On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

TL;DR

This paper classifies piecewise smooth rationally integrable projective billiards, revealing new minimal degrees of integrals and extending prior classifications of billiard systems with polynomial integrals.

Contribution

It provides a comprehensive classification of piecewise smooth rationally integrable projective billiards, including cases with dual pencils of conics and new minimal degrees of integrals.

Findings

Classified piecewise smooth rationally integrable projective billiards.

Discovered integrals of minimal degree 2, 4, or 12 for certain billiards.

Extended previous results to include dual multibilliards with new integral degrees.

Abstract

The billiard flow in a planar domain acts on its tangent bundle as geodesic flow with reflections from the boundary. Its trivial first integral is the squared velocity. Bolotin's Conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those planar billiards whose flow admits an integral polynomial in the velocity whose restriction to the unit tangent bundle is non-constant. It states that 1) if the boundary of such a billiard is $C^2$-smooth, nonlinear and connected, then it is a conic; 2) if it is piecewise $C^2$-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of "admissible lines" for the pencil; 3) the minimal degree of the integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, defining reflection of oriented lines and the projective billiard flow. They are common generalization of billiards on constant curvature surfaces, but in general may have no canonical integral. In a previous paper the author classified those $C^4$-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational $0$-homogeneous function of the velocity (with coefficients depending on the position): they are called rationally $0$-homogeneously integrable. It was shown that: 1) the underlying curve is a conic; 2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; 3) otherwise it can be arbitrary even number. In the present paper we classify piecewise $C^4$-smooth rationally $0$-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards.