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

Alberto A Abbondandolo (1) , Luca A Asselle (1) , Gabriele Benedetti (2) , Marco Mazzucchelli (3) , Iskander A Taimanov (4)

1 Ruhr-Universitat Bochum, Fakultat fur Mathematik
2 Mathematisches Institut der Universität Leipzig
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
4 Sobolev Institute of Mathematics

We consider magnetic Tonelli Hamiltonian systems on the cotan-gent 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).

Domaines: Mathématiques [math]

Alberto A Abbondandolo, Luca A Asselle, Gabriele Benedetti, Marco Mazzucchelli, Iskander A Taimanov. The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere. Advanced Nonlinear Studies, 2017, 17, ⟨10.1515/ans-2016-6003⟩. ⟨ensl-01475309⟩