ON THE APPLICATION TO DYNAMICS OF A GENERAL MATHEMATICAL METHOD

PREVIOUSLY APPLIED TO OPTICS

By

William Rowan Hamilton

(British Association Report, Edinburgh 1834, pp. 513–518.)

Edited by David R. Wilkins On the Application to Dynamics of a General Mathematical Method previously applied to Optics. By W. R. Hamilton, M.R.I.A., Astronomer Royal for Ireland.

[Report of the Fourth Meeting of the British Association for the Advancement of Science; held at Edinburgh in 1834. (John Murray, London, 1835), pp. 513–518.]

The method is founded on a combination of the principles of variations with those of partial differentials, and may suggest to analysts a separate branch of algebra, which may be called, perhaps, the Calculus of Principal Functions; because, in all the chief applications of algebra to physics, and in a very extensive class of purely mathematical questions, it reduces the determination of many mutually connected functions to the search and study of one principal or central relation. In applying this method to Dynamics, (having previously applied it to Optics,) Professor Hamilton has discovered the existence of a principal function, which, if its form were fully known, would give, by its partial differential coefficients, all the intermediate and all the final integrals of the known equations of motion.

Professor Hamilton is of opinion that the mathematical explanation of all the phænomena of matter distinct from the phænomena of life, will ultimately be found to depend on the properties of systems of attracting and repelling points. And he thinks that those who do not adopt this opinion in all its extent, must yet admit the properties of such systems to be more highly important in the present state of science, than any other part of the application of mathematics to physics. He therefore accounts it the capital problem of Dynamics “to determine the 3n rectangular coordinates, or other marks of position, of a free system of n attracting or repelling points as functions of the time,” involving also 6n initial constants, which depend on the initial circumstances of the motion, and involving besides, n other constants called the masses, which measure, for a standard distance, the attractive or repulsive energies.

Denoting these n masses by m_{1} m_{2} … m_{n} and their 3n rectangular coordinates by x_{1} y_{1} z_{1} … x_{n} y_{n} z_{n}, and also the 3n component accelerations, or second differential coefficients of these coordinates, taken with respect to the time, by x′′_{1} y′′_{1} z′′_{1} … x′′_{n} y′′_{n} z′′_{n}, he adopts Lagrange’s statement of this problem; namely, a formula of the following kind,

Σ.m(x′′ δx + y′′ δy + z′′ δz) = δU,  (1.)

in which U is the sum of the products of the masses, taken two by two, and then multiplied by each other and by certain functions of their mutual distances, such that their first derived functions express the laws of their mutual repulsion, being negative in the case of attraction. Thus, for the solar system, each product of two masses is to be multiplied by the reciprocal of their distance and the results are to be added in order to compose the function U.

Mr. Hamilton next multiplies this formula of Lagrange by the element of the time dt, and integrates from the time 0 to the time t, considering the time and its element as not subject at present to the variation δ. He denotes the initial values, or values at the time 0, of the coordinates x y z, and of their first differential coefficients x′ y′ z′, by a b c and a′ b′ c′; and thus he obtains, from Lagrange’s formula (1.), this other important formula,

Σ.m(x′ δx − a′ δa + y′ δy − b′ δb + z′ δz − c′ δc) = δS,  (2.)

S being the definite integral

S = ∫_{0}^{t} \{U + Σ.\frac{m}{2}(x^{′2} + y^{′2} + z^{′2})\} dt.  (3.)

If the known equations of motion, of the forms

m_{i}x′′_{i} = \frac{δU}{δx_{i}} , m_{i}y′′_{i} = \frac{δU}{δy_{i}} , m_{i}z′′_{i} = \frac{δU}{δz_{i}} ,  (4.)

had been completely integrated, they would give the 3n coordinates x y z, and therefore also S, as a function of the time t, the masses m_{1} … m_{n} and the 6n initial constants a b c a′ b′ c′; so that, by eliminating the 3n initial components of velocities a′ b′ c′ we should in general obtain a relation between the 7n + 2 quantities S, t, m, x, y, z, a, b, c, which would give S as a function of the time, the masses, and the final and initial coordinates. We do not yet know the form of this last function, but we know its variation (2.), taken with respect to the 6n coordinates; and on account of the independence of their 6n variations, we can resolve this expression (2.) into two groups, containing each 3n equations: namely

\frac{δS}{δx_{i}} = m_{i}x′_{i}, \frac{δS}{δy_{i}} = m_{i}y′_{i}, \frac{δS}{δz_{i}} = m_{i}z′_{i},  (5.)

and

\frac{δS}{δa_{i}} = −m_{i}a′_{i}, \frac{δS}{δb_{i}} = −m_{i}b′_{i}, \frac{δS}{δc_{i}} = −m_{i}c′_{i};  (6.)

the first members being partial differential coefficients of the function S, which Mr. Hamilton calls the Principal Function of motion of the attracting or repelling system. He thinks that if analysts had perceived this principal function S, and these groups of equations (5.) and (6.), they must have preceived their importance. For the group (5.) expresses the 3n intermediate integrals of the known equations of motion (4.) under the form of 3n relations between the time t, the masses m, the varying coordinates x,y, z, the varying components of velocities x′ y′ z′, and the 3n initial constants a b c; while the group (6.) expresses the 3n final integrals of the same known differential equations, as 3n relations, with 6n initial and arbitrary constants a b c a′ b′ c′, between the time, the masses, and the 3n varying coordinates. These 3n intermediate and 3n final integrals, it was the problem of dynamics to discover. Mathematicians had found seven intermediate, and none of the final integrals.

Professor Hamilton’s solution of this long celebrated problem contains, indeed, one unknown function, namely, the principal function S, to the search and study of which he has reduced mathematical dynamics. This function must not be confounded with that so beautifully conceived by Lagrange for the more simple and elegant expression of the known differential equations. Lagrange’s function states, Mr. Hamilton’s function would solve the problem.

The one serves to form the differential equations of motion, the other would give their integrals. To assist in pursuing this new track, and in discovering the form of this new function, Mr. Hamilton remarks that it must satisfy the following partial differential equation of the first order and second degree, (the time being now made to vary,)

\frac{δS}{δt} + Σ.\frac{1}{2m} \{(\frac{δS}{δx} )^{2} + (\frac{δS}{δy} )^{2} + (\frac{δS}{δz} )^{2}\} = U;  (7.)

which may rigorously be thus transformed, by the help of the equations (5.),

S = S_{1} + ∫_{0}^{t} (U − \frac{δS_{1}}{δt} − Σ.\frac{1}{2m} \{(\frac{δS_{1}}{δx} )^{2} + (\frac{δS_{1}}{δy} )^{2} + (\frac{δS_{1}}{δz} )^{2}\}) dt

+ ∫_{0}^{t} Σ.\frac{1}{2m} \{(\frac{δS}{δx} − \frac{δS_{1}}{δx} )^{2} + (\frac{δS}{δy} − \frac{δS_{1}}{δy} )^{2} + (\frac{δS}{δz} − \frac{δS_{1}}{δz} )^{2}\} dt,  (8.)

S_{1} being any arbitrary function of the same quantities t, m, x, y, z, a, b, c, supposed only to vanish (like S) at the origin of time. If this arbitrary function S_{1} be so chosen as to be an approximate value of the sought function S, (and it is always easy so to choose it,) then the two definite integrals in the formula (8.) are small, but the second is in general much smaller than the first; it may, therefore, be neglected in passing to a second approximation, and in calculating the first definite integral, the following appproximate forms of the equations (6.) may be used,

\frac{δS_{1}}{δa} = −ma′, \frac{δS_{1}}{δb} = −mb′, \frac{δS_{1}}{δc} = −mc′.  (9.)

In this manner, a first approximation may be successively and indefinitely corrected. And for the practical perfection of the method nothing further seems to be required, except to make this process of correction more easy and rapid in its applications.

Professor Hamilton has written two Essays on this new method in Dynamics, and one of them is already printed in the second part of the Philosophical Transactions (of London) for 1834. The method did not at first present itself to him under quite so simple a form. He used at first a Characteristic Function V, more closely analogous to that optical function which he had discovered, and denoted by the same letter, in his Theory of Systems of Rays. In both optics and dynamics this function was the quantity called Action, considered as depending (chiefly) on the final and initial coordinates. But when this Action-Function was employed in dynamics, it involved an auxiliary quantity H, namely the known constant part in the expression of half the living force of a system; and many troublesome eliminations were required in consequence, which are avoided by the new form of the method.

Mr. Hamilton thinks it worth while, however, to point out briefly a new property of this constant H, which suggests a new manner of expressing the differential and integral equations of motion of an attracting or repelling system. It is often useful to express the 3n rectangular coordinates x_{1} y_{1} z_{1} … x_{n} y_{n} z_{n}, as functions of 3n other marks of position, which may be thus denoted, η_{1} η_{2} … η_{3n}; and if 3n other new variables ϖ_{1} ϖ_{2} … ϖ_{3n}, be introduced, and defined as follows,

ϖ_{i} = Σ.m (x′ \frac{δx}{δη_{i}} + y′ \frac{δy}{δη_{i}} + z′ \frac{δz}{δη_{i}} ) ,  (10.)

it is, in general, possible to express, reciprocally, the 6n variables x y z x′ y′ z′ as functions of these 6n new variables η ϖ; it is, therefore, possible to express, as such a function, the quantity

H = Σ.\frac{m}{2}(x^{′2} + y^{′2} + z^{′2}) − U  (11.)

under the form

H = F (ϖ_{1}, … ϖ_{3n}, η_{1}, … η_{3n}) − U(η_{1}, …, η_{3n}),  (12.)

in which the part F is rational, integer and homogeneous of the second dimension with respect to the variables ϖ. Now Mr. Hamilton has found that when the quantity H is expressed in this last way as a function of these 6n new variables, η ϖ, its variation may be put under this form,

δH = Σ(η′ δϖ − ϖ′ δη),  (13.)

η′ ϖ′ denoting the first differential coefficients of these new variables η ϖ, considered as functions of the time. The 3n differential equations of motion of the second order, (4.), between the rectangular coordinates and the time, for any attracting or repelling system, may therefore be generally transformed into twice that number of equations of the first order between these 6n variables and the time, of the forms

η′_{i} = \frac{δH}{δϖ_{i}} , ϖ′_{i} = −\frac{δH}{δη_{i}} .  (14.)

To integrate this system of equations is to assign, from them, 6n relations between the time t, the 6n variables η_{i} ϖ_{i}, and their 6n initial values which may be called e_{i} p_{i}. Mr. Hamilton resolves the problem, under this more general form, by the same principal function S as before, regarding it, however, as depending now on the new marks η e of final and initial positions of the various points of the system. For, putting in this new notation,

S = ∫_{0}^{t} (Σ.ϖ \frac{δH}{δϖ} − H) dt,  (15.)

and considering the time as given, he finds now the formula of variation

δS = Σ(ω δη − p δe),  (16.)

and therefore the 6n separate equations

ϖ_{i} = \frac{δS}{δη_{i}} , p_{i} = −\frac{δS}{δe_{i}} ,  (17.)

which are forms for the sought relations.

Professor Hamilton thinks that these two formulæ of variation, (13.) and (16.), namely

δH = Σ(η′ δϖ − ϖ′ δη),  (A.)

and

δS = Σ(ω δη − p δe),  (B.)

are worthy of attention, as expressing, under concise and simple forms, the one the differential and the other the integral equations of motion, of an attracting or repelling system. They may be extended to other problems of dynamics, besides this capital problem. The expression H can always easily be found, and the function S can be determined with indefinite accuracy by a method of successive approximation of the kind already explained.

The properties of his Principal Function are treated of more fully in his “Second Essay on a General Method in Dynamics*”; in which he has introduced several forms of a certain Function of Elements, connected with the Principal Function, and with each other, and adapted to questions of perturbation; and has shown that for the perturbations of a ternary or multiple system with any laws of attraction or repulsion and with one predominant mass, the differential equations of the varying elements of all the smaller masses may be expressed together, and as simply as in the usual way, by the coefficients of one disturbing function, (namely, the disturbing part of the whole expression H,) and may be integrated rigorously by a corollary of his general method.

* This essay will be found in the Philosophical Transactions for 1835.