Integrable elliptic billiards and ballyards

Abstract

The billiard problem concerns a point particle moving freely in a region of the horizontal plane bounded by a closed curve Γ, and reflected at each impact with Γ. The region is called a ‘billiard’, and the reflections are specular: the angle of reflection equals the angle of incidence. We review the dynamics in the case of an elliptical billiard. In addition to conservation of energy, the quantity L1 L2 is an integral of the motion, where L1 and L2 are the angular momenta about the two foci. We can regularize the billiard problem by approximating the flat-bedded, hard-edged surface by a smooth function. We then obtain solutions that are everywhere continuous and differentiable. We call such a regularized potential a ‘ballyard’. A class of ballyard potentials will be defined that yield systems that are completely integrable. We find a new integral of the motion that corresponds, in the billiards limit, to L1 L2. Just as for the billiard problem, there is a separation of the orbits into boxes and loops. The discriminant that determines the character of the solution is the sign of L1 L2 on the major axis.