Periodic orbits of reversible Lagrangian systems without self-intersections and Ma\~n\'e genericity

TL;DR

This paper demonstrates that for a generic potential in classical Lagrangian systems on higher-dimensional compact manifolds, the prime periodic orbits are simple, non-intersecting, and do not self-intersect, extending previous genericity results.

Contribution

It establishes that generic potentials lead to prime periodic orbits without self-intersections or intersections, generalizing Mañé's genericity results to Lagrangian systems.

Findings

Prime periodic orbits are simple and non-intersecting

Results hold for manifolds of dimension n ≥ 3

Extends Mañé genericity to Lagrangian systems

Abstract

Bernard [3] showed that a Ma\~n\'e generic convex Hamiltonian has only non-degenerate periodic orbits on a given energy level. We show that one can use this result to prove that for a generic potential the prime periodic orbits of fixed energy of a Lagrangian system of classical type on a compact manifold of dimension $n\ge 3$ do not have self-intersections and do not intersect each other.