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

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 $H$.