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 |
Keywords:
perturbation of Hamiltonian systems; perturbation theory for critical points; T-periodic solutionsReferences:
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