On Tonelli periodic orbits with low energy on surfaces

TL;DR

This paper proves the existence of low-energy periodic orbits on closed surfaces for Tonelli Lagrangians, showing that such systems have infinitely many periodic orbits across a range of energy levels.

Contribution

It extends previous results by establishing the existence and abundance of periodic orbits for general Tonelli Lagrangians on closed surfaces, beyond electromagnetic cases.

Findings

Existence of a local minimizer periodic orbit on each low energy level.

Almost every energy level in the specified range has infinitely many periodic orbits.

Extension of prior electromagnetic Lagrangian results to general Tonelli Lagrangians.

Abstract

We prove that, on a closed surface, a Lagrangian system defined by a Tonelli Lagrangian $L$ possesses a periodic orbit that is a local minimizer of the free-period action functional on every energy level belonging to the low range of energies $(e_0(L),c_{\mathrm{u}}(L))$. We also prove that almost every energy level in $(e_0(L),c_{\mathrm{u}}(L))$ possesses infinitely many periodic orbits. These statements extend two results, respectively due to Taimanov and Abbondandolo-Macarini-Mazzucchelli-Paternain, valid for the special case of electromagnetic Lagrangians.