Classical solutions for a free particle in a confocal elliptic billiard

The classical dynamics of a free particle constrained to move in an integrable two-dimensional confocal elliptic billiard is investigated. We derive the characteristic equations for periodic orbits, classify the orbits, present the Poincare maps, give expressions for the lengths of the trajectories, and do a stability analysis of special orbits. We also explore some interesting geometrical constructions for the billiard which can be extended to the confocal elliptic billiard. The latter provides a well-motivated and relatively straightforward example of Hamilton-Jacobi theory, elliptic integrals, and Jacobi elliptic functions in a way that is seldom discussed in the undergraduate curriculum. (C) 2004 American Association of Physics Teachers.