Brake orbits for Hamiltonian systems of classical type via Finsler geodesics

TL;DR

This paper links the existence of brake orbits in classical Hamiltonian systems to orthogonal geodesic chords in Finsler manifolds, providing a new approach to multiplicity problems in Hamiltonian dynamics.

Contribution

It reduces the multiplicity problem of brake orbits to the problem of orthogonal geodesic chords in Finsler geometry, enabling potential generalizations of Seifert's conjecture.

Findings

Reduces brake orbit multiplicity to Finsler geodesic chords

Establishes a connection between Hamiltonian dynamics and Finsler geometry

Provides groundwork for generalizing Seifert's conjecture

Abstract

We consider Hamiltonian functions of classical type, namely even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton's equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of classical type to the multiplicity problem of orthogonal geodesic chords in a concave Finslerian manifold with boundary. This paper will be used for a generalization of a Seifert's conjecture about the multiplicity of brake orbits to Hamiltonian functions of classical type.