Closed trajectories of a slow charge in a magnetic field

The motion of a unit charge on a Riemannian manifold (N,g) subject to a magnetic field 𝜎 can be described as the Hamiltonian flow of the metric Hamiltonian (p,q) ↦ 1/2 |p|² on the twisted cotangent bundle (𝓣 * 𝓝, 𝜔_𝜊 + π*𝜎) where 𝜔_𝜊 is the standard symplectic form and 𝜎 (the magnetic field) is a closed 2-form on N. In contrast to the geodesic flow, the dynamics of a charge in a magnetic field depends on its energy.

We shall explain two recent results on closed trajectories of a slow charge.

• Given any magnetic field 𝜎 ≠ 0, for a dense set of sufficiently small energies the corresponding energy level carries a closed orbit projecting to a contractible trajectory on N.
• If 𝜎 is exact (i.e., the magnetic field has a potential), then almost every sufficiently low energy level carries a closed orbit projecting to a contractible trajectory on N.

While the proof of a) relies on results from Hofer-geometry, the proof of b) uses an explicit isomorphism between the Floer homology of (𝓣 * 𝓝, 𝜔_𝜊 + π*𝜎) and the Morse homology of 𝓣 * 𝓝.

This is joint work with Urs Frauenfelder (Hokkaido University).