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Periodic solutions of non-autonomous Hamiltonian systems with symmetries. (English) Zbl 0794.58037
Consider the non-autonomous Hamiltonian system \(\dot u= J\nabla H(t,u)\) where the Hamiltonian \(H\in C^ 1 (\mathbb{R}\times \mathbb{R}^{2N}, \mathbb{R})\) is 1-periodic in \(t\) and satisfies certain growth conditions with respect to \(u\); in particular, \(H\) is superquadratic. We prove the existence of an unbounded sequence of 1-periodic solutions provided \(H\) is invariant under a fairly general class of linear symplectic group actions on \(\mathbb{R}^{2N}\). For the proof we use variational methods. Critical points of the corresponding strongly indefinite functional are obtained via a Galerkin type approximation leading to weakly indefinite functionals. This makes the argument rather elementary compared with related approaches of V. Benci [Trans. Am. Math. Soc. 274, 533-572 (1982; Zbl 0504.58014)].

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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