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Closed trajectories of a slow charge in a magnetic field

  • Felix Schlenk (ETH ZΓΌrich)
A3 01 (Sophus-Lie room)

Abstract

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, πœ”πœŠ + Ο€*𝜎) where πœ”πœŠ 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 β‰  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, πœ”πœŠ + Ο€*𝜎) and the Morse homology of 𝓣 * 𝓝.

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

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail