Linking and closed orbits

TL;DR

This paper demonstrates the existence of periodic solutions in classical mechanics on Riemannian manifolds under specific potential conditions, linking geometric properties with dynamical behavior.

Contribution

It establishes new conditions under which periodic orbits exist in classical mechanics on manifolds, connecting potential growth, sign changes, and topological properties.

Findings

Existence of periodic solutions under specified potential conditions

Link between potential asymptotics and orbit existence

Application of topological methods to classical mechanics

Abstract

We show that the Lagrangian of classical mechanics on a Riemannian manifold of bounded geometry carries a periodic solution of motion with rescribed energy, provided the potential satisfies an asymptotic growth condition, changes sign, and the negative set of the potential is non-trivial in the relative homology.