Implementation of Integrable Systems by Topological, Geodesic Billiards with Potential and Magnetic Field

Abstract

In the paper, eight classes of integrable billiards are studied; in particular, classes introduced by the authors: elementary, topological, billiard books, billiards on the Minkowski plane, geodesic billiards on quadrics in three-dimensional Euclidean space, billiards in a magnetic field, and also a class containing all of the ones above. It turns out that, in the class of billiard books, topological obstacles to implementation occurred, for example, for the “twisted” Lagrange top (as we conventionally call a modification of the usual Lagrange top which we had discovered) for one of energy zones. We indicate this obstacle explicitly. It turns out further that this system can still be implemented in the class of magnetic billiards.