Topological Obstacles to the Realizability of Integrable Hamiltonian Systems by Billiards

Abstract

We introduce the following classes of integrable billiards: elementary billiards, topological billiards, billiard books, billiards with a potential, with a magnetic field, and geodesic billiards. These classes are used to test Fomenko’s conjecture about the realizability, up to Liouville equivalence, of integrable nondegenerate Hamiltonian systems with two degrees of freedom by billiards. In the class of book billiards, topological obstacles to realizability are found.