Projective integrable mechanical billiards

Abstract

In this paper, we use the projective dynamical approach to integrable mechanical billiards as in (Zhao 2021 Commun. Contemp. Math. 24 2150085) to establish the integrability of natural mechanical billiards with the Lagrange problem, which is the superposition of two Kepler problems and a Hooke problem, with the Hooke center at the middle of the Kepler centers, as the underlying mechanical systems, and with any finite combinations of confocal conic sections with foci at the Kepler centers as the reflection wall, in the plane, on the sphere, and in the hyperbolic plane. This covers many previously known integrable mechanical billiards, especially the integrable Hooke, Kepler and two-center billiards in the plane, as has been investigated in (Takeuchi and Zhao 2021 arXiv:2110.03376), as subcases. The approach of (Takeuchi and Zhao 2021 arXiv:2110.03376) based on conformal correspondence has been also applied to integrable Kepler billiards in the hyperbolic plane to illustrate their equivalence with the corresponding integrable Hooke billiards on the hemisphere and in the hyperbolic plane as well.