The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere

TL;DR

This paper proves that for magnetic Hamiltonian systems on the 2-sphere, most energy levels within a specific intermediate range have infinitely many periodic orbits, extending understanding of orbit multiplicity in such systems.

Contribution

It establishes that almost all energy levels in a certain range contain infinitely many periodic orbits, even when the magnetic form is not exact.

Findings

Most energy levels in the range $(e_0,e_1)$ have infinitely many periodic orbits.

Low and high energy levels may have only finitely many orbits.

The result applies to physically relevant cases with oscillating magnetic forms.

Abstract

We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range $(e_0,e_1)$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, $e_0=0$ is the minimal energy of the system).