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Nelin. Dinam., 2019, Volume 15, Number 4, Pages 397–407 (Mi nd669)  

Elliptical Billiards in the Minkowski Plane and Extremal Polynomials

A. K. Adabraha, V. Dragovićba, M. Radnovićcb

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

Abstract: We derive necessary and sufficient conditions for periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlying elliptic curve. Equivalent conditions are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. Similarities and differences with respect to the previously studied Euclidean case are indicated.

Keywords: Minkowski plane, elliptical billiards, elliptic curve, Akhiezer polynomials

Funding Agency Grant Number
Australian Research Council DP190101838
Serbian Ministry of Science and Technological Development 174020
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.


DOI: https://doi.org/10.20537/nd190401

Full text: PDF file (266 kB)
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MSC: 14H70, 41A10, 70H06, 37J35, 26C05
Received: 05.06.2019
Accepted:26.08.2019

Citation: A. K. Adabrah, V. Dragović, M. Radnović, “Elliptical Billiards in the Minkowski Plane and Extremal Polynomials”, Nelin. Dinam., 15:4 (2019), 397–407

Citation in format AMSBIB
\Bibitem{AdaDraRad19}
\by A. K. Adabrah, V. Dragovi\'c, M. Radnovi\'c
\paper Elliptical Billiards in the Minkowski Plane and Extremal Polynomials
\jour Nelin. Dinam.
\yr 2019
\vol 15
\issue 4
\pages 397--407
\mathnet{http://mi.mathnet.ru/nd669}
\crossref{https://doi.org/10.20537/nd190401}


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