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 Regul. Chaotic Dyn., 2019, Volume 24, Issue 1, Pages 1–35 (Mi rcd387)

Caustics of Poncelet Polygons and Classical Extremal Polynomials

a Department for Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, 75080 Richardson TX, USA
b Mathematical Institute SANU, Kneza Mihaila 36, 11001 Beograd, p.p. 367, Serbia
c The University of Sydney, School of Mathematics and Statistics, Carslaw F07, 2006 NSW, Australia

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.

Keywords: Poncelet polygons, elliptical billiards, Cayley conditions, extremal polynomials, elliptic curves, periodic trajectories, caustics, Pell’s equations, Chebyshev polynomials, Zolotarev polynomials, Akhiezer polynomials, discriminantly separable polynomials

 Funding Agency Grant Number Serbian Ministry of Science and Technological Development 174020 Australian Research Council DP190101838 This research was supported by the Serbian Ministry of Education, Science, and Technological Development, Project 174020 Geometry and Topology of Manifolds, Classical Mechanics, and Integrable Dynamical Systems; and the Australian Research Council, Project DP190101838 Billiards within confocal quadrics and beyond.

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

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MSC: 14H70, 41A10, 70H06, 37J35, 26C05
Accepted:17.12.2018
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Citation: Vladimir Dragović, Milena Radnović, “Caustics of Poncelet Polygons and Classical Extremal Polynomials”, Regul. Chaotic Dyn., 24:1 (2019), 1–35

Citation in format AMSBIB
\Bibitem{DraRad19} \by Vladimir Dragovi\'c, Milena Radnovi\'c \paper Caustics of Poncelet Polygons and Classical Extremal Polynomials \jour Regul. Chaotic Dyn. \yr 2019 \vol 24 \issue 1 \pages 1--35 \mathnet{http://mi.mathnet.ru/rcd387} \crossref{https://doi.org/10.1134/S1560354719010015} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457880700001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85061087764} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. K. Adabrah, V. Dragović, M. Radnović, “Elliptical Billiards in the Minkowski Plane and Extremal Polynomials”, Nelineinaya dinam., 15:4 (2019), 397–407
2. Anani Komla Adabrah, Vladimir Dragović, Milena Radnović, “Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials”, Regul. Chaotic Dyn., 24:5 (2019), 464–501
3. Dragovic V., Radnovic M., “Periodic Ellipsoidal Billiard Trajectories and Extremal Polynomials”, Commun. Math. Phys., 372:1 (2019), 183–211