Closed orbits of a charge in a weakly exact magnetic field

Abstract

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let (M, g) denote a closed connected Riemannian manifold and σ ∈ 2 (M) a weakly exact 2-form. Let φ t : T M → T M denote the magnetic flow determined by σ , and let c(g, σ ) ∈ ‫ޒ‬ ∪ {∞} denote the Mañé critical value of the pair (g, σ ). We prove that if k > c(g, σ ), then for every nontrivial free homotopy class of loops on M there exists a closed orbit of φ t with energy k whose projection to M belongs to that free homotopy class. We also prove that for almost all k < c(g, σ ) there exists a closed orbit of φ t with energy k whose projection to M is contractible. In particular, when c(g, σ ) = ∞ this implies that almost every energy level has a contractible closed orbit. As a corollary we deduce that a weakly exact magnetic flow with [σ ] = 0 on a manifold with amenable fundamental group (which implies c(g, σ ) = ∞) has contractible closed orbits on almost every energy level.

Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level

“…The main result known so far about the critical points of S k in this generality is Theorem 1.1 of [Mer10], which we now state.…”

“…We refer to [Mer10] and references therein, in particular [Pat06], for a proof and a discussion of this result.…”

“…The main idea to overcome this problem in our case is due to Will Merry [Mer10]. He shows that, if the integral of σ vanishes on every 2-torus T 2 → M , the closed 1-form dS k is actually exact on Λ and a primitive S k : Λ → R can be explicitly written.…”

“…Following [Mer10] we now introduce a second functional S k : Λ → R when c(g, σ) < ∞. The key observation is the following Lemma 2.2.…”

“…Theorem 2.5 (W. Merry, 2010). Let (M, g) be a closed connected Riemannian manifold and let σ ∈ Ω 2 (M ) be a closed weakly exact 2-form.…”