Multiple brake orbits in $\mathbf m$-dimensional disks

TL;DR

This paper proves the existence of at least two distinct orthogonal geodesics in a concave disk on a Riemannian surface and applies this to establish the existence of two brake orbits in certain Hamiltonian systems.

Contribution

It introduces new geometric conditions ensuring multiple orthogonal geodesics and applies these results to demonstrate brake orbit multiplicity in Hamiltonian dynamics.

Findings

At least two orthogonal geodesics exist in certain concave disks.

Existence of two brake orbits in specific Hamiltonian systems.

Utilizes recent deformation techniques for the proof.

Abstract

Let $(M,g)$ be a (complete) Riemannian surface, and let $\Omega\subset M$ be an open subset whose closure is homeomorphic to a disk. We prove that if $\partial\Omega$ is smooth and it satisfies a strong concavity assumption, then there are at least two distinct orthogonal geodesics in $\overline\Omega=\Omega \bigcup\partial\Omega$. Using the results given in [6], we then obtain a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. In our proof we shall use recent deformation results proved in [7].