Elliptic islands in generalized Sinai billiards

Abstract

AbstractWe find necessary conditions for a generalized Sinai billiard to be ergodic, independent of the geometry of the disks; i.e. if the conditions are not satisfied, then we can always construct special geometries for which the system is not ergodic. We prove non-ergodicity by constructing an elliptic periodic orbit and showing that this orbit satisfies the non-degeneracy condition of KAM theory. By a combination of theory and numerical calculation involving a computer algebra system (Mathematica), we show that the first Birkhoff invariant is non-zero. We apply this result to produce non-ergodic Hamiltonian flows that are generated by smooth potentials of finite range (repelling potentials and Lennard–Jones potentials) and by geodesic flows on the torus.