Lectures on the free period Lagrangian action functional

TL;DR

This paper explores the existence of periodic orbits at specific energy levels in classical Hamiltonian systems using a variational approach centered on the free period Lagrangian action functional and Mañé critical values.

Contribution

It provides an expository analysis of how the free period Lagrangian action functional's behavior relates to the existence of periodic orbits at prescribed energies.

Findings

Identifies conditions for the existence of periodic orbits at certain energy levels.

Analyzes the role of Mañé critical values in the variational framework.

Provides insights into the structure of the action functional near critical energy values.

Abstract

In this expository article we study the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces. We use a variational approach, by studying how the behavior of the free period Lagrangian action functional changes when the energy crosses certain values, known as the Ma\~n\'e critical values.