PROBLEM 2. Determine the Poisson brackets formed from the components of M. SOLUTION. A direct calculation from formula (42.5) gives [Mx, My] = -M2, [My, M] = -Mx, [Mz, Mx] = -My. Since the momenta and co-ordinates of different particles are mutually independent variables, it is easy to see that the formulae derived in Problems 1 and 2 are valid also for the total momentum and angular momentum of any system of particles. PROBLEM 3. Show that [, M2] = 0, where is any function, spherically symmetrical about the origin, of the co-ordinates and momentum of a particle. SOLUTION. Such a function can depend on the components of the vectors r and p only through the combinations r2, p2, r. p. Hence and similarly for The required relation may be verified by direct calculation from formula (42.5), using these formulae for the partial derivatives. PROBLEM 4. Show that [f, M] = n xf, where f is a vector function of the co-ordinates and momentum of a particle, and n is a unit vector parallel to the z-axis. SOLUTION. An arbitrary vector f(r,p) may be written as f = where 01, O2, 03 are scalar functions. The required relation may be verified by direct calculation from formulae (42.9), (42.11), (42.12) and the formula of Problem 3. $43. The action as a function of the co-ordinates In formulating the principle of least action, we have considered the integral (43.1) taken along a path between two given positions q(1) and q(2) which the system occupies at given instants t1 and t2. In varying the action, we compared the values of this integral for neighbouring paths with the same values of q(t1) and q(t2). Only one of these paths corresponds to the actual motion, namely the path for which the integral S has its minimum value. Let us now consider another aspect of the concept of action, regarding S as a quantity characterising the motion along the actual path, and compare the values of S for paths having a common beginning at q(t1) = q(1), but passing through different points at time t2. In other words, we consider the action integral for the true path as a function of the co-ordinates at the upper limit of integration. The change in the action from one path to a neighbouring path is given (if there is one degree of freedom) by the expression (2.5): 8S = Since the paths of actual motion satisfy Lagrange’s equations, the integral in 8S is zero. In the first term we put Sq(t1) = 0, and denote the value of §43 The action as a function of the co-ordinates 139 8q(t2) by 8q simply. Replacing 0L/dq by p, we have finally 8S = pdq or, in the general case of any number of degrees of freedom, ES==Pisqu- (43.2) From this relation it follows that the partial derivatives of the action with respect to the co-ordinates are equal to the corresponding momenta: = (43.3) The action may similarly be regarded as an explicit function of time, by considering paths starting at a given instant t1 and at a given point q(1), and ending at a given point q(2) at various times t2 = t. The partial derivative asiat thus obtained may be found by an appropriate variation of the integral. It is simpler, however, to use formula (43.3), proceeding as follows. From the definition of the action, its total time derivative along the path is dS/dt = L. (43.4) Next, regarding S as a function of co-ordinates and time, in the sense des- cribed above, and using formula (43.3), we have dS A comparison gives asid = L- or (43.5) Formulae (43.3) and (43.5) may be represented by the expression (43.6) for the total differential of the action as a function of co-ordinates and time at the upper limit of integration in (43.1). Let us now suppose that the co- ordinates (and time) at the beginning of the motion, as well as at the end, are variable. It is evident that the corresponding change in S will be given by the difference of the expressions (43.6) for the beginning and end of the path, i.e. dsp (43.7) This relation shows that, whatever the external forces on the system during its motion, its final state cannot be an arbitrary function of its initial state; only those motions are possible for which the expression on the right-hand side of equation (43.7) is a perfect differential. Thus the existence of the principle of least action, quite apart from any particular form of the Lagran- gian, imposes certain restrictions on the range of possible motions. In parti- cular, it is possible to derive a number of general properties, independent of the external fields, for beams of particles diverging from given points in 140 The Canonical Equations