Wire billiards, the first steps

Abstract

Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature $k$ in $\mathbb{R}^n$ (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In this paper we are investigating whether the basic features of conventional planar billiards extend to this more general situation. In particular, we extend Lazutkin's KAM result, as well as Mather's converse KAM result, to wire billiards. We address the notion of caustics: for wire billiards, it corresponds to striction curve of the ruled surface spanned by the chords of the invariant curve. If the ruled surface is developable this is a genuine caustic. We found remarkable examples of the wires which are closed orbits of 1-parameter subgroup of $SO(n)$. These wire billiards are totally integrable. Using the theory of interpolating Hamiltonians, we prove that the distribution of impact points of the wire becomes uniform with respect to the measure $k^{2/3} dx$ (where $x$ is the arc length parameter), as the length of the chords tends to zero. Applying this result, we prove that the billiard transformation in an ellipsoid commutes with the reparameterized geodesic flow on a confocal ellipsoid: the speed of the foot point of the line tangent to a geodesic equals $k^{-2/3}$, where $k$ is the curvature of the geodesic in the ambient space. We also discuss perspectives and open problems of this new class of billiards.