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

Abstract

Abstract. Let (T 2 , g) be a Riemannian two-torus and let σ be an oscillating 2-form on T 2 . We show that for almost every small positive number k the magnetic flow of the pair (g, σ) has infinitely many periodic orbits with energy k. This result complements the analogous statement for closed surfaces of genus at least 2 [AB15] and at the same time extends the main theorem of [AMMP14] to the non-exact oscillating case.

On the existence of closed magnetic geodesics via symplectic reduction

“…In the following we will prove the existence of critical points of S k using variational methods, even though the functional S k might fail to satisfy a crucial compactness property (namely the Palais-Smale condition). To overcome this difficulty we will use a monotonicity argument, better known as the Struwe monotonicity argument, which is originally due to Struwe [37] and has been already successfully applied [1,8,10,9,11,16] to the existence of closed magnetic geodesics.…”

“…If one tries to generalize this approach dropping the exactness assumption, then one has to overcome the difficulty given by the fact that the action functional is not well-defined but rather "multi-valued". Nevertheless, following ideas contained in [30,31,38], progresses in this direction have been recently made in [9,10] by studying the existence of zeros of the action 1-form.…”

On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

“…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]).…”

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

“…In recent years, such methods have also been used to prove the existence of infinitely many (not necessarily contractible) periodic orbits on almost all low energy levels when M is a closed surface, σ is exact, and H = H kin (see [AMP15,AMMP14]). In a recent paper, the authors extended this latter result to the case in which σ is oscillating but not necessarily exact, under the further assumption that the surface M is not S 2 (see [AB15,AB] Finally, we analyze the existence of contractible periodic orbits with energy below e 0 (H) without assuming any additional condition on M or on σ. To this purpose we recall that, if (W, ω) is a symplectic manifold, a set U ⊂ W is said to be displaceable in (W, ω) if there exists a compactly supported Hamiltonian diffeomorphism ϕ : W → W , i.e.…”