Periodic orbits of Tonelli Lagrangian systems

Abstract

This final chapter is devoted to the proof of multiplicity results for periodic orbits of Lagrangian systems given by Tonelli Lagrangians L : T × TM → R with global Euler-Lagrange flow. As in the previous chapters, the approach that we follow is based on the critical point theory for the action functional. However, under the Tonelli assumptions, the machinery of abstract critical point theory is not directly applicable. The main problem is that a functional setting in which the Tonelli action is both regular and satisfies the Palais-Smale condition (the minimal requirements needed for critical point theory) is not known. The breakthrough required to overcome these difficulties was found by Abbondandolo and Figalli [AF07].