Projective Integrable Mechanical Billiards

TL;DR

This paper proves the integrability of a broad class of mechanical billiards combining Kepler, Hooke, and two-center problems using a projective dynamical approach, extending known results to various geometries.

Contribution

It introduces a unified method to establish integrability of complex billiards with confocal conic reflections in multiple geometries, generalizing previous specific cases.

Findings

Established integrability of combined Kepler-Hooke billiards.

Extended integrability results to spherical and hyperbolic geometries.

Unified approach applicable to various confocal conic billiards.

Abstract

In this paper, we use the projective dynamical approach to integrable mechanical billiards as in [Zhao, 2021] 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 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-Zhao, 2021], as subcases. The approach of [Takeuchi-Zhao, 2021] 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.