Elliptical Billiards in the Minkowski Plane and Extremal Polynomials


    2019, Vol. 15, no. 4, pp.  397-407

    Author(s): Adabrah A. K., Dragović V., Radnović M.

    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
    Citation: Adabrah A. K., Dragović V., Radnović M., Elliptical Billiards in the Minkowski Plane and Extremal Polynomials, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 4, pp.  397-407
    DOI:10.20537/nd190401


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    References
    [1] Adabrah, A. K., Dragović, V., and Radnović, M., “Periodic Billiards within Conics in the Minkowski Plane and Akhiezer Polynomials”, Regul. Chaotic Dyn., 24:5 (2019), 464–501  mathnet  crossref  mathscinet  adsnasa
    [2] Achyezer, N. I., “Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen: 1”, Izv. Akad. Nauk SSSR. Ser. 7, 1932, no. 9, 1163–1202
    [3] Achyezer, N. I., “Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen: 2”, Izv. Akad. Nauk SSSR. Ser. 7, 1933, no. 3, 309–344
    [4] Achyezer, N. I., “Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen: 3”, Izv. Akad. Nauk SSSR. Ser. 7, 1933, no. 4, 499–536
    [5] Birkhoff, G. and Morris, R., “Confocal Conics in Space-Time”, Amer. Math. Monthly, 69:1 (1962), 1–4  crossref  mathscinet  zmath
    [6] Dragović, V. and Radnović, M., “Ellipsoidal Billiards in Pseudo-Euclidean Spaces and Relativistic Quadrics”, Adv. Math., 231 (2012), 1173–1201  crossref  mathscinet  zmath
    [7] Dragović, V. and Radnović, M., “Minkowski Plane, Confocal Conics, and Billiards”, Publ. Inst. Math. (Beograd) (N. S.), 94(108) (2013), 17–30  crossref  mathscinet  zmath
    [8] Dragović, V. and Radnović, M., “Periodic Ellipsoidal Billiard Trajectories and Extremal Polynomials”, Comm. Math. Phys., 372:1 (2019), 183–211  crossref  mathscinet  zmath  adsnasa
    [9] Dragović, V. and Radnović, M., “Caustics of Poncelet Polygons and Classical Extremal Polynomials”, Regul. Chaotic Dyn., 24:1 (2019), 1–35  mathnet  crossref  mathscinet  zmath  adsnasa
    [10] Genin, D., Khesin, B., and Tabachnikov, S., “Geodesics on an Ellipsoid in Minkowski Space”, Enseign. Math. (2), 53:3–4 (2007), 307–331  mathscinet  zmath
    [11] Khesin, B. and Tabachnikov, S., “Pseudo-Riemannian Geodesics and Billiards”, Adv. Math., 221:4 (2009), 1364–1396  crossref  mathscinet  zmath  elib
    [12] Kreĭn, M. G., Levin, B. Ya., and Nudel'man, A. A., “On Special Representations of Polynomials That Are Positive on a System of Closed Intervals, and Some Applications”, Functional Analysis, Optimization, and Mathematical Economics: A Collection of Papers Dedicated to the Memory of L. V. Kantorovich, ed. L. J. Leifman, Oxford Univ. Press, New York, 1990, 56–114  mathscinet  zmath
    [13] Wang, Y.-X., Fan, H., Shi, K.-J., Wang, Ch., Zhang, K., and Zeng, Y., “Full Poncelet Theorem in Minkowski dS and AdS Spaces”, Chin. Phys. Lett., 26:1 (2009), 010201, 4 pp.  crossref  adsnasa



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