On the existence of periodic orbits for magnetic systems on the two-sphere

TL;DR

This paper proves the existence of periodic orbits in magnetic systems on the two-sphere for almost all energy levels, extending to magnetic geodesics on Riemannian two-spheres.

Contribution

It establishes the existence of periodic orbits on almost all energy levels for magnetic systems on the two-sphere, including magnetic geodesics on Riemannian spheres.

Findings

Periodic orbits exist on almost all energy levels of the system.

Existence of closed magnetic geodesics for almost all kinetic energies.

Results apply to a broad class of magnetic systems on the two-sphere.

Abstract

We prove that there exist periodic orbits on almost all compact regular energy levels of a Hamiltonian function defined on a twisted cotangent bundle over the two-sphere. As a corollary, given any Riemannian two-sphere and a magnetic field on it, there exists a closed magnetic geodesic for almost all kinetic energy levels.