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

2023/02/19 by Zhang, Duanzhi, Zhao, Zhihao
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2302.09472

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.

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