Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds

TL;DR

This paper proves the existence of periodic orbits with prescribed energy levels for convex Lagrangian systems on complete Riemannian manifolds, extending previous results to noncompact cases using a modified minimax approach.

Contribution

It develops a new minimax principle for $ ext{ extlsh}$ Lagrangian systems on noncompact manifolds, establishing existence of contractible periodic orbits for almost every energy level.

Findings

Existence of periodic orbits for almost every energy level in a specified range.

Extension of previous results to noncompact Riemannian manifolds.

Discussion on the existence of closed geodesics on product manifolds.

Abstract

In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by developing a modified minimax principal to a class of Lagrangian systems on noncompact Riemannian manifolds, namely the so called $\lsh$ Lagrangian systems. In particular, we prove that for almost every $k\in(0,c_u(L))$ the exact magnetic flow associated to a $\lsh$ Lagrangian has a contractible periodic orbit with energy $k$. We also discuss the existence and non-existence of closed geodesics on the product Riemannian manifold $\R\times M$.