Abstract
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 ) ${(e_{0},e_{1})}$ 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 ${e_{0}=0}$ is the minimal energy of the system).
Highlights
This paper is the last chapter of a work started in [AMP15] and further developed in [AMMP14, AB15a, AB15b, AM16] devoted to studying the multiplicity of periodic orbits on generic low energy levels in magnetic Tonelli Lagrangian systems on surfaces
The original waist Theorem says that a Riemannian 2-sphere possesses infinitely many closed geodesics provided it possesses a waist. Such a statement is a crucial ingredient for the proof that, every Riemannian 2-sphere possesses infinitely many closed geodesics [Ban[93], Fra[92], Hin93]
If M is a closed smooth manifold, a Tonelli Lagrangian L : TM → R is a smooth function whose restriction to any fiber of TM is superlinear with positive definite Hessian, see e.g. [Mat[91], Fat[08], Abb13]
Summary
This paper is the last chapter of a work started in [AMP15] and further developed in [AMMP14, AB15a, AB15b, AM16] devoted to studying the multiplicity of periodic orbits on generic low energy levels in magnetic Tonelli Lagrangian systems on surfaces. The combination of Theorem 1.1 together with the above mentioned results in [AB15a, AB15b, AM16], yields the following statement about the multiplicity of periodic orbits on general closed surfaces.
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