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Lyusternik–Schnirelman Theory for Flows and Periodic Orbits for Hamiltonian Systems on T n × R n
Author(s) -
Josellis Frank W.
Publication year - 1994
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-68.3.641
Subject(s) - mathematics , hamiltonian system , integrable system , torus , hilbert space , differentiable function , multiplicity (mathematics) , pure mathematics , morse theory , hamiltonian (control theory) , manifold (fluid mechanics) , mathematical analysis , rotation number , periodic orbits , mathematical physics , geometry , mechanical engineering , mathematical optimization , invariant (physics) , engineering
We present a Lyusternik–Schnirelman approach for flows which generalizes the classical result concerning the number of critical points for a differentiable function on a compact manifold. This method can also be extended to special gradient‐like flows on non‐compact manifolds. Our main goal is the application to the existence and multiplicity of critical points for certain strongly indefinite functions of the form f : M × E → R, where M is a compact manifold and E is a Hilbert space. The case M = T n of the n ‐dimensional torus arises in the study of periodic solutions of Hamiltonian systems which are global perturbations of completely integrable systems. For a large class of Hamiltonian systems on T * T n we prove the existence of at least n + 1 forced oscillations in every homotopy class of loops in T * T n . Moreover, there exist at least n + 1 periodic solutions having an arbitrarily prescribed rational rotation vector.