Authors: Dragović, Vladimir 
Radnović, Milena
Title: Caustics of Poncelet Polygons and Classical Extremal Polynomials
Journal: Regular and Chaotic Dynamics
Volume: 24
Issue: 1
Issue Date: 1-Jan-2019
Rank: M21
ISSN: 1560-3547
DOI: 10.1134/S1560354719010015
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.
Keywords: 14H70 | 26C05 | 37J35 | 41A10 | 70H06 | Akhiezer polynomials | caustics | Cayley conditions | Chebyshev polynomials | discriminantly separable polynomials | elliptic curves | elliptical billiards | extremal polynomials | Pell’s equations | periodic trajectories | Poncelet polygons | Zolotarev polynomials
Publisher: Springer Link
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Australian Research Council, Project DP190101838 Billiards within confocal quadrics and beyond

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