Poncelet property of planar elliptic integrable Kepler billiards

TL;DR

This paper studies the integrable dynamics of planar Kepler billiards bounded by conic sections, revealing a geometric property related to tangent lines and analyzing the system's elliptic curve structure.

Contribution

It introduces a novel geometric property of Kepler billiards involving tangent lines to a fixed circle and analyzes the integrable dynamics via elliptic curves and Cayley's criteria.

Findings

Lines of consecutive orbital foci are tangent to a fixed circle for non-zero-energy orbits.

The dynamics can be linearized on an elliptic curve with explicit shift.

Conditions for n-periodicity are derived using Cayley's criteria.

Abstract

We consider the integrable dynamics of a Kepler billiard in the plane bounded by a branch of a conic section focused at the Kepler center. We show that in this case, for non-zero-energy orbits, the lines of consecutive second orbital foci along a billiard trajectory are all tangent to a fixed circle. Based on this observation we analyse in details the integrable dynamics of a planar Kepler billiard inside or outside an elliptic reflection wall, with the Kepler center occupying one of its foci. We identify the associated elliptic curve on which the dynamics is linearized, and the shift defined thereon. We also discuss explicit conditions on $n$-periodicity using Cayley's criteria.