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.
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