A variational approach to the study of the existence of invariant Lagrangian graphs

TL;DR

This paper reviews results on the existence of invariant Lagrangian graphs in Tonelli Hamiltonian systems, highlighting variational methods and their implications for dynamical systems theory.

Contribution

It provides a survey of the author's work on invariant Lagrangian graphs using variational techniques in the context of Tonelli Hamiltonian systems.

Findings

Existence results for invariant Lagrangian graphs in specific Hamiltonian systems

Application of variational methods to study invariant structures

Connections between invariant graphs and Hamiltonian dynamics

Abstract

This paper surveys some results by the author and collaborators on the existence of invariant Lagrangian graphs for Tonelli Hamiltonian systems. The presentation is based on an invited talk by the author at XIX Congresso Unione Matematica Italiana (Bologna, 12-17 Sept. 2011).