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

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).

arXiv comment17 pages, 1 figure

References in corpus (4)

• Symplectic topology of Mañé's critical values
• Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level
• The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles
• Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level

Cited by in corpus (6)

• Trace formula for the magnetic Laplacian
• On Tonelli periodic orbits with low energy on surfaces
• On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus
• Periodic orbits in virtually contact structures
• Existence of curves with constant geodesic curvature in a Riemannian 2-sphere
• Waist theorems for Tonelli systems in higher dimensions