Integrability and Ergodicity of Classical Billiards in a Magnetic Field

Nils Berglund and Hervé Kunz
J. Statist. Phys. 83, 81-126 (1996)

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane.
The part of dynamics which is not trivially integrable can be described by a ``bouncing map''. We compute a general expression for the Jacobian matrix of this map, which allows to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits.
We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of non-convex billiards.
On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.

Keywords and phrases: billiards, magnetic field, twist map, integrability, adiabatic invariant, ergodicity.

 

Journal Homepage Published article: 10.1007/BF02183641 MR1382763 Zbl1081.37523

PDF file (520 Kb) hal-00130567 arXiv/chao-dyn/9501009