Abstract:
We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe a relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. Similarities and differences with respect to the previously studied Euclidean case are highlighted.
The research of V.D. and M.R. was supported by the Discovery Project #DP190101838 Billiards
within confocal quadrics and beyond from the Australian Research Council and Project #174020
Geometry and Topology of Manifolds, Classical Mechanics and Integrable Systems of the Serbian
Ministry of Education, Technological Development and Science. V.D. would like to thank Sydney
Mathematics Research Institute and their International Visitor Program for kind hospitality.
Citation:
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
\Bibitem{AdaDraRad19}
\by Anani Komla Adabrah, Vladimir Dragovi\'c, Milena Radnovi\'c
\paper Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 5
\pages 464--501
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https://www.mathnet.ru/eng/rcd/v24/i5/p464
This publication is cited in the following 8 articles:
Vladimir Dragović, Milena Radnović, “Billiards Within Ellipsoids in the 4-Dimensional
Pseudo-Euclidean Spaces”, Regul. Chaotic Dyn., 28:1 (2023), 14–43
Vladimir Dragović, Milena Radnović, “Resonance of ellipsoidal billiard trajectories and extremal rational functions”, Advances in Mathematics, 424 (2023), 109044
V. Dragović, S. Gasiorek, M. Radnović, “Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology”, Sb. Math., 213:9 (2022), 1187–1221
V. Dragovic, V. Shramchenko, “Deformations of the Zolotarev polynomials and Painleve VI equations”, Lett. Math. Phys., 111:3 (2021), 75
Andrews G.E., Dragovic V., Radnovic M., “Combinatorics of Periodic Ellipsoidal Billiards”, Ramanujan J., 2021
Corentin Fierobe, “Complex Caustics of the Elliptic Billiard”, Arnold Math J., 7:1 (2021), 1
Vladimir Dragović, Milena Radnović, Springer Proceedings in Mathematics & Statistics, 338, Asymptotic, Algebraic and Geometric Aspects of Integrable Systems, 2020, 159
A. K. Adabrah, V. Dragović, M. Radnović, “Elliptical Billiards in the Minkowski Plane and Extremal Polynomials”, Rus. J. Nonlin. Dyn., 15:4 (2019), 397–407
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