to system of

ally in

particles,

of two of the particles. It serves to show how the influence of Application an infinitely small change in one of those positions, on the di- free muturection of the other particle passing through the other position, fluencing is related to the influence on the direction of the former particle passing through the former position produced by an infinitely small change in the latter position. A corresponding statement, and to gein terms of generalized co-ordinates, may of course be adapted system. to a system of rigid bodies or particles connected in any way. All such statements are included in the following very general proposition:

The rate of increase of any one component momentum, corresponding to any one of the co-ordinates, per unit of increase of any other co-ordinate, is equal to the rate of increase of the component momentum corresponding to the latter per unit increase or diminution of the former co-ordinate, according as the two coordinates chosen belong to one configuration of the system, or one of them belongs to the initial configuration and the other to the final.

Let and X be two out of the whole number of co-ordinates constituting the argument of the Hamiltonian characteristic function A; and έ, n the corresponding momentums. We have [§ 330 (18)]

dA
dy

dA

the upper or lower sign being used according as it is a final or
an initial co-ordinate that is concerned.

neralized

Hence

if one belongs to the initial configuration, and the other to the
final, which is the second proposition. The geometrical inter-
pretation of this statement for the case of a free particle, and two
co-ordinates both belonging to one position, its final position, for

Application to system of free mutually in

fluencing particles,

and to generalized system.

Slightly disturbed

instance, gives merely the proposition of § 332 above, for the
case of particles projected from one point, with equal velocities
in all directions; or, in other words, the case of the arbitrary
surface of that enunciation, being reduced to a point. To com-
plete the set of variational equations derived from § 330 we have
dt dn
which expresses another remarkable property of con-

dx dE

servative motion.

337. By the help of Lagrange's form of the equations of equilibrium. motion, § 318, we may now, as a preliminary to the consideration of stability of motion, investigate the motion of a system infinitely little disturbed from a position of equilibrium, and left free to move, the velocities of its parts being initially infinitely small. The resulting equations give the values of the independent co-ordinates at any future time, provided the displacements continue infinitely small; and the mathematical expressions for their values must of course show the nature of the equilibrium, giving at the same time an interesting example of the coexistence of small motions, § 89. The method consists simply in finding what the equations of motion, and their integrals, become for co-ordinates which differ infinitely little from values corresponding to a configuration of equilibriumand for an infinitely small initial kinetic energy. The solution of these differential equations is always easy, as they are linear and have constant coefficients. If the solution indicates that these differences remain infinitely small, the position is one of stable equilibrium; if it shows that one or more of them may increase indefinitely, the result of an infinitely small displacement from or. infinitely small velocity through the position of equilibrium may be a finite departure from it—and thus the equilibrium is unstable.

Since there is a position of equilibrium, the kinematic relations must be invariable. As before,

T = { {(4, 4) ¿2 + ($, $) $2 + 2 (4, 6) $$ + etc........} .....(1), which cannot be negative for any values of the co-ordinates. Now, though the values of the coefficients in this expression are not generally constant, they are to be taken as constant in the approximate investigation, since their variations, depending on

disturbed

the infinitely small variations of 4, 4, etc., can only give rise to Slightly terms of the third or higher orders of small quantities. Hence equilibrium. Lagrange's equations become simply

and the first member of each of these equations is a linear func-
tion of 4, 4, etc., with constant coefficients.

Now, since we may take what origin we please for the gene-
ralized co-ordinates, it will be convenient to assume that 4, 4, 0,
etc., are measured from the position of equilibrium considered;
and that their values are therefore always infinitely small.

Hence, infinitely small quantities of higher orders being neglected, and the forces being supposed to be independent of the velocities, we shall have linear expressions for V, , etc., in terms of 4, 4, etc., which we may write as follows :—

Equations (2) consequently become linear differential equations of the second order, with constant coefficients; as many in number as there are variables 4, 4, etc., to be determined.

The regular processes explained in elementary treatises on dif ferential equations, lead of course, independently of any particular relation between the coefficients, to a general form of solution (§ 343 below). But this form has very remarkable characteristics in the case of a conservative system; which we therefore examine particularly in the first place. In this case we have

where is, in our approximation, a homogeneous quadratic
function of,,... if we take the origin, or configuration of
equilibrium, as the configuration from which ($ 273) the poten-
tial energy is reckoned. Now, it is obvious*, from the theory

*For in the first place any such assumption as

¥=A¥, +BQ, +
$=A'¥,+B'4, + ...
etc., etc.

gives equations for 4, 4, etc., in terms of 4, 4,, etc., with the same coefficients, A, B, etc., if these are independent of t. Hence (the co-ordinates being i in

Simultaneous transformation of two quadratic functions to sums of squares.

Slightly disturbed equilibrium.

ous transformation

of the transformation of quadratic functions, that we may, by a determinate linear transformation of the co-ordinates, reduce the

Simultane number) we have i2 quantities A, A', A", ... B, B′, B", ... etc., to be determined by i equations expressing that in 27 the coefficients of 43, 3, etc. are each equal to unity, and of 4, etc. each vanish, and that in V the coefficients of 44, etc. each vanish. But, particularly in respect to our dynamical problem, the following process in two steps is instructive:

of two quadratic functions to sums of squares.

2

(1) Let the quadratic expression for T in terms of Ÿ3, ¿a, ¿¿, etc., be reduced to the form ++... by proper assignment of values to A, B, etc. This may be done arbitrarily, in an infinite number of ways, without the solution of any algebraic equation of degree higher than the first; as we may easily see by working out a synthetical process algebraically according to the analogy of finding first the conjugate diametral plane to any chosen diameter of an ellipsoid, and then the diameter of its elliptic section, conjugate to any i (i − 1) chosen diameter of this ellipse. Thus, of the equations expressing that the coefficients of the products,, 4,0,, ¿,0,, etc. vanish in T, take first the one expressing that the coefficient of ¿, vanishes, and by it find the value of one of the B's, supposing all the A's and all the B's but one to be known. Then take the two equations expressing that the coefficients of 4, and 6,0, vanish, and by them find two of the C's supposing all the C's but two to be known, as are now all the A's and all the B's: and so on. Thus, in terms of all the A's, all the B's but one, all the C's but two, all the D's but three, and so on, supposed known, we find by the solution of linear equations the remaining B's, C's, D's, etc. Lastly, using the values thus found for the unassumed quantities, B, C, D, etc., and equating to unity the coefficients of 42, 03, 03, etc. in the transformed expression for 27, we have i equations among the squares (i + 1) i and products of the assumed quantities, (i) A's, (i −1) B's, (i − 2) C's, 2 etc., by which any one of the A's, any one of the B's, any one of the C's, and so i (i − 1) on, are given immediately in terms of the ratios of the others to them.

2

ing that also in the transformed quadratic V the coefficients of 4, 4,0,, 4,0,, etc. vanish.

Or, having made the first transformation as in (1) above, with assumed values i (i-1) for disposable ratios, make a second transformation determinately thus:

2

and

expressions

kinetic and

expression for 27, which is essentially positive, to a sum of Simplified
squares of generalized component velocities, and at the same for the
time to a sum of the squares of the corresponding co-ordi- potential
energies.
nates, each multiplied by a constant, which may be either positive
or negative, but is essentially real. [In the case of an equality
or of any number of equalities among the values of these con-
stants (a, B, etc. in the notation below), roots as they are of a
determinantal equation, the linear transformation ceases to be
wholly determinate; but the degree or degrees of indeterminacy
which supervene is the reverse of embarrassing in respect to
either the process of obtaining the solution, or the interpretation.
and use of it when obtained.] Hence 4, 4, may be so chosen
T= (3¿2 + etc.)

that

and

...

V = }} (a¥3 + ß&2 + etc.)...........

·(4),

.(5),

a, ẞ, etc., being real positive or negative constants. Hence
Lagrange's equations become

4=-a¥, $=-ß4, etc.......

The solutions of these equations are

.(6).

Integrated equations of motion, expressing

mental modes of

↓ = A cos (t√a − e), += A' cos (t√ß– e'), etc. ..(7), A, e, A', e', etc., being the arbitrary constants of integration. the fundaHence we conclude the motion consists of a simple harmonic variation of each co-ordinate, provided that a, ß, etc., are all vibration. positive. This condition is satisfied when V is a true minimum at the configuration of equilibrium; which, as we have seen ($292), is necessarily the case when the equilibrium is stable. If any one or more of a, ß, ... vanishes, the equilibrium might

12+ m2 + ... =1, l'2+m22+ = 1, etc.,

leavingi (i − 1) disposables.

We shall still have, obviously, the same form for 27, that is :-

Simultaneous transformation of two quadratic functions

to sums of

And, according to the known theory of the transformation of quadratic functions, squares. we may determine the i (i-1) disposables of l, m,

l', m',

...9

so as to make the i(i-1) products of the co-ordinates, P., etc. disappear from the expression for V, and give

2V=ay+Bp+....

where a, ẞ, y, etc., are the roots, necessarily real, of an equation of the ith degree of which the coefficients depend on the coefficients of the squares and products in the expression for V in terms of ,,, etc. Later [(7′), (8) and (9) of § 343 f], a single process for carrying out this investigation will be worked out.