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Caustics of Poncelet Polygons and Classical Extremal Polynomials

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Abstract

A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line and ellipsoidal billiards in d-dimensional space. Even in the planar case systematically studied in the present paper, it leads to new results in characterizing n periodic trajectories vs. so-called n elliptic periodic trajectories, which are n-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell’s equation. This new approach also sheds light on some classical results. In particular, we connect the search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which, as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for n = 3, 4, 5, 6 along with an effective search for caustics. As a byproduct, an intriguing connection between Cayley-type conditions and discriminantly separable polynomials has been observed for all those small periods.

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Authors and Affiliations

  1. Department for Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080, USA

    Vladimir Dragović

  2. Mathematical Institute SANU, Kneza Mihaila 36, 11001, Beograd, p.p. 367, Serbia

    Vladimir Dragović & Milena Radnović

  3. The University of Sydney, School of Mathematics and Statistics, Carslaw F07, 2006, NSW, Australia

    Milena Radnović

Authors
  1. Vladimir Dragović
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  2. Milena Radnović
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Correspondence to Vladimir Dragović.

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Dragović, V., Radnović, M. Caustics of Poncelet Polygons and Classical Extremal Polynomials. Regul. Chaot. Dyn. 24, 1–35 (2019). https://doi.org/10.1134/S1560354719010015

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  • DOI: https://doi.org/10.1134/S1560354719010015

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