Lecture notes on closed orbits for twisted autonomous Tonelli Lagrangian flows

TL;DR

These lecture notes explore the existence and stability of periodic orbits in twisted autonomous Tonelli Lagrangian systems, emphasizing magnetic flows on surfaces and introducing a method for low-energy orbit detection.

Contribution

The notes unify old and new results on periodic orbits for these systems and highlight a specific method for magnetic flows, including stability analysis.

Findings

A general theorem on periodic orbits for twisted Tonelli Lagrangians.

Application of Taimanov's method to magnetic flows on surfaces.

Analysis of stability properties of energy levels.

Abstract

These notes were prepared in occasion of a mini-course given by the author at the "CIMPA Research School - Hamiltonian and Lagrangian Dynamics" (10-19 March 2015 - Salto, Uruguay). The talks were meant as an introduction to the problem of finding periodic orbits of prescribed energy for autonomous Tonelli Lagrangian systems on the twisted cotangent bundle of a closed manifold. In the first part of the lecture notes, we put together in a general theorem old and new results on the subject. In the second part, we focus on an important class of examples: magnetic flows on surfaces. For such systems, we discuss a special method, originally due to Taimanov, to find periodic orbits with low energy and we study in detail the stability properties of the energy levels.