Billiards and Nonholonomic Distributions

In this note, we consider billiards with full families of periodic orbits. It is shown that the construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to an (N - 1)-dimensional distribution on a (2N - 1)-dimensional manifold. We describe the properties of this distribution, as well as some important consequences for billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist. Bibliography: 13 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 56–64.