Geometry of integrable billiards and pencils of quadrics

Journal de Mathématiques Pures et Appliquées - Tập 85 - Trang 758-790 - 2006
Vladimir Dragović1, Milena Radnović1,2
1Mathematical Institute SANU, Kneza Mihaila 35, 11000 Belgrade, Serbia and Montenegro
2Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel

Tài liệu tham khảo

Abenda Appell, 1880, Sur les fonctions hypergéométriques de deux variables et sur des équations lineaires aux derivées partielles, Comptes Rendus, 90, 296 Appell, 1926, Fonctions hypergéométriques et hypersphériques Arnol'd, 1978 Barth, 1996, Poncelet theorems, Expo. Math., 14, 125 Barth, 1993, Modular curves and Poncelet polygons, Math. Ann., 295, 25, 10.1007/BF01444875 Berger, 1987 Bertrand, 1852, J. Math., 17, 121 Bos, 1987, Poncelet's closure theorem, Expo. Math., 5, 289 Cayley, 1854, Developments on the porism of the in-and-circumscribed polygon, Philosophical Magazine, 7, 339 Chang, 1993, Elliptical billiard systems and the full Poncelet's theorem in n dimensions, J. Math. Phys., 34, 2242, 10.1063/1.530115 Chang, 1989, Billiard systems on quadric surfaces and the Poncelet theorem, J. Math. Phys., 30, 798, 10.1063/1.528399 Darboux, 1870, Sur les polygônes inscrits et circonscrits à l'ellipsoïde, Bulletin de la Société philomathique, 7, 92 Darboux, 1901, Archives Néerlandaises (2), 6, 371 Darboux, 1914 Dragović, 1996, On integrable potential perturbations of the Jacobi problem for the geodesics on the ellipsoid, J. Phys. A: Math. Gen., 29, L317, 10.1088/0305-4470/29/13/002 Dragovich, 1914, Integrable perturbations of the Birkgof billiard inside an ellipse, Prikladnaya Matematika i Mekhanika, 62, 166 Dragović, 2002, The Appell hypergeometric functions and classical separable mechanical systems, J. Phys A: Math. Gen., 35, 2213, 10.1088/0305-4470/35/9/311 Dragović, 1998, Conditions of Cayley's type for ellipsoidal billiard, J. Math. Phys., 39, 355, 10.1063/1.532317 Dragović, 1998, On periodical trajectories of the billiard systems within an ellipsoid in Rd and generalized Cayley's condition, J. Math. Phys., 39, 5866, 10.1063/1.532600 Dragović, 2004, Cayley-type conditions for billiards within k quadrics in Rd, J. Phys. A: Math. Gen., 37, 1269, 10.1088/0305-4470/37/4/014 V. Dragović, M. Radnović, Algebra of pencils of quadrics, billiards and closed geodesics, in press Griffiths, 1977, A Poncelet theorem in space, Comment. Math. Helvetici, 52, 145, 10.1007/BF02567361 Griffiths, 1978, On Cayley's explicit solution to Poncelet's porism, Enseign. Math., 24, 31 Gunning, 1966 Hermite, 1912 C. Jacobi, Vorlesungen über Dynamic. Gesammelte Werke, Supplementband, Berlin, 1884 Jakob, 1993, Moduli of Poncelet polygons, J. Reine Angew. Math., 436, 33 Knörrer, 1980, Geodesics on the ellipsoid, Invent. Math., 59, 119, 10.1007/BF01390041 Kozlov, 1991 Lebesgue, 1942 Moser, 1991, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139, 217, 10.1007/BF02352494 Moser, 1980, Geometry of quadrics and spectral theory, 147 J.V. Poncelet, Traité des propriétés projectives des figures, Mett-Paris, 1822 Vilenkin, 1995, Representations of Lie Groups and Special Functions, 10.1007/978-94-017-2885-0 Whittaker, 1927