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
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. 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.Comment: Explanations added in Section 5, Section 6 enlarged, small errors
corrected; Large figures have been bitmapped; 40 pages LaTeX, 15 figures,
uuencoded tar.gz. file. To appear in J. Stat. Phys. 8