MECHANICS 723 retain the same character throughout the motion they may be ex pressed by a (generally finite) number of equations of the form Each of these gives rise to a purely kinematical relation affecting some one or more of the quantities Sx, &c., of the form = 0. 2 J- dx By introducing, as usual, a set of undetermined multipliers u, one for each of the conditions of constraint, we obtain on adding all these equations to the general equation above 3/ii(x8x+ ...) = ". If there be p particles of the system, there are Bp coordinates x, y, z. connected by (say) q equations of constraint, so that there are Sp-q degrees of freedom, and therefore Sp-q independent co ordinates. Equating separately to zero the multipliers of Sx lf Sy 1} &c., in the resultant equation above, we have Bp equations of which we write only one as a type, viz. , Taken along with the q equations of the form /=0 these form a group of Bp + q equations, theoretically necessary and sufficient to determine the Bp quantities x v y v Sj, &c., and the q quantities fj., in terms of t. Thus we have the complete analytical statement of the conditions, and the rest of the solution is a question of pure mathematics.
- T on- "When we deal with a non-conservative system (vhich_is cquiva-
- onser- lent in nature to saying " when we take an incomplete view of the
, ative question "), some of the conditions may vary in character during jystem. the motion. This will be expressed analytically by the entrance of t explicitly into one or more of the equations of condition /. But, if we think of the mode of formation of Lagrange s equation, we see that it was built up of separate equations, such as which are true whether the equations of condition involve t explicitly or not. Each of these was multiplied by a quantity Sx, &c., the only limitation on which was that it should be consistent with the conditions of the system at the instant considered, what ever instant that might be. Hence equation (2) still holds good. "When, however, we introduce in that equation multipliers cor responding to the actual motion of the system, so that Sx = xSt, &c., we find a remarkably simple expression for the energy given to, or withdrawn from, the system in consequence of the varying con ditions. For the uuintegrated equation (2) now becomes (&m(& + &* + s 2 )) = 2(Xz + Yy + Zz) - where the differential coefficient of / is partial. This follows at once from equations of the form dt which are obtained by differentiating the equations of condition with regard to t. When the conditions do not vary, the quantities , &c. , all vanish, and we see that the constraint does not alter dt the energy of the system. Least and Varying Action. Action. 200. To complete our sketch of kinetics of a particle we will now briefly consider the important quantity called " action." This, for a single particle, may be defined either as the space integral of the momentum or as double the time integral of the kinetic energy, calculated from any assumed position of the moving particle, or from an assigned epoch. For a system its value is the sum of its separate values for the various particles of the system. No one has, as yet, pointed out (in the simple form in which it is all but certain that they can be expressed) the true relations of this quantity. It was originally introduced into kinetics to suit the metaphysical necessity that something should be a minimum in the path of a luminous corpuscle (see an extract from Hamilton in the article LIGHT, vol. xiv. p. 598). But there can be little doubt that it is destined to play an important part in the final systematizing of the fundamental laws of kinetics. The importance of the quantity called action, so far as is at present known, depends upon the two principles of " least action " and of " varying action," the first as old as Maupertuis, the other discovered by Hamilton about half a century ago. The first is IftJie sum of the potential and kinetic energies Least of a system is the same in all its configurations, then, of all the sets of paths by which the jiarts of the system can be guided by frictionless constraint to pass from one given configuration to another, that one for which the action is least is the natural one or requires no constraint. 201. Unfortunately it is not easy to give examples of this important principle which can be satisfactorily treated by elementary methods, except, indeed, the very simplest, such as those furnished by the corpuscular theory of light. Thus it is obvious that, as long as a medium is homogeneous and isotropic, the speed of a corpuscle in it is constant. The action is thus reduced to the product of the constant speed of the corpuscle by the length of its path. Hence the principle at once shows that the path must be a straight line. When the corpuscle is refracted from one such medium into another, the path is a broken line such that the product of each of its parts by the corresponding speed of the corpuscle is the least possible. This gives the law of the sines, but to agree with experiment the speed would have to be greater in the denser medium than in the rarer. 202. The problem to find change of action as depending on Change change (nowhere finite) of the mode of passage from one given con- of action. figuration to another (restricted by the condition already men tioned), is expressed mathematically by 5 A = Sfemsds = 8j^m(xdx + ydy + ids) , while T = |2ms 2 = |2m(x 2 + y" + z 1 ) = H - V , H being the constant energy of the system, and the integral being taken between limits supplied by the two given configurations. The first equation gives 8A =fs,m(dx$x + dySy + dzSz + xdSx + ydSy + zdSz) by partial integration. But the integrated part obviously vanishes at both limits, because the initial and final configurations are given. If we now take the corresponding variation of the expressions for the kinetic energy, we have ST = Sm from which we have f2m(dxSx + dySy + dzSz) =/S Also we have dxSx + dySy + dzSz = (xSx + y8y + z8z)dt ; so that finally 5A =fdt[sT - -Sm(xSx + i/Sy + 282)] , which so far is a mere kinematical result. But it can be rendered physical by piitting - 8V for 8T, in accordance with the above con dition. This we will suppose done. If now we desire to make 8A vanish, so as to obtain what is called the "stationary condition," we must make the factor in square brackets in the integral vanish ; i.e., we must have 2m(x8x + y8y + zSz) + 8 V = for all admissible simultaneous values of Sx, Sy, 8~ for the various particles of the system. But this is precisely the general equation which, as we found in 199 (1), determines the undisturbed motion of the system. 203. The expression SA-0 really signifies that any infinites!- Station mal change from the natural mode of passage produces an infinitely ary smaller change in the corresponding amount of the action between action. the terminal configurations. 204. It will be noticed that the essential characteristic of the modes of passage considered in this investigation is that all shall
have the same terminal configurations, and that the system shall