Phase space of billiards with and without magnetic field as symplectic quotient.

The system considered is the motion of a charged point mass within a compact, connected subset of the euclidean plane with non-empty interior, bounded by a curve diffeomorphic to the unit circle, with constant magnetic field perpendicular to the plane and with reflection at the boundary according to the law of reflection. Two models are introduced, which both describe the motion in terms of the iteration of a smooth map preserving a symplectic stucture. Both models are discussed for zero and non-zero magnetic field. For the Birkhoff billiards model, the related concept of generating functions is discussed. The orbit dynamics model is generalized to higher dimensions. The equivalence of both models in the two-dimensional case is shown.