The Palais-Smale condition for contact type energy levels for convex lagrangian systems

TL;DR

This paper establishes the existence of periodic orbits on most energy levels for convex Lagrangian systems and proves the Palais-Smale condition for contact type energy levels, advancing understanding in dynamical systems.

Contribution

It demonstrates that almost all energy levels contain periodic orbits and satisfies the Palais-Smale condition for contact type levels, extending previous results in Lagrangian dynamics.

Findings

Almost all energy levels contain a periodic orbit.

Energy levels below Ma ne's critical value have conjugate points.

The free time action functional satisfies Palais-Smale condition on contact type levels.

Abstract

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma ne's critical value of the lift of the Lagrangian to the universal cover, almost all energy levels have conjugate points. We prove that if the energy level [E=k] is of contact type and M is not the 2-torus then the free time action functional of L+k satisfies the Palais-Smale condition.