Journal of Symplectic Geometry

Volume 15 (2017)

Number 3

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

Pages: 905 – 936



Jinxin Xue (Department of Mathematics, University of Chicago, Illinois, U.S.A.)


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].

Received 25 August 2014

Published 8 September 2017