin 0 = r 0 , cos 0 + sin 0 = r 0 , cos 0 + sin 0 = r cos ,

(98.)

33


36/65

and therefore, by the intermediate and nal integrals, (89.), (90.),

r = , r 0 = 0 ; (99.)

results which evidently agree with the condition (T 2 .), and which give by (79.) and (81.), forall directions of coordinates,

r 2 + h2

r 2 2(m 1 + m 2)f (r ) = r 20 + h2

r 20 2(m 1 + m 2 )f (r 0) = 2 H 1m 1

+ 1m 2

; (100.)

the other auxiliary quantity H is therefore also a constant, independent of the time, and

enters as such into the constant part in the expression for r 2 + h2

r 2 the square of the

relative velocity. The equation of condition (I 2 .), connecting these two constants h, H , withthe extreme lengths of the radius vector r , and with the angle described by this radiusin revolving from its initial to its nal direction, is the equation of the plane relative orbit;and the other equation of condition (T 2 .), connecting the same two constants with the sameextreme distances and with the time, gives the law of the velocity of mutual approach orrecess.

We may remark that the part V of the whole characteristic function V , which representsthe relative action and determines the relative motion in the system, namely,

V = m1m 2m 1 + m 2

h + r

r 0 dr , (V2 .)

may be put, by (I 2 .), under the form

V = m1m 2m 1 + m 2

r

r 0 h

h

dr, (W2 .)

or nally, by (79.)V = 2

r

r 0

m 1m 2f (r ) + H

dr ; (X2 .)

the condition (I 2 .) may also itself be transformed, by (79.), as follows:

= h r

r 0

drr 2

: (Y2 .)

results which all admit of easy verications. The partial differential equations connected withthe law of relative living force, which the characteristic function V of relative motion mustsatisfy, may be put under the following forms:

V r

2+ 1

r 2V

2= 2m 1m 2

m 1 + m 2(U + H ),

V r 0

2

+ 1r 20

V

2

= 2m 1m 2m 1 + m 2

(U 0 + H );(Z2 .)

and if the rst of the equations of this pair have its variation taken with respect to r and, attention being paid to the dynamical meanings of the coefficients of the characteristic

function, it will conduct (as in former instances) to the known differential equations of motionof the second order.

34


37/65

On the undisturbed Motion of a Planet or Comet about the Sun: Dependence of the Charac-teristic Function of such Motion, on the chord and the sum of the Radii.

15. To particularize still further, let

f (r ) = 1r , (101.)

that is, let us consider a binary system, such as a planet or comet and the sun, with theNewtonian law of attraction; and let us put, for abridgement,

m 1 + m 2 = , h2

= p, m 1 m 2

2H = a . (102.)

The characteristic function V of relative motion may now be expressed as follows

V = m1 m 2

p +

r

r 0 2r

1

a

pr 2

.dr ; (A3 .)

in which p is to be considered as a function of the extreme radii vectores r , r0 , and of theirincluded angle , involving also the quantity a, or the connected quantity H , and determinedby the condition

= r

r 0

drr 2 2rp 1a p 1r 2

(B3 .)

that is, by the derivative of the formula (A 3 .), taken with respect to p; the upper sign beingtaken in each expression when the distance r is increasing, and the lower sign when thatdistance is diminishing, and the quantity p being treated as constant in calculating the twodenite integrals. It results from the foregoing remarks, that this quantity p is constant alsoin the sense of being independent of the time, so as not to vary in the course of the motion;and that the condition (B 3 .), connecting this constant with r r 0 a, is the equation of theplane relative orbit; which is therefore (as it has long been known to be) an ellipse, hyperbola,or parabola, according as the constant a is positive, negative, or zero, the origin of r beingalways a focus of the curve, and p being the semiparameter. It results also, that the time of motion may be thus expressed:

t = V H

= 2a2

m 1m 2

V a

, (C3 .)

and therefore thus:t =

r

r 0

dr

2r a pr 2; (D3 .)

which latter is a known expression. Conning ourselves at present to the case a > 0, andintroducing the known auxiliary quantities called excentricity and excentric anomaly, namely,

e = 1 pa , (103.)35


38/65

and = cos 1

a rae

, (104.)

which give

2ar r2

pa = a e sin , (105.) being considered as continually increasing with the time; and therefore, as is well known,r = a(1 e cos ), r 0 = a(1 e cos 0),

= 2 tan 1 1 + e1 e tan 2 2tan

1 1 + e1 e tan 02

,(106.)

and

t =

a3

. (

0 e sin + e sin

0); (107.)

we nd that this expression for the characteristic function of relative motion,

V = m1m 2

r

r 0

2r

1a

dr

2r 1a pr 2, (E3 .)

deduced from (A 3 .) and (B 3 .), may be transformed as follows:

V = m 1m 2 a ( 0 + e sin e sin 0) : (F3 .)in which the excentricity e, and the nal and initial excentric anomalies , 0 , are to beconsidered as functions of the nal and initial radii r , r0 , and of the included angle ,determined by the equations (106.). The expression (F 3 .) may be thus written:

V = 2m 1m 2 a ( + e sin ), (G3 .)if we put, for abridgement,

= 0

2 , e = e cos

+ 02

; (108.)

for the complete determination of the characteristic function of the present relative motion,it remains therefore to determine the two variables and e , as functions of r r 0 , or of some other set of quantities which mark the shape and size of the plane triangle bounded bythe nal and initial elliptic radii vectores and by the elliptic chord.

36


39/65

For this purpose it is convenient to introduce this elliptic chord itself, which we shall call

, so that 2 = r 2 + r 20 2rr 0 cos ; (109.)because this chord may be expressed as a function of the two variables , e , (involving also

the mean distance a ,) as follows. The value (106.) for the angle , that is, by (95.), for 0 , gives 2tan

1 1 + e1 e tan 2

= 0 2tan 1 1 + e1 e tan

02

= , (110.)

being a new constant independent of the time, namely, one of the values of the polar angle, which correspond to the minimum of radius vector; and therefore, by (106.),

r cos( ) = a(cos e), r sin( ) = a 1 e2 sin ,r 0 cos(0

) = a(cos 0

e), r 0 sin(0

) = a

1

e2 sin 0 ;

(111.)

expressions which give the following value for the square of the elliptic chord:

2 = {r cos( ) r 0 cos(0 )}2 + {r sin( ) r 0 sin(0 )}2= a 2{(cos cos 0)2 + (1 e2)(sin sin 0)2}= 4a 2 sin 2 sin

+ 02

2

+ (1 e2) cos + 0

2

2

= 4a 2(1 e2 ) sin 2 :

(112 .)

we may also consider as having the same sign with sin , if we consider it as alternately

positive and negative, in the successive elliptic periods or revolutions, beginning with theinitial position.

Besides, if we denote by the sum of the two elliptic radii vectores, nal and initial, sothat

= r + r 0 , (113.)

we shall have, with our present abridgements,

= 2a(1 e cos ); (114.)the variables e are therefore functions of , , a, and consequently the characteristicfunction V is itself a function of those three quantities. We may therefore put

V = m1m 2wm 1 + m 2

, (H3 .)

w being a function of , , a, of which the form is to be determined by eliminating ebetween the three equations,

w = 2 a( + e sin ), = 2a(1 e cos ), = 2a(1 e2 )

12 sin ;

(I3 .)

37


40/65

and we may consider this new function w as itself a characteristic function of elliptic motion;the law of its variation being expressed as follows, in the notation of the present essay:

w = + + + t a

2a2 . (K3 .)

In this expression are the relative coordinates of the point m1 , at the time t, referredto the other attracting point m2 as an origin, and to any three rectangular axes; aretheir rates of increase, or the three rectangular components of nal relative velocity; are the initial values, or values at the time zero, of these relative coordinates andcomponents of relative velocity; a is a quantity independent of the time, namely, the meandistance of the two points m 1 , m2 ; and is the sum of their masses. And all the propertiesof the undisturbed elliptic motion of a planet or comet about the sun may be deduced in anew way, from the simplied characteristic function w, by comparing its variation (K 3 .) withthe following other form,

w = w

+ w

+ w a

a; (L3 .)

in which we are to observe that = 2 + 2 + 2 + 2 + 2 + 2 , = ( )2 + ( )2 + ( )2 .

(M3 .)

By this comparison we are brought back to the general integral equations of the relativemotion of a binary system, (89.) and (90.); but we have now the following particular valuesfor the coefficients A, B , C :

A = 1r

w

+ 1

w

, B = 1

w

, C = 1r 0

w

+ 1

w

; (N3 .)

and with respect to the three partial differential coefficients, w

,

w

,

w

a , we have the

following relation between them:

aw a

+ w

+ w

= w2

, (O3 .)

the function w being homogeneous of the dimension 12 with respect to the three quantities a, , ; we have also, by (I3 .),

w

= a . sin e cos , w

= a . 1 e

2

cos e, (P 3 .)

and thereforew

w

= 2

2 2,

w

2

+w

2

+

a =

4

2 2, (Q3 .)

from which may be deduced the following remarkable expressions:w

+ w

2

= 4 +

a

,

w

w

2

= 4

a

.

(R 3 .)

These expressions will be found to be important in the application of the present method tothe theory of elliptic motion.

38


41/65

16. We shall not enter, on this occasion, into any details of such application; but we mayremark, that the circumstance of the characteristic function involving only the elliptic chordand the sum of the extreme radii, (besides the mean distance and the sum of the masses,)affords, by our general method, a new proof of the well-known theorem that the elliptic timealso depends on the same chord and sum of radii; and gives a new expression for the law of this dependence, namely,

t = 2a2

w a

. (S3 .)

We may remark also, that the same form of the characteristic function of elliptic motionconducts, by our general method, to the following curious, but not novel property, of theellipse, that if any two tangents be drawn to such a curve, from any common point outside,these tangents subtend equal angles at one focus; they subtend also equal angles at the other.Reciprocally, if any plane curve possess this property, when referred to a xed point in itsown plane, which may be taken as the origin of polar coordinates r , , the curve must satisfythe following equation in mixed differences:

cotan 2

. 1r

= ( + 2) dd

1r

, (115.)

which may be brought to the following form,

dd

+ d3

d31r

= 0 , (116.)

and therefore gives, by integration,

r = p

1 + e cos( ); (117.)

the curve is, consequently, a conic section, and the xed point is one of its foci.The properties of parabolic are included as limiting cases in those of elliptic motion, and

may be deduced from them by making

H = 0 , or a = ; (118.)and therefore the characteristic function w and the time t, in parabolic as well as in ellipticmotion, are functions of the chord and of the sum of the radii. By thus making a innite inthe foregoing expressions, we nd, for parabolic motion, the partial differential equations

w

+ w

2

= 4 +

,w

w

2

= 4

; (T3 .)

and in fact the parabolic form of the simplied characteristic function w may easily be shownto be

w = 2 ( + ), (U3 .)

39


42/65

being, as before, the chord, and the sum of the radii; while the analogous limit of theexpression (S 3 .), for the time, is

t = 16 {( + )

32 ( )

32 }: (V3 .)

which latter is a known expression.The formul (K 3 .) and (L 3 .), to the comparison of which we have reduced the study of

elliptic motion, extend to hyperbolic motion also; and in any binary system, with Newton slaw of attraction, the simplied characteristic function w may be expressed by the deniteintegral

w =

+ 4a . d, (W3 .)this function w being still connected with the relative action V by the equation (H 3 .); whilethe time t , which may always be deduced from this function, by the law of varying action, is

represented by this other connected integral,

t = 14

+

4a

12

d : (X3 .)

provided that, within the extent of these integrations, the radical does not vanish nor becomeinnite. When this condition is not satised, we may still express the simplied characteristicfunction w, and the time t, by the following analogous integrals:

w =

2

a

d , (Y3 .)

and

t =

2

a

12

d , (Z3 .)

in which we have put for abridgement

= +

2 , =

2

, (119.)

and in which it is easy to determine the signs of the radicals. But to treat fully of these varioustransformations would carry us too far at present, for it is time to consider the properties of systems with more points than two.

On Systems of three Points, in general; and on their Characteristic Functions.

17. For any system of three points, the known differential equations of motion of the2nd order are included in the following formula:

m 1(x 1 x1 + y1 y1 + z1 z1 ) + m 2(x 2 x2 + y2 y2 + z2 z2)+ m 3(x 3 x3 + y3 y3 + z3 z3) = U,

(120.)

40


43/65

the known force-function U having the form

U = m 1m 2 f (1 ,2) + m 1m 3 f (1 ,3) + m 2m 3 f (2 ,3) , (121.)

in which f (1 ,2) , f (1 ,3) , f (2 ,3) , are functions respectively of the three following mutual distancesof the points of the system:

r (1 ,2) = (x 1 x 2)2 + ( y1 y2)2 (z1 z2)2 ,r (1 ,3) = (x 1 x 3)2 + ( y1 y3)2 (z1 z3)2 ,r (2 ,3) = (x 2 x 3)2 + ( y2 y3)2 (z2 z3)2 :(122.)

the known differential equations of motion are therefore, separately, for the point m1 ,

x1 = m 2f (1 ,2)

x1

+ m 3f (1 ,3)

x1

,

y1 = m 2f (1 ,2)

y1+ m 3

f (1 ,3)

y1,

z1 = m 2f (1 ,2)

z1+ m 3

f (1 ,3)

z1,

(123.)

with six other analogous equations for the points m 2 and m 3 : x1 , &c., denoting the componentaccelerations of the three points m1 m2 m3 , or the second differential coefficients of theircoordinates, taken with respect to the time. To integrate these equation is to assign, bytheir means, nine relations between the time t , the three masses m1 m2 m3 , the nine varying

coordinates x1 y1 z1 x2 y2 z2 x3 y3 z3 , and their nine initial values and nine initial rates of increase, which may be thus denoted, a1 b1 c1 a2 b2 c2 a3 b3 c3 a1 b1 c1 a2 b2 c2 a3 b3 c3 .The known intermediate integral containing the law of living force, namely,

12 m 1(x

21 + y

21 + z

21 ) + 12 m 2(x

22 + y

22 + z

22 ) + 12 m 3(x

23 + y

23 + z

23 )

= m 1m 2 f (1 ,2) + m 1m 3 f (1 ,3) + m 2m 3 f (2 ,3) + H, (124.)

gives the following initial relation:

12 m 1(a

21 + b

21 + c

21 ) + 12 m 2(a

22 + b

22 + c

22 ) + 12 m 3(a

23 + b

23 + c

23 )

= m 1m 2 f (1 ,2)0 + m 1m 3 f

(1 ,3)0 + m 2m 3f

(2 ,3)0 + H,

(125.)

in which f (1 ,2)0 , f (1 ,3)0 , f

(2 ,3)0 are composed of the initial coordinates, in the same manner

as f (1 ,2) f (1 ,3) f (2 ,3) are composed of the nal coordinates. If then we knew the nine nalintegrals of the equations of motion of this ternary system, and combined them with theinitial form (125.) of the law of living force, we should have ten relations to determine theten quantities t a 1 b1 c1 a2 b2 c2 a3 b3 c3 , namely, the time and the nine initial componentsof the velocities of the three points, as functions of the nine nal and nine initial coordinates,and of the quantity H , involving also the masses; we could therefore determine whatever else

41


44/65

depends on the manner and time of motion of the system, from its initial to its nal position,as a function of the same extreme coordinates, and of H . In particular, we could determinethe action V , or the accumulated living force of the system, namely,

V = m 1

t

0(x 2

1 + y 2

1 + z 2

1 ) dt + m 2

t

0(x 2

2 + y 2

2 + z 2

2 ) dt + m 3

t

0(x 2

3 + y 2

3 + z 2

3 ) dt, (A4 .)

as a function of these nineteen quantities, x1 y1 z1 x 2 y2 z2 x 3 y3 z3 a 1 b1 c1 a 2 b2 c2 a 3 b3 c3 H ;and might then calculate the variation of this function,

V = V x 1

x 1 + V y1

y1 + V z1

z1 + V a 1

a 1 + V b1

b1 + V c1

c1

+ V x 2

x 2 + V y2

y2 + V z2

z2 + V a 2

a 2 + V b2

b2 + V c2

c2

+ V x 3

x 3 + V y3

y3 + V z3

z3 + V a 3

a 3 + V b3

b3 + V c3

c3

+ V H

H.

(B4 .)

But the law of varying action gives, previously , the following expression for this variation:

V = m 1(x 1 x 1 a 1 a 1 + y1 y1 b1 b1 + z1 z1 c1 c1)+ m 2(x 2 x 2 a 2 a 2 + y2 y2 b2 b2 + z2 z2 c2 c2)+ m 3(x 3 x 3 a 3 a 3 + y3 y3 b3 b3 + z3 z3 c3 c3)+ t H ;

(C 4 .)

and shows, therefore, that the research of all the intermediate and all the nal integral equa-tions, of motion of the system, may be reduced, reciprocally, to the search and differentiationof this one characteristic function V ; because if we knew this one function, we should havethe nine intermediate integrals of the known differential equations, under the forms

V x 1

= m 1x 1 , V

y1= m 1y1 ,

V z1

= m 1z1 ,

V x 2

= m 2x 2 , V

y2= m 2y2 ,

V z2

= m 2z2 ,

V

x 3= m 3x 3 ,

V

y3= m 3y3 ,

V

z3= m 3z3 ,

(D4 .)

and the nine nal integrals under the forms

V a 1

= m 1a 1 , V

b1= m 1 b1 ,

V c1

= m 1 c1 ,V a 2

= m 2a 2 , V

b2= m 2 b2 ,

V c2

= m 2 c2 ,V a 3

= m 3a 3 , V

b3= m 3 b3 ,

V c3

= m 3 c3 ,(E 4 .)

42


45/65

the auxiliary constant H being to be eliminated, and the time t introduced, by this otherequation, which has often occurred in this essay,

t = V H

. (E .)

The same law of varying action suggests also a method of investigating the form of thischaracteristic function V , not requiring the previous integration of the known equations of motion; namely, the integration of a pair of partial differential equations connected with thelaw of living force; which are,

12m 1

V x 1

2

+V y1

2

+V z1

2

+ 12m 2

V x 2

2

+V y2

2

+V z2

2

+ 12m

3

V x

3

2

+V y

3

2

+V z

3

2

= m 1m 2 f (1 ,2) + m 1m 3 f (1 ,3) + m 2m 3 f (2 ,3) + H,

(F 4 .)

and

12m 1

V a 1

2

+V b1

2

+V c1

2

+ 12m 2

V a 2

2

+V b2

2

+V c2

2

+ 12m 3

V a 3

2

+V b3

2

+V c3

2

= m 1m 2 f (1 ,2)0 + m 1m 3 f (1 ,3)0 + m 2m 3 f (2 ,3)0 + H.

(G 4 .)

And to diminish the difficulty of thus determining the function V , which depends on 18coordinates, we may separate it, by principles already explained, into a part V dependingonly on the motion of the centre of gravity of the system, and determined by the formula(H1 .), and another part V , depending only on the relative motions of the points of thesystem about this internal centre, and equal to the accumulated living force, connected withthis relative motion only. In this manner the difficulty is reduced to determining the relativeaction V ; and if we introduce the relative coordinates

1 = x 1 x3 , 1 = y1 y3 , 1 = z1 z3 , 2 = x 2 x3 , 2 = y2 y3 , 2 = z2 z3 , (126.)

and 1 = a 1 a 3 , 1 = b1 b3 , 1 = c1 c3 , 2 = a 2 a 3 , 2 = b2 b3 , 2 = c2 c3 ,

(127.)

we easily nd, by the principles of the tenth and following numbers, that the function V may be considered as depending only on these relative coordinates and on a quantity H

43


46/65


47/65


48/65


49/65

0 = s21s 23 + s22s 24 + s 23s 25 + s 24s 21 + s 25s 22+ s 21d23 + s22d24 + s 23d25 + s 24d21 + s 25d22+ d21d22 + d22d23 + d23d24 + d24d25 + d25d212s 21s 3s4 2s 22s4s 5 2s 23s 5s 1 2s24s 1s 2 2s 25s 2s32s 21s 3d3 2s 22s4d4 2s 23s 5d5 2s24s 1d1 2s 25s 2d22s 21s 4d3 2s 21s5d4 2s 21s 1d5 2s21s 2d1 2s 21s 3d22s 1d2d23 2s 2d3d24 2s 3d4d25 2s4d5d21 2s 5d1d222s 1d23d4 2s 2d24d5 2s 3d25d1 2s4d21d2 2s 5d22d32d1d22d3 2d2d23d4 2d3d24d5 2d4d25d1 2d5d21d24s 1s 3s4d3 4s 2s4s 5d4 4s 3s 5s 1d5 4s4s 1s 2d1 4s 5s 2s3d24s 1d2d3d4 4s 2d3d4d5 4s 3d4d5d1 4s4d5d1d2 4s 5d1d2d32s 1s 2s3d4 2s 2s3s 4d5 2s 3s 4s 5d1 2s4s 5s 1d2 2s 5s 1s2d32s 1s 3d1d2 2s 2s4d2d3 2s 3s 5d3d4 2s4s 1d4d5 2s 5s 2d5d12s 1d1d3d5 2s 2d2d4d1 2s 3d3d5d2 2s4d4d1d3 2s 5d5d2d4+2 s 1s 2s3s 4 +2 s 2s3s 4s 5 +2 s 3s 4s 5s 1 +2 s4s 5s 1s2 +2 s 5s 1s2s3+2 s 1s 2s4d3 +2 s

on a general method in dynamics - william rowan hamilton

Documents