Tangent map for classical billiards in magnetic fields

Abstract

A charged particle is moving inside a planar billiard embedded in a uniform constant magnetic field which is directed perpendicular to the plane. The stability of classical trajectories is studied by the area-preserving tangent map which is derived for any billiard shape having a smooth convex boundary. As an example, two generic billiards are considered here, namely the ellipse and the stadium, classical trajectories of which are well known to be regular or chaotic, respectively, for an uncharged particle. Lyapunov exponents and Poincare sections are studied as a function of the field strength and of the billiard deformation. Regular motion is restored by increasing the magnetic field and/or the deformation.