Integrability of Kepler Billiards at Zero-Energy

2025/07/18 by Lei Zhao, Zhao, Lei
Physics and Astronomy · #Cosmology and Gravitation Theories #Black Holes and Theoretical Physics #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2507.15879

Abstract

We consider a Kepler billiard with zero-energy in the plane defined inside a smooth closed connected simple curve which intersects all focused parabola at at most two points. We show that if has an invariant curve consisting of 2-periodic orbits and there exists a C¹-first integral with non-vanishing gradient in the region between the invariant curve and the boundary curve, then the system is defined actually inside an ellipse with the Kepler center occupying one of the foci. This statement is obtained as a simple ``translation'' of the theorem of Bialy-Mironov with Levi-Civita transformation.

Citations

• On the Birkhoff conjecture for Kepler billiards
• Exploration of billiards with Keplerian potential
• Conformal Transformations and Integrable Mechanical Billiards
• The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables

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