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 $\Gamma$, and reflected at each impact with $\Gamma$. 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 $L_1 L_2$ is an integral of the motion, where $L_1$ and $L_2$ 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 $N\to\infty$, to $L_1 L_2$. 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 $L_1 L_2$ on the major axis.