726 MECHANICS mcnt of the value of a, may be made to lie as close together as is found necessary for tracing the curve of action through them libcra manu. In fig. 56 portions of three separate sets of mutually orthogonal logarithmic spirals have been traced, by using the intersections of the fundamental straight lines and circles. Fig. 56. We may pursue the subject much farther, by combining parti cular solutions like those given but taken from different origins. We can afford space for one only. Let P, Q (fig. 57) be points on the axis of x, distant a and - a from the origin, R any point in the plane of the figure. Let PR = r, ^.RPa; = 0, RQ = r 1; <^RQx = d 1 . Then if A = log?--logr 1) A -e-flj, we must have, not only the equation d?A <^A = dx 2 dy- satisfied by each of A and A , but also the conditions d_A = dh!_ dA_ dA dx dy dy dx This follows at once from the fact that all the equations are linear. Fig. 57. 214. In fig. 57 we have g A = PR/QR . Hence the locus of R is a circle, whose centre, B, is on QP produced. Again A = ^RP*-^iRQa; = ^PR.Q . The locus of ^R is, in this case, any circle passing through P and Q. These_ circles evidently cut one another orthogonally in R ; for BR, which is a radius of the one, is a tangent to the other. Thus particles moving in a plane, so that the speed at any point R is inversely as PR . RQ, may describe circles in which PQ is a chord. In this case the curves of equal action are circles de- lined by the condition that the ratio PR : RQ is constant. Or they may move in the latter system of circles, in which case the former system gives the lines of equal action. For the equations in the preceding section give dxj dijj dx from which the conclusion is obvious. We may easily extend this example to other sets of orthogonal curves whose equations are Or we may extend the where A and A have their recent values example by assuming at starting A = ?,logr - TO! log^ , in which case it will be found that we must have A = m9-m 1 e 1 . These pairs may again be combined into pA + A and - A --pA , and so on. It will be noticed that in these examples the curves of equal action Analo" and the paths of the particles correspond in steady fluid motion betweu to curves of equal pressure and lines of flow, and in electric con- action duction to equipotential lines and current lines. In such cases in and fact, where there is no vortex-motion, the action is closely analogous velocity to what is called the " velocity potential" in a fluid. potent;* Genera lized Coordin a tes. 215. By the help of the result already obtained in con- Genera: nexion with least action, we may easily obtain in a simple, i 1(l > though indirect, way the remarkable transformation of the orclinal1 equations of motion of a system which was first given by Lagrange. We are not prepared to give here the trans formation to Generalized Coordinates in its most general form ; but, even in the restricted form to which we pro ceed, it is almost invaluable in the treatment of the motion of conservative systems of particles in which the number of degrees of freedom is less than three times that of the particles. The one point to be noticed is that, when we restrict ourselves to a system of this kind, the expression for the kinetic energy, T, is necessarily a pure quadratic function of the rates of increase of the generalized coordinates. This is obvious from 19. Kepeating with generalized coordinates the investigation of 202, we have A = 2/T<ft =/(T + H - V)* . Hence SA =/(ST + SH - 5V)dt . JSTow let 0, 0, $, &c., be the generalized coordinates, and we have where P, Q, R, . . . are in general functions of 0, </>, x|>, . . . Of course V is a function of 0, <p, f> . . . alone, and does not involve p, 0, ^, &C. Thus we have, writing, for one only of the generalized coordinates, dT d0 But we saw that, for any natural motion, the unintegrated part of SA necessarily vanishes. Thus, as 6, <j>, ij/ . . . . , and, therefore, their variations, are by their very nature independent of one another, the vanishing of the unintegrated part gives us one equation of motion for each degree of freedom, the type being in all of them the same, viz., _ dtd6 de r/V To exemplify the use of these equations we will take again a few Ex- of the more important cases of constraint already treated, and will amples. then proceed to some others of interest as well as of somewhat greater complexity. In the simple pendulum, I being again the length of the string, and the inclination to the vertical at time t, we have obviously Hence or dej Thus the equation of motion is ml-i) + ; 9 + -
dO ) . sin = 0, as in 134. Suppose the same pendulum to be moving anyhow, still denot ing its inclination to the vertical, and </> denoting the azimuth of the plane in which it is displaced, we have T = iml 2 (fi + sin 2 9. </>-), V = C-^cos0 These give at once ~ de Hence the two equations an
0- .sin cos 0.