Numerical study of a billiard in a gravitational field

Abstract

Billiards have always been used as models for mechanical systems. In this paper we describe a very simple billiard which, over a range of one continous parameter only, exhibits the characteristics of Hamiltonian systems having two degrees of freedom and a discontinuity. The relationship between this billiard and the well-known one-dimensional self-gravitating system (with N = 3) is given. This billiard consists of a mass point moving in a symmetric wedge of angle 2θ under the influence of a constant gravitational field. For θ<45° KAM and chaotic regions coexist in the phase space. A specific family of curves, related to collisions at the wedge vertex, limits the expansion of near-integrable regions. For θ=45°, the motion is strictly integrable. Finally, for θ>;45°, complete chaos is obtained, suggesting K-system behavior. The general properties of the mapping and some numerical results obtained are discussed. Of special interest are invariant curves which cross a line of discontinuity, and a new “universality” class for Lyapunov numbers.