Existence of noncontractible periodic orbits of Hamiltonian systems separating two Lagrangian tori on $T^{*} \mathbb{T}^n$ with application to nonconvex systems

In this paper, we show the existence of non-contractible periodic orbits in Hamiltonian systems defined on $T^{*} \mathbb{T}^n$ separating two Lagrangian tori under a certain cone assumption. Our result gives a positive answer to a question of Polterovich in [P]. As an application, we find periodic orbits in almost all the homotopy classes on a dense set of energy levels in Lorentzian type mechanical Hamiltonian systems defined on $T^{*} \mathbb{T}^2$. This solves a problem of Arnold in [A].

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Received 25 August 2014

Published 8 September 2017