Abstract
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.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 56–64.
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Baryshnikov, Y., Zharnitsky, V. Billiards and Nonholonomic Distributions. J Math Sci 128, 2706–2710 (2005). https://doi.org/10.1007/s10958-005-0220-1
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DOI: https://doi.org/10.1007/s10958-005-0220-1