Dynamics of a gravitational billiard with a hyperbolic lower boundary

Gravitational billiards provide a simple method for the illustration of the dynamics of Hamiltonian systems. Here we examine a new billiard system wit h two parameters, which exhibits, in two limiting cases, the behaviors of t wo previously studied one-parameter systems, namely the wedge and parabolic billiard. The billiard consists of a point mass moving in two dimensions u nder the influence of a constant gravitational field with a hyperbolic lowe r boundary. An iterative mapping between successive collisions with the low er boundary is derived analytically. The behavior of the system during tran sformation from the wedge to the parabola is investigated for a few specifi c cases. It is surprising that the nature of the transformation depends str ongly on the parameter values. (C) 1999 American Institute of Physics. [S10 54-1500(99)00904-0].