On the Birkhoff conjecture for Kepler billiards

TL;DR

This paper studies the integrability of Kepler billiards in convex domains, showing that only elliptical domains with the center at a focus exhibit high-energy integrability, and it constructs symbolic dynamics indicating chaos.

Contribution

It provides a partial proof of a Keplerian analogue of the Birkhoff-Poritsky Conjecture, linking integrability to domain shape and center placement, and introduces symbolic dynamics for chaotic regimes.

Findings

Ellipses with the center at a focus are the only integrable Kepler billiard domains.

Chaotic behavior is demonstrated through symbolic dynamics with positive topological entropy.

Non-elliptic domains can have a focal point of the second kind, with examples of such domains.

Abstract

We investigate the integrability of Kepler billiards-mechanical billiard systems in which a particle moves under the influence of a Keplerian potential and reflects elastically at the boundary of a strictly convex planar domain. Our main result establishes that, except possibly for one location of the gravitational center, analytic integrability at high energies occurs only when the domain is an ellipse and the center is placed at one of its foci. This provides a partial affirmative answer to a Keplerian analogue of the classical Birkhoff-Poritsky Conjecture. Our approach is based on the construction of symbolic dynamics arising from chaotic subsystems that emerge in the high-energy regime. Depending on the geometric configuration of the boundary and the location of the attraction center, we construct three types of symbolic dynamics by shadowing chains of punctured Birkhoff-type trajectories. These constructions yield subsystems conjugated to Bernoulli shifts, implying positive topological entropy and precluding analytic integrability. We further analyze the notion of focal points of the second kind, showing that in real-analytic, non-elliptic domains there can be at most one such point, while ellipses are the only domains admitting two-coinciding with their classical foci. Finally, we demonstrate the existence of an infinite-dimensional family of non-elliptic domains possessing a focal point of the second kind, and conclude with numerical simulations illustrating chaotic behavior in such cases.