The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

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 ) 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).

Trace formula for the magnetic Laplacian

“…The magnetic flow Φ t : T * M → T * M associated with (g, F ) is the Hamiltonian flow given by the Hamiltonian (1) H(x, p) = 1 2 |p| 2 g −1 = 1 2 n j,k=1 g jk p j p k , with respect to the twisted symplectic form on T * M :…”

“…One can apply to such fields an approach suggested in [46]. The exposition of the state of the art in the variational theory of closed magnetic geodesics on surfaces can be found in [1,2] (see also the references therein). In [14], it was established that, for exact magnetic fields on two-dimensional surfaces, the energy levels, with respect to which the magnetic field is "strong" or "weak", are separated by one constant, which equals the Mañé level of this Hamiltonian system.…”

Infinitely many periodic solutions of planar Hamiltonian systems via the Poincaré–Birkhoff theorem

“…A generalized form of the Poincaré-Birkhoff fixed point theorem. Let A be an annular region bounded by two strictly star-shaped curves around the origin, 1 and 2 , 1 ⊂ int( 2 ), where int( 2 ) denotes the interior domain bounded by 2 …”

“…1 School of Mathematical Sciences, Capital Normal University, Beijing, People's Republic of China. 2 Editorial Department of Journal, Capital Normal University, Beijing, People's Republic of China.…”

Non-resonant circles for strong magnetic fields on surfaces

“…When b ≡ 0, the existence of one periodic magnetic geodesic for all low speeds follows from [22] (see also [13]). If the function b assumes both positive and negative values, which happens for instance if the two-form bµ is exact, then the existence of infinitely many periodic magnetic geodesics for almost every speed s < s was proved in the series of papers [1,3,2,11,12] by employing the variational characterization of periodic solutions to (1.1) as critical points of the free-period action functional. The value s can be explicitly characterized and, when bµ is exact, 1 2 s2 coincides with the Mañé critical value of the universal cover.…”