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Duality Between Period and Energy of Certain Periodic Hamiltonian Motions
Author(s) -
van Groesen E.
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-34.3.435
Subject(s) - convexity , monotone polygon , mathematics , monotonic function , energy functional , hamiltonian (control theory) , mathematical analysis , saddle point , hamiltonian system , duality (order theory) , lagrangian , pure mathematics , mathematical physics , mathematical optimization , geometry , financial economics , economics
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 L T and a functional J E that appears in a recent modification of the Euler‐Maupertuis principle. With T and E as parameters, the values of L T and J E 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.