action.

If there are any real solutions, there is one of them for which Least the action is less than for any other real solution, and less than for any constrainedly guided motion with proper sum of potential and kinetic energies. Compare §§ 346-366 below.

Let x, y, z be the co-ordinates of a particle, m, of the system, at time 7, and V the potential energy of the system in its particular configuration at this instant; and let it be required to find the way to pass from one given configuration to another with velocities at each instant satisfying the condition

Σ { m (x2 + y2 + z) + V= E, a constant....... ...(5),

so that A, or

Jam (d+jdy + d)

may be the least possible.

By the method of variations we must have SA = 0, where

.......

SA = ƒ Σm (xdồx + ÿddy + żdồz + dxdx + Sýdy + Sżdz) ................... (6).
Taking in this dx = xdτ, dy = ýdτ, dz=żdτ, and remarking that
Σm (x8x + ÿồÿ + żdż) = ST..................

we have

STdr

....(7),

ƒźm (ôædæ+ ôÿdy + ôîdî) = [" 8Td.
..............

Also by integration by parts,

ƒΣm(xd8x + ...) = {Σm (¿dx + ...)} − [Σm (xdx + .....)]−ƒΣm(xdx+.....)dτ,

where [...] and {...} denote the values of the quantities enclosed,
at the beginning and end of the motion considered, and where,
further, it must be remembered that dx = xdτ, etc.
from above,

81= {2m (cô + yông + đô: )} – [2m (8+ jôy+8z)]

Hence,

This, it may be observed, is a perfectly general kinematical expres-
sion, unrestricted by any terminal or kinetic conditions. Now
in the present problem we suppose the initial and final positions
to be invariable. Hence the terminal variations, dx, etc., must
all vanish, and therefore the integrated expressions {...}, [...] dis-
appear. Also, in the present problem 87'=-8V, by the equation
of energy (5). Hence, to make 84 = 0, since the intermediate
variations, dx, etc., are quite arbitrary, subject only to the con-

Least action.

Principle of
Least Action
applied
to find
Lagrange's

generalized
equations
of motion.

ditions of the system, we must have

(10),

Σm (ïdx + ÿdy + ïdz) + § V = 0............. which [(4), § 293 above] is the general variational equation of motion of a conservative system. This proves the proposition.

It is interesting and instructive as an illustration of the principle of least action, to derive directly from it, without any use of Cartesian co-ordinates, Lagrange's equations in generalized co-ordinates, of the motion of a conservative system [§ 318 (24)]. We have

A = √2Tdt,

where T denotes the formula of § 313 (2). If now we put

+

dt

Say

((4, 6)
‚4) d¥ + (4,4) dô + etc. dd + etc. + fdt8(4,4,etc.) T,

dt

where 8,, etc.) denotes variation dependent on the explicit appearance of, 4, etc. in the coefficients of the quadratic function T. The second chief term in the formula for SA is clearly

equal to faz doy, and this, integrated by parts, becomes

dỳ

where [] denotes the difference of the values of the bracketed expression, at the beginning and end of the time fdt. Thus we have finally

T-C-V.

From it we find

So far we have a purely kinematical formula. Now introduce Principle of the dynamical condition [§ 293 (7)]

Least Action
applied
to find
Lagrange's
generalized
equations
of motion.

..(10)".

which are the required equations [§ 318 (24)].

From the proposition that SA=0 implies the equations of motion, it follows that

"stationary

Hamilton.

328. In any unguided motion whatever, of a conservative why called system, the Action from any one stated position to any other, action" by though not necessarily a minimum, fulfils the stationary condition, that is to say, the condition that the variation vanishes, which secures either a minimum or maximum, or maximumminimum.

action.

This can scarcely be made intelligible without mathematical Stationary language. Let (x1, Y1, ≈ ̧), (X, Y, Z), etc., be the co-ordinates of particles, m,, m, etc., composing the system; at any time ▾ of the actual motion. Let V be the potential energy of the system, in this configuration; and let E denote the given value of the sum of the potential and kinetic energies. The equation of energy is

2

2

1

} {m, (x ̧3 + ÿ ̧2 + ż ̧2) + m ̧ (¿‚2 + ÿ ̧2 + ¿ ̧2) + etc.} + V = E. (5) bis.
Choosing any part of the motion, for instance that from time 0
to time t, we have, for the action during it,

Let now the system be guided to move in any other way possible for it, with any other velocities, from the same initial to the same final configuration as in the given motion, subject only to the condition, that the sum of the kinetic and potential energies shall still be E. Let (x, y, z,'), etc., be the co-ordinates, and V' the corresponding potential energy; and let (', ÿ,', '), etc., be the component velocities, at time in this arbitrary motion; equation (2) still holding, for the accented letters, with only E unchanged. For the action we shall have

where t' is the time occupied by this supposed motion. Let now ‣ denote a small numerical quantity, and let έ,, 7,, etc., be finite lines such that

Varying action.

V'-V
0

vanishes

The "principle of stationary action" is, that

when is made infinitely small, for every possible deviation
(0, 0, etc.) from the natural way and velocities, subject only
to the equation of energy and to the condition of passing through
the stated initial and final configurations: and conversely, that if
V' - V
vanishes with for every possible such deviation from a
certain way and velocities, specified by (x, y, z,), etc., as the
co-ordinates at t, this way and these velocities are such that the
system unguided will move accordingly if only started with
proper velocities from the initial configuration.

Ө

329. From this principle of stationary action, founded, as we have seen, on a comparison between a natural motion, and any other motion, arbitrarily guided and subject only to the law of energy, the initial and final configurations of the system being the same in each case, Hamilton passes to the consideration of the variation of the action in a natural or unguided motion of the system produced by varying the initial and final configurations, and the sum of the potential and kinetic energies. The result is, that

action.

330. The rate of decrease of the action per unit of increase Varying of any one of the free (generalized) co-ordinates (§ 204) specifying the initial configuration, is equal to the corresponding (generalized) component momentum [§ 313, (c)] of the actual motion from that configuration: the rate of increase of the action per unit increase of any one of the free co-ordinates specifying the final configuration, is equal to the corresponding component momentum of the actual motion towards this second configuration: and the rate of increase of the action per unit increase of the constant sum of the potential and kinetic energies, is equal to the time occupied by the motion of which the action is reckoned.

To prove this we must, in our previous expression (9) for SA, now suppose the terminal co-ordinates to vary; ST to become SE - SV, in which SE is a constant during the motion; and each Action set of paths and velocities to belong to an unguided motion of a a functhe system, which requires (10) to hold. Hence

8A = {2m(i8x + yông+:8)} – [2m (28+ jôy+8z)]+tôE...(13).

If, now, in the first place, we suppose the particles constituting the system to be all free from constraint, and therefore (x, y, z) for each to be three independent variables, and if, for distinctness, we denote by (x, y, z,') and (x, y, z) the co-ordinates of m in its initial and final positions, and by (x', ÿ,', ž‚'), (x,, ÿ,, ž‚) the components of the velocity it has at those points, we have, from the preceding, according to the ordinary notation of partial differential coefficients,

expressed

tion of initial and final coordinates and the

energy;

its diffe. rential coefficients equal respectively to initial and final momen

tums, and to the time from be

ginning to

In these equations we must suppose A to be expressed as a func- end.
tion of the initial and final co-ordinates, in all six times as many
independent variables as there are of particles; and E, one more
variable, the sum of the potential and kinetic energies.

If the system consist not of free particles, but of particles connected in any way forming either one rigid body or any number