Remarks on Joachimsthal integral and Poritsky property

TL;DR

This paper characterizes conic billiard tables through Joachimsthal integrals, extending the concept to spherical, hyperbolic geometries, and higher dimensions, and links it to the Poritsky property of billiard curves.

Contribution

It establishes that the Joachimsthal integral uniquely characterizes conics and extends this characterization to other geometries and dimensions, connecting it with the Poritsky property.

Findings

Joachimsthal integral characterizes conics in billiard systems.

Extension of the integral characterization to spherical and hyperbolic geometries.

Connection between Joachimsthal integral and the Poritsky property.

Abstract

The billiard in an ellipse has an integral linear in momentum, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and hyperbolic geometries and to higher dimensions. We connect the existence of Joachimsthal integral with the Poritsky property, a property of billiard curves, called so after H. Poritsky whose important 1950 paper was one of the early studies of the billiard problem.