Infinitely many Brake orbits of Tonelli Hamiltonian systems on the cotangent bundle

TL;DR

This paper proves the existence of infinitely many symmetric brake orbits in Tonelli Hamiltonian systems on twisted cotangent bundles, extending previous results and providing a complete proof of a conjecture by G. Lu.

Contribution

It establishes the existence of infinitely many brake orbits for time-dependent Tonelli Hamiltonian systems on twisted cotangent bundles, filling a gap in the literature.

Findings

Infinitely many brake orbits exist on twisted cotangent bundles with exact magnetic forms.

Symmetric orbits of the dual Euler-Lagrange system are infinitely many.

Complete proof of G. Lu's conjecture on symmetric orbits in Tonelli systems.

Abstract

We prove that on the twisted cotangent bundle of a closed manifold with an exact magnetic form, a Hamiltonian system of a time-dependent Tonelli Hamiltonian function possesses infinitely many brake orbits. More precisely, by applying Legendre transform we show that there are infinitely many symmetric orbits of the dual Euler-Lagrange system on the configuration space. This result contains an assertion for the existence of infinitely many symmetric orbits of Tonelli Euler-Lagrange systems given by G. Lu at the end of [Lu09a, Remark 6.1]. In this paper, we will present a complete proof of this assertion.