Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries (Q6610204)

The interplay between the geometry of boundaries, potentials, and ergodic properties of billiards as models of simple mechanical systems goes back at least as far as the foundational work of \textit{L. Boltzmann} [Wien. Ber. 58, 1035--1044 (1868; JFM 01.0348.01)] and remains of lively current interest. Indeed, the model he studied of a point mass moving under a Keplerian~\(1/r\) potential with a straight reflection wall was shown to possess an additional integral relatively recently by \textit{G. Gallavotti} and \textit{I. Jauslin} [``A Theorem on Ellipses, an Integrable System and a Theorem of Boltzmann'', Preprint, \url{arXiv:2008.01955}]. This led to the discovery of an associated elliptic curve and the computation of periodic orbits, and a relation of an additional first integral to the energy of a Kepler problem on the sphere. In the plane, the Kepler problem is dual to the Hooke problem, and this connection was extended to Kepler and Hooke billiards by \textit{A. A. Panov} [Math. Notes 55, No. 3, 1 (1994; Zbl 0821.70011); translation from Mat. Zametki 55, No. 3, 139--140 (1994)]. \N\NThe authors pursue these ideas with a particular emphasis on geometrical properties of the second foci in similar integrable Hooke and Kepler billiards reflecting at a conic section in the plane. After setting up the geometrical setting, the additional first integrals of the Kepler billiards are given a geometrical interpretation, and consecutive Kepler orbits are described in terms of consecutive foci. Other geometric properties such as focal reflection and envelope curves of the directrices of consecutive orbits are found. Finally, the conformal duality between the Hooke and Kepler systems is used to relate these discussions to Hooke billiards.