Duality between period and energy of certain periodic hamiltonian motions

For a one-parameter family of periodic solutions of a second-order, autonomous, Hamiltonian system, it is shown that the minimal period T and the energy E are related in a monotone way if the even potential satisfies certain convexity and monotonicity conditions. The results are obtained using variational methods by considering the usual Lagrange functional LT and a functional JE that appears in a recent modification of the Euler–Maupertuis principle. With T and E as parameters, the values of LT and JE at certain critical points (in general, of saddle point type) define functions of T and E respectively. These functions turn out to be related by duality, from which the results follow