TY - JOUR

T1 - The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere

AU - Abbondandolo, A.

AU - Asselle, L.

AU - Benedetti, G.

AU - Mazzucchelli, M.

AU - Taimanov, I.A.

PY - 2017/1/2

Y1 - 2017/1/2

N2 - © 2017 by De Gruyter.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 (e0,e1) 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).

AB - © 2017 by De Gruyter.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 (e0,e1) 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).

U2 - 10.1515/ans-2016-6003

DO - 10.1515/ans-2016-6003

M3 - Article

VL - 17

SP - 17

EP - 30

JO - Advanced Nonlinear Studies

JF - Advanced Nonlinear Studies

SN - 1536-1365

IS - 1

ER -