A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium |
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Authors: | Thomas Bartsch |
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Institution: | Mathematisches Institut, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany |
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Abstract: | We study the Hamiltonian system (HS)
where H ε C2 (
2N,
) satisfies H (0) = 0, H′ (0) = 0 and the quadratic form
) is non-degenerate. We fix τ0 > 0 and assume that
2N E F decomposes into linear subspaces E and F which are invariant under the flow associated to the linearized system (LHS)
= JH″ (0) x and such that each solution of (LHS) in E is τ0-periodic whereas no solution of (LHS) in F − 0 is τ0-periodic. We write σ(τ0) = σQ(τ0) for the signature of the quadratic form Q restricted to E. If σ(τ0) ≠ 0 then there exist periodic solutions of (HS) arbitrarily close to 0. More precisely we show, either there exists a sequence xk → 0 of τk-periodic orbits on the energy level H−1 (0) with τk → τ0; or for each λ close to 0 with λσ(τ0) > 0 the energy level H−1 (λ) contains at least
distinct periodic orbits of (HS) near 0 with periods near τ0. This generalizes a result of Weinstein and Moser who assumed QE to be positive definite. |
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Keywords: | |
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