space. The study of these properties forms a part of the subject of geometrical optics.+ It is of interest to note that Hamilton’s equations can be formally derived from the condition of minimum action in the form (43.8) which follows from (43.6), if the co-ordinates and momenta are varied inde- pendently. Again assuming for simplicity that there is only one co-ordinate and momentum, we write the variation of the action as = dt - (OH/dp)8p dt]. An integration by parts in the second term gives At the limits of integration we must put 8q = 0, so that the integrated term is zero. The remaining expression can be zero only if the two integrands vanish separately, since the variations Sp and 8q are independent and arbitrary dq = (OH/OP) dt, dp = - (dH/dq) dt, which, after division by dt, are Hamilton’s equations. $44. Maupertuis’ principle The motion of a mechanical system is entirely determined by the principle of least action: by solving the equations of motion which follow from that principle, we can find both the form of the path and the position on the path as a function of time. If the problem is the more restricted one of determining only the path, without reference to time, a simplified form of the principle of least action may be used. We assume that the Lagrangian, and therefore the Hamilton- ian, do not involve the time explicitly, SO that the energy of the system is conserved: H(p, q) = E = constant. According to the principle of least action, the variation of the action, for given initial and final co-ordinates and times (to and t, say), is zero. If, however, we allow a variation of the final time t, the initial and final co-ordinates remaining fixed, we have (cf.(43.7)) 8S = -Hot. (44.1) We now compare, not all virtual motions of the system, but only those which satisfy the law of conservation of energy. For such paths we can replace H in (44.1) by a constant E, which gives SS+Est=0. (44.2) t See The Classical Theory of Fields, Chapter 7, Pergamon Press, Oxford 1962. §44 Maupertuis’ principle 141 Writing the action in the form (43.8) and again replacing H by E, we have (44.3) The first term in this expression, (44.4) is sometimes called the abbreviated action. Substituting (44.3) in (44.2), we find that 8S0=0. (44.5) Thus the abbreviated action has a minimum with respect to all paths which satisfy the law of conservation of energy and pass through the final point at any instant. In order to use such a variational principle, the momenta (and so the whole integrand in (44.4)) must be expressed in terms of the co-ordinates q and their differentials dq. To do this, we use the definition of momentum: (44.6) and the law of conservation of energy: E(g) (44.7) Expressing the differential dt in terms of the co-ordinates q and their differen- tials dq by means of (44.7) and substituting in (44.6), we have the momenta in terms of q and dq, with the energy E as a parameter. The variational prin- ciple so obtained determines the path of the system, and is usually called Maupertuis’ principle, although its precise formulation is due to EULER and LAGRANGE. The above calculations may be carried out explicitly when the Lagrangian takes its usual form (5.5) as the difference of the kinetic and potential energies: The momenta are and the energy is The last equation gives dt (44.8) 142 The Canonical Equations §44 substituting this in Epides we find the abbreviated action: (44.9) In particular, for a single particle the kinetic energy is T = 1/2 m(dl/dt)2, where m is the mass of the particle and dl an element of its path; the variational principle which determines the path is ${/[2m(B-U)]dl=0 (44.10) where the integral is taken between two given points in space. This form is due to JACOBI. In free motion of the particle, U = 0, and (44.10) gives the trivial result 8 I dl = 0, i.e. the particle moves along the shortest path between the two given points, i.e. in a straight line. Let us return now to the expression (44.3) for the action and vary it with respect to the parameter E. We have substituting in (44.2), we obtain (44.11) When the abbreviated action has the form (44.9), this gives = (44.12) which is just the integral of equation (44.8). Together with the equation of the path, it entirely determines the motion. PROBLEM Derive the differential equation of the path from the variational principle (44.10). SOLUTION. Effecting the variation, we have f In the second term we have used the fact that dl2 = dr2 and therefore dl d8l = dr. d&r. Integrating this term by parts and then equating to zero the coefficient of Sr in the integrand, we obtain the differential equation of the path: