Main Results

We have studied the properties of planar classical billiards in a constant magnetic field perpendicular to the plane. In some cases, we have added an in-plane potential. Our main analytical results are the following:

• Periodic orbits: We have developed a variational method which allows to find periodic orbits in billiards with magnetic field and potential, and to determine their linear and nonlinear stability. The method is based on a generating function, which has a simple geometric interpretation, and admits stationary points on periodic orbits. See [1,3,4].

• Invariant curves near the boundary: We proved that billiards in a magnetic field always admit invariant curves near the boundary, corresponding to skipping orbits, provided the boundary is sufficiently smooth (C ⁶) and either convex, or with a curvature radius not smaller than the Larmor radius. This result is based on Kolmogorov-Arnol'd-Moser (KAM) theory, and generalises a well-known result by Lazutkin. See [1,4].

• Adiabatic invariants in strong magnetic field: We proved that billiards in a strong magnetic field B admit an adiabatic invariant conserved during a time of order B ^{k - 3} if the boundary is C ^k. When the boundary is analytic, there is an invariant conserved during a time of order e ^B. This result is based on Nekhoroshev-type estimates, and implies that magnetic billiards are, in a certain sense, integrable in the large B limit. See [1,4].

• Magnetic billiard in a square: We analysed the stability of specific families of periodic orbits of the billiard in a square with a magnetic field. These orbits become unstable for certain arbitrarily high values of the magnetic field, so that we conjectured that the billiard is ergodic for some of these values. See [1].

• Bound states of a scattering billiard: We studied the planar motion of a charged particle in crossed electromagnetic fields, outside a circular scatterer. We showed that for certain initial conditions close to the scatterer, which form a set of positive measure, the particle keeps colliding with the scatterer an infinite number of times, forming a classical bound state. This happens for sufficiently small electric field, and not too large magnetic field (a Larmor radius larger than the radius of the scatterer). See [2,3,4].