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Journal of the London Mathematical Society 1986 s2-34(3):435-448; doi:10.1112/jlms/s2-34.3.435
© 1986 by London Mathematical Society
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© Oxford University Press

Duality Between Period and Energy of Certain Periodic Hamiltonian Motions

E. van Groesen

Department of Applied Mathematics, University of Twente PO Box 217, Enschede, The Netherlands

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.


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