The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

Abstract

Abstract. Let M be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian H : T * M → R and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if M is not aspherical then contractible periodic orbits exist for almost all energies above the maximum critical value of H.

On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

“…The main result of the present paper is the following generalization of the main theorem in [8]. In the statement below π orb ℓ (M ) denote the orbifold-theoretic homotopy groups as defined in [5, Def.…”

“…As it turns out, this is enough to show existence of critical points of S k -for almost every k -by means of a clever monotonicity argument, better known as the Struwe monotonicity argument [36] (for other applications we refer e.g. to [1,2,3,4,6,7,8,9,19]).…”

“…[3,10,13,14,19,25,26,33,34,35,37,38]). For generalities about magnetic flows on manifolds we refer to [8] and references therein.…”

On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus

“…We already showed in [AB15] that the result proved in [AMMP14] for exact forms extends to oscillating forms when M has genus at least 2 and c u (L ϑ ) is replaced by some τ * + (g, σ) ∈ (0, c u (L ϑ )] (observe that c u (L ϑ ) is still well-defined since the lift of σ to the universal cover is exact). Implementing ideas contained in [AB16], we are now able to treat the case in which M = T 2 is the two-torus. After we submitted our manuscript, the case of the two-sphere has also been solved by the authors in collaboration with Abbondandolo, Mazzucchelli and Taimanov [AAB + 16].…”

Periodic orbits in virtually contact structures

“…Hamiltonians with vanishing potentials) are studied. For more existence results of periodic solutions in magnetic Hamiltonian mechanics inspired by the work of Ginzburg [34], Polterovich [51], and Taȋmanov [56] confer [2,3,4,5,9,8,11,12,43,53] and the citations therein.…”