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

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.