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Symmetry breaking in Hamiltonian systems. (English) Zbl 0606.58043

This paper is devoted to the question whether there exist T-periodic solutions of the Hamiltonian system \[ \dot p=-\frac{\partial H}{\partial q}+\epsilon h_ 1(t)\quad \dot q=\frac{\partial H}{\partial p}+\epsilon h_ 2(t), \] where \(\epsilon >0\) is small, \(h_ 1\) and \(h_ 2\) are T- periodic functions and the unperturbed system \((H_ 0)\) is autonomous. The authors give several theorems concerning the question, provided \((H_ 0)\) fulfills certain additional assumptions (i.e. convexity, growth conditions, integrability conditions etc.). The method they use to prove these theorems depends on an ”abstract” perturbation result for critical points. This result is proved in the first two sections of the paper.
Reviewer: N.Jacob

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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