The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

TL;DR

This paper proves the existence of contractible periodic orbits for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles over non-aspherical manifolds, for almost all energies above a critical value.

Contribution

It extends the Lusternik-Fet theorem to autonomous Tonelli Hamiltonian systems on twisted cotangent bundles, establishing existence results for periodic orbits.

Findings

Contractible periodic orbits exist for almost all energies above the maximum critical value.

The result applies to non-aspherical manifolds.

The study generalizes classical results to twisted symplectic forms.

Abstract

Let $M$ be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian $H:T^*M\rightarrow \mathbb R$ and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if $M$ is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of $H$.