Periodic billiard trajectories in a magnetic field

The problem of the existence of periodic trajectories of a charged particle in a magnetic field, when the particle moves inside a closed convex region and is elastically reflected from its boundary, is considered. The presence of an infinite number of different periodic trajectories at low magnetic field strengths is established using Poincaré's geometric theorem. The conditions for two-link trajectories to be stable in the case of a uniform magnetic field are obtained.