Algebraic non-integrability of magnetic billiards

TL;DR

This paper investigates the algebraic integrability of magnetic billiards in convex domains, proving that only circular boundaries can be algebraically integrable and establishing non-integrability for elliptical cases.

Contribution

It demonstrates that magnetic billiards are algebraically non-integrable in most cases, except potentially for circles, and introduces the concept of Outer magnetic billiards.

Findings

Magnetic billiards in elliptical domains are not algebraically integrable.

Only circular boundaries may be algebraically integrable in magnetic billiards.

The paper introduces and analyzes Outer magnetic billiards with similar non-integrability results.

Abstract

We consider billiard ball motion in a convex domain of the Euclidean plane bounded by a piece-wise smooth curve influenced by the constant magnetic field. We show that if there exists a polynomial in velocities integral of the magnetic billiard flow then every smooth piece $\gamma$ of the boundary must be algebraic and either is a circle or satisfies very strong restrictions. In particular in the case of ellipse it follows that magnetic billiard is algebraically not integrable for all magnitudes of the magnetic field. We conjecture that circle is the only integrable magnetic billiard not only in the algebraic sense, but for a broader meaning of integrability. We also introduce the model of Outer magnetic billiards. As an application of our method we prove analogous results on algebraically integrable Outer magnetic billiards.