Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on $T^*\T^n$ with application to non convex Hamiltonian systems

TL;DR

This paper proves the existence of noncontractible periodic orbits in Hamiltonian systems on cotangent bundles of tori, addressing longstanding questions and applying results to Lorentzian Hamiltonian systems.

Contribution

It establishes the existence of noncontractible periodic orbits under cone assumptions and applies this to find orbits of various homotopy types in Lorentzian systems, solving open problems.

Findings

Existence of noncontractible periodic orbits separating Lagrangian tori.

Periodic orbits of almost all homotopy types on dense energy levels.

Resolution of a question posed by Polterovich and a problem by Arnold.

Abstract

In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*\T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in \cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*\T^2$. This solves a problem of Arnold in \cite{A}.