Algebraic non-integrability of magnetic billiards on the Sphere and Hyperbolic plane

TL;DR

This paper proves that magnetic billiard systems on spheres and hyperbolic planes generally lack polynomial integrals of motion unless the domain boundary is a perfect circle, highlighting their non-integrability.

Contribution

It establishes a link between polynomial integrals of motion and algebraic properties of boundary curves, demonstrating non-integrability for most magnetic fields.

Findings

Polynomial integrals imply algebraic boundary curves.

Most magnetic billiards lack polynomial integrals except for circular domains.

Non-integrability holds for all but finitely many magnetic field strengths.

Abstract

We consider billiard ball motion in a convex domain on a constant curvature surface influenced by the constant magnetic field. We examine the existence of integral of motion which is polynomial in velocities. We prove that if such an integral exists then the boundary curve of the domain determines an algebraic curve in $\mathbf{C}^3$ which must be nonsingular. Using this fact we deduce that for any domain different from round disc for all but finitely many values of the magnitude of the magnetic field billiard motion does not have Polynomial in velocities integral of motion.