Integrable Mechanical Billiards in Higher-Dimensional Space Forms

In this article, we consider mechanical billiard systems defined with Lagrange’s integrable extension of Euler’s two-center problems in the Euclidean space, the sphere, and the hyperbolic space of arbitrary dimension $n \geqslant 3$. In the three-dimensional Euclidean space, we show that the billiard systems with any finite combinations of spheroids and circular hyperboloids of two sheets having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and in the three- dimensional hyperbolic space by means of central projection. Using the same approach, we also extend these results to the $n$-dimensional cases.