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Polygons, conics and billiards

  • Ronaldo A. Garcia [1]
    1. [1] Universidade Federal de Goiás

      Universidade Federal de Goiás

      Brasil

  • Localización: Materials matemàtics, ISSN-e 1887-1097, Nº. 0, 2020
  • Idioma: inglés
  • Enlaces
  • Resumen

    • In this expository article we will describe some elementary properties of billiards and Poncelet maps and special attention is dedicated to N-periodic orbits. In general, problems involving billiards are easy to state and under-standing, and difficult or laborious to solve.

  • Referencias bibliográficas
    • Alkoumi, Naeem; Schlenk, Felix. Shortest closed billiard orbits on convex tables. Manuscripta Math. 147 (2015), no. 3–4, 365–380.
    • Arseniy Akopyan and Richard Schwartz and Serge Tabachnikov, Billiards in Ellipses Revisited, Eur. J. Math., 9. 2020, http://doi.org/ 10.1007/s40879-020-00426-9...
    • Akhmet, Marat. Principles of discontinuous dynamical systems. Springer Verlag, New York, (2010) xii+176 pp.
    • Amiran, Edoh Lazutkin coordinates and invariant curves for outer billiards. J. Math. Phys. 36 (1995), no. 3, 1232–1241.
    • Arnold,V. I., Singularities of caustics and wave fronts, Math. Appl. 62, Kluwer, Dordrecht, 1990.
    • Avila, A., Kaloshin, V., de Simoi, J. An integrable deformation of an ellipse of small eccentricity is an ellipse, Ann. of Math., Vol. 184...
    • Arnold, M. Bialy, M. Nonsmooth convex caustics for Birkhoff billiards. Pacific J. Math. 295 (2018), no. 2, 257–269.
    • Barth, W. and Bauer, T. Poncelet Theorems, Exp. Math., 14 (1996), 125–144.
    • Berger, Marcel, Geometry Revealed: a Jacob’s Ladder to Higher Geometry. Springer Verlag, (2005) 831pp.
    • Berger, Marcel, A panoramic view of Riemannian geometry. Springer Verlag, (2002).
    • Bezdek, K., Connelly, R. Covering curves by translates of a convex set. Am. Math. Monthly 96(9),789–806 (1989).
    • Bialy, M. and Mronovthe, E. The Birkhoff-Porstsky conjecture for centrally-symmetric billiard tables. arXiv:2008.03566v2 (2020).
    • Birkhoff, G. On the periodic motions of dynamical systems. Acta Mathematica, 50(1):359–379 (1927).
    • Birkhoff, George D. Dynamical systems. With an addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. IX American...
    • Brogliato, Bernard.Nonsmoothimpactmechanics.Models, dynamics and control. Lecture Notes in Control and Information Sciences, 220. Springer-Verlag...
    • Boyland, P. Dual billiards, twist maps and impact oscillators Nonlinearity 9 (1996) 1411–1438. Bryant R. Poncelet’s theorem. http://arimoto.lolipop.jp/...
    • Bryant R. Poncelet’s theorem. http://arimoto.lolipop.jp/ PonceletforBMC.pdf
    • Bunimovich, Leonid A. Mechanisms of chaos in billiards: dispersing, defocusing and nothing else. Nonlinearity 31 (2018), no. 2, R78–R92.
    • Chernov, Nikolai; Markarian, Roberto Introduction to the ergodic theory of chaotic billiards. Monografías del Instituto de Matemática y Ciencias...
    • Cima, Anna; Gasull, Armengol; Mañosa, Víctor.OnPoncelet’smaps. Comput. Math. Appl. 60 (2010), no. 5, 1457–1464.
    • Coriolis, G.-G. Gaspard-Gustave de Coriolis,G.G. Théorie Mathématique des Effets du Jeu de Billiard, Carilian-Goeury, Paris (1835).
    • H. T. Croft, H. T.; Swinnerton-Dyer, H. P. F. On the Steinhaus billiard table problem. Volume 59(1) (1963), pp. 37–41
    • Darboux, G. Principes de Géométrie Analytique. Paris: GauthierVillars (1917).
    • Del Centina, Andrea Poncelet’s porism: a long story of renewed discoveries, I. Arch. Hist. Exact Sci. 70 (2016), no. 1, 1–122.
    • Del Centina, Andrea Poncelet’s porism: a long story of renewed discoveries, II. Arch. Hist. Exact Sci. 70 (2016), no. 2, 123–173.
    • Del Centina, Andrea Pascal’s mystichexagram, and a conjectural restoration of his lost treatise on conic sections. Arch. Hist. Exact Sci....
    • Cieślak, W. ; Mozgawa, W. In search of a measure in Poncelet’s porism. Acta Math. Hungar. 149 (2016), no. 2, 338–345.
    • Charles, M. Propriétés genérales des arcs d’une section conique dont la différence est rectifiable. Comptes Rendus Hebdomadaires des Séances...
    • Connes, A., Zagier, D. A property of parallelograms inscribed in ellipses. The American Math. Monthly, 114(10): 909?914. (2007).
    • Coxeter, H. Projective Geometry. Second Editon. Springer Verlag (1987).
    • Dias Carneiro, Mário Jorge; Oliffson Kamphorst, Sylvie; Pinto de Carvalho, Sônia. Elliptic islands in strictly convex billiards. Ergodic Theory...
    • Dragović, V.(2011).Poncelet-Darbouxcurves, theircompletedecomposition and Marden theorem. Int. Math. Res. Not. IMRN no. 15, 3502–3523.
    • Dragović, V., Radnović, M. (2014). Bicentennial of the great Poncelet theorem (1813–2013): current advances. Bull. Amer. Math. Soc. (N.S.),...
    • Dragović, Vladimir; Radnović, Milena Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Frontiers...
    • M. Farber and S. Tabachnikov, Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards, Topology 41...
    • Fierobe, C. Complex caustics of the elliptic billiard. arXiv:1904.03706v6. Arnold Math J. (2020). https://doi.org/10. 1007/s40598-020-00152-w...
    • Flatto, Leopold. Poncelet’s theorem. Chapter 15 by S. Tabachnikov. American Mathematical Society, Providence, RI, (2009) xvi+240 pp.
    • Fokicheva, V. V. Topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics. Sb. Math. 206 (2015),...
    • Fuchs, Dmitry; Tabachnikov, Serge. Mathematical omnibus. Thirty lectures on classic mathematics. American Mathematical Society, Providence,...
    • Gaivão, José Pedro. Asymptotic periodicity in outer billiards with contraction. Ergodic Theory Dynam. Systems 40 (2020), no. 2, 402– 417.
    • G. Galperin, A. Stepin, and Y. Vorobets. Periodic orbits in polygons; generating mechanisms, Russian Math. Surueys 47 (1992), 5–80.
    • Garcia, R., EllipticBilliardsandEllipsesAssociatedtothe3-Periodic Orbits, American Mathematical Monthly, 126(06), (2019) 491–504.
    • Garcia, R., Reznik, D. and Koiller J. Loci of 3-periodics in Ell. Billiard: why so many ellipses? Arxiv. (2020).
    • Garcia, R., Reznik, D. Invariants of Simple and Self-Intersected NPeriodics in the Elliptic Billiard. Arxiv (2020).
    • Garcia, R., Reznik, D. and Koiller J. Loci of 3-periodics in elliptic billiard: why so many ellipses? Arxiv (2020).
    • Garcia, R., Reznik, D. and Koiller J. New properties of elliptic billiards. Amer. Math. Monthly. To appear (2021).
    • Garcia, Ronaldo; Sotomayor, Jorge Differential equations of classical geometry, a qualitative theory. Publicações Matemáticas do IMPA. 27o...
    • Genin, Daniel. Hyperbolic outer billiards: a first example. Nonlinearity 19 (2006), no. 6, 1403–1413.
    • M. Ghomi, Shortest periodic billiard trajectories in convex bodies. Geom. Funct. Anal. 14(2), 295–302 (2004).
    • Glaeser, G., Odehnal B., Stachel, H. (2016). The Universe of Conics: From the Ancient Greeks to 21st Century Developments. New York: Springer...
    • Guillemin, Victor; Melrose, Richard An inverse spectral result for elliptical regions in R2. Adv. in Math. 32 (1979), no. 2, 128–148.
    • Glutsyuk, A., Shustin, E. On polynomially integrable planar outer billiards and curves with symmetry property. Math. Ann. 372, 1481– 1501...
    • Griffiths, P., Harris, J. On Cayley’s explicit solution to Poncelet’s porism, Enseign. Math. 24(1-2): 31–40 (1978).
    • Gruber, Peter M. Convex billiards. Geom. Dedicata 33 (1990), no. 2, 205–226.
    • Gutkin, E. Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn. 8(1):1–13 (2003).
    • Gutkin, E. Billiards in Polygons. Physica D 19, 311–333, (1986).
    • Gutkin, E. ; Katok, A. Caustics for inner and outer billiards. Communications in Mathematical Physics 173,(1995) pp. 101–133.
    • Gutkin, E. Two Applications of Calculus to Triangular Billiards. Amer. Math. Monthly. 104(7) pp. 618–622 (1997).
    • Hasselblatt,Boris; Katok,Anatole.Afirstcourseindynamics.Witha panorama of recent developments. Cambridge University Press, New York, (2003)....
    • Hassenblatt B. e Katok A. Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge (1995).
    • Halbeisen L., Hungerbühler N. (2015). A Simple Proof of Poncelet’s Theorem(ontheOccasionofItsBicentennial).Amer. Math. Monthly, 122(6): 537–551....
    • Halbeisen, Lorenz, Hungerbühler, Norbert. Closed chains of conics carrying Poncelet triangles. Beitr. Algebra Geom. 58 (2017), no. 2, 277–302....
    • Hénon, M. Chaotic scattering modelled by an inclined billiard. Progress in chaotic dynamics. Phys. D 33 (1988), no. 1–3, 132–156.
    • Holmes, P. J. The dynamics of repeated impacts with a sinusoidally vibrating table. J. Sound Vibration 84 (1982), no. 2, 173–189.
    • Kaloshin V, Sorrentino A. On the integrability of Birkhoff billiards. Phil. Trans. R. Soc. A 376: 20170419 (2018)
    • Kaloshin, Vadim; Sorrentino, Alfonso. On the local Birkhoff conjectureforconvexbilliards.Ann.ofMath.(2)188(2018),no.1,315–380.
    • Kozlova, T. V. Nonintegrability of a rotating elliptic billiard. J. Appl. Math. Mech. 62 (1998), no. 1, 81–85 .
    • Kamphorst, Sylvie Oliffson; Pinto-de-Carvalho, Sônia. The first Birkhoff coefficient and the stability of 2-periodic orbits on billiards. Experiment....
    • Katok, A. B. Billiard table as a playground for a mathematician. In V. Prasolov & Y. Ilyashenko (Eds.), Surveys in modern mathematics,...
    • Kimberling, C.: Encyclopedia of triangle centers (2019). https:// faculty.evansville.edu/ck6/encyclopedia/ETC.html
    • J. L. King. Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609–628.
    • Klingenberg, Wilhelm P. A. Riemannian geometry. Second edition. De Gruyter Studies in Mathematics, 1. Walter de Gruyter & Co., Berlin,...
    • Kołodziej, Rafał. The rotation number of some transformation related to billiards in an ellipse. Studia Math. 81 (1985), no. 3, 293– 302.
    • Kozlov, Valeri˘ı V.; Treshchëv, Dmitri˘ı V. Billiards. A genetic introduction to the dynamics of systems with impacts. Translations of Mathematical...
    • Langevin, Remi ; Levitt, Gilbert ; Rosenberg, Harold. Classes d’homotopiedesurfacesavecrebroussementsetqueuesd’arondedans R3. Canad. J. Math....
    • Lebesgue, H. Les coniques. Gauthier Villars. Paris. (1942).
    • Levi, M., Tabachnikov, S. . The Poncelet Grid and Billiards in Ellipses. American Math. Monthly, 114:895–908 (2007).
    • Lopes, Artur O. ; Sebastiani, Marcos. Poncelet pairs and the twist map associated to the Poncelet billiard. Real Anal. Exchange 35 (2010),...
    • Marco, Jean-Pierre.Entropyofbilliardmapsandadynamicalversion of the Birkhoff conjecture. J. Geom. Phys. 124 (2018), 413–420.
    • Marò, Stefano. Chaotic dynamics in an impact problem. Ann. Henri Poincaré 16 (2015), no. 7, 1633–1650.
    • Mather, J. Non-existence of invariant circles. Ergod. Th. Dyn. Syst. 4 (1984), 301–309.
    • Melo, W. and Palis, J. Introdução aos Sistemas Dinâmicos, Projeto Euclides, IMPA, 2a. edição. (2018).
    • Mirman, Boris. Explicit solutions to Poncelet’s porism. Linear Algebra Appl. 436 (2012), no. 9, 3531–3552.
    • Moser, J.K. Is the solar system stable? Math. Intelligencer 1 no. 2, 65–71 (1978/79).
    • Moser, J. K. On Invariant Curves of Area–Preserving Mappings of an Annulus , Nachr. Akad. Wiss. Göttingen, Math.–Phys. Kl.II, (1962), 1–20.
    • Poncelet, J.-V. Traité des propriétés projectives des figures. Metz, Paris. (1822).
    • Popov, Georgi. Invariants of the length spectrum and spectral invariants of planar convex domains. Comm. Math. Phys. 161 (1994), no. 2, 335–364....
    • Poritsky, H. The billiard ball problem on a table with a convex boundary–an illustrative dynamical problem. Ann. of Math. (2) 51 (1950), 446–470.
    • Previato, E. Poncelet’s theorem in space. Proc. Amer. Math. Soc. 127(9) (1999), 2547–2556.
    • Ragazzo, Clodoaldo; Dias Carneiro, Mário Jorge; Adda-Zanata, Salvador. Introdução à Dinâmica de Aplicações do Tipo Twist, 25o Colóquio Brasileiro...
    • Reznik, Dan; Garcia, Ronaldo and Koiller, Jair. Can an elliptic billiard still surprise us? Math. Intelligencer vol.42, (2020) 6–17.
    • Reznik, Dan; Garcia, Ronaldo and Koiller, Jair. Eighty New Invariants in the Elliptic Billiard. ArXiv:2004.12497v11.
    • Romaskevich, O. On the incenters of triangular orbits on elliptic billiard. Enseign. Math. 60(3-4):247–255 (2014)
    • Romaskevich, O. Dynamique des systèmes physiques, formes normales et chaînes de Markov. These de Doctarat de L’Université de Lyon. (2016).
    • Rozikov, Utkir A. An Introduction to Mathematical Billiards. World Scientific Publishing Company, Singapore (2019)
    • M. R. Rychlik, Periodic points of the billiard ball map in a convex domain. J. Differential Geom. 30 (1):91–205 (1989).
    • Schwartz, Richard E. The plaid model. Annals of Mathematics Studies, 198. Princeton University Press, Princeton, NJ, (2019).
    • R.E. Schwartz, R. Outer billiards on kites. Annals of Mathematics Studies. 171. Princeton University Press (2009).
    • Schwartz, Richard E. Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. Experiment. Math. 18 (2009), no....
    • Schwartz, R. The Poncelet grid. Adv. Geom. 7(2): 157-175 (2007).
    • Schwartz, R., Tabachnikov, S. Centers of Mass of Poncelet Polygons, 200 Years After. Math. Intelligencer 38(2): 29–34 (2016).
    • I. J. Schoenberg, On Jacobi-Bertrand’s proof of a theorem of Poncelet. Studies in pure mathematics, 623–627, Birkhäuser, Basel, (1983).
    • Sinai, Ya. G. Introduction to ergodic theory. Princeton University Press, Princeton, NJ, (1977).
    • Sinai, Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Usp. Mat. Nauk 25(2 (152)), 141– 192 (1970)....
    • Sorrentino, Alfonso. Mathematicians play... billiards! Mat. Cult. Soc. Riv. Unione Mat. Ital. (I) 4 (2019), no. 2, 131–144.
    • Sotomayor, J. Lições de Equações Diferenciais Ordinárias. Projeto Euclides, IMPA (1979).
    • Tabachnikov, S. Geometry and Billiards, vol. 30 of Student Mathematical Library. Providence, RI: American Mathematical Society (2005).
    • Tabachnikov, S. Outer billiards. Russian Math. Surveys 48 (1993), no. 6, 81–109.
    • Tabachnikov, Serge. A baker’s dozen of problems. Arnold Math. J. 1 (2015), no. 1, 59–67.
    • Tumanov, A. Scarcity of Periodic Orbits in Outer Billiards. J Geom Anal 30, 2479–2490 (2020).
    • Vedyushkina, V. V. ; Fomenko, A. T. Integrable topological billiards and equivalent dynamical systems. Izv. Math. 81 (2017), no. 4, 688– 733....
    • Vorobets, Ya. B. On the measure of the set of periodic points of a billiard. Math. Notes 55 (1994), no. 5-6, 455–460
    • Weisstein, E.: Mathworld (2019). http://mathworld.wolfram.com
    • Wojtkowski, Maciej. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys. 105 (1986), no. 3, 391–414.
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