Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane
- Authors: Bialy M.L.1, Mironov A.E.2,3
-
Affiliations:
- Tel Aviv University, School of Mathematical Sciences
- Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
- Novosibirsk State University
- Issue: Vol 74, No 2 (2019)
- Pages: 3-26
- Section: Articles
- URL: https://journal-vniispk.ru/0042-1316/article/view/133551
- DOI: https://doi.org/10.4213/rm9871
- ID: 133551
Cite item
Abstract
About the authors
Misha L. Bialy
Tel Aviv University, School of Mathematical Sciences
Andrei Evgen'evich Mironov
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences; Novosibirsk State University
Email: mironov@math.nsc.ru
Doctor of physico-mathematical sciences, no status
References
- P. Albers, G. D. Banhatti, M. Herrmann, Numerical simulations of magnetic billiards in a convex domain in ${mathbb R}^2$, 2017, 10 pp.
- A. Avila, J. De Simoi, V. Kaloshin, “An integrable deformation of an ellipse of small eccentricity is an ellipse”, Ann. of Math. (2), 184:2 (2016), 527–558
- N. Berglund, H. Kunz, “Integrability and ergodicity of classical billiards in a magnetic field”, J. Statist. Phys., 83:1-2 (1996), 81–126
- M. Bialy, “On totally integrable magnetic billiards on constant curvature surface”, Electron. Res. Announc. Math. Sci., 19 (2012), 112–119
- M. Bialy, A. E. Mironov, “Algebraic non-integrability of magnetic billiards”, J. Phys. A, 49:45 (2016), 455101, 18 pp.
- М. Бялый, А. Е. Миронов, “О полиномиальных интегралах четвертой степени бильярда Биркгофа”, Современные проблемы механики, Сборник статей, Тр. МИАН, 295, МАИК “Наука/Интерпериодика”, М., 2016, 34–40
- M. Bialy, A. E. Mironov, “Angular billiard and algebraic Birkhoff conjecture”, Adv. Math., 313 (2017), 102–126
- M. Bialy, A. E. Mironov, “Algebraic Birkhoff conjecture for billiards on sphere and hyperbolic plane”, J. Geom. Phys., 115 (2017), 150–156
- M. Bialy, A. E. Mironov, “A survey on polynomial in momenta integrals for billiard problems”, Philos. Trans. Roy. Soc. A, 376:2131 (2018), 20170418, 19 pp.
- С. В. Болотин, “Интегрируемые биллиарды Биркгофа”, Вестн. Моск. ун-та. Сер. 1. Матем., мех., 1990, № 2, 33–36
- С. В. Болотин, “Интегрируемые бильярды на поверхностях постоянной кривизны”, Матем. заметки, 51:2 (1992), 20–28
- A. Glutsyuk, “On polynomially integrable Birkhoff billiards on surfaces of constant curvature”, J. Eur. Math. Soc. (JEMS) (to appear)
- А. А. Глуцюк, “О двумерных полиномиально интегрируемых бильярдах на поверхностях постоянной кривизны”, Докл. РАН, 481:6 (2018), 594–598
- B. Gutkin, “Hyperbolic magnetic billiards on surfaces of constant curvature”, Comm. Math. Phys., 217:1 (2001), 33–53
- E. Gutkin, S. Tabachnikov, “Billiards in Finsler and Minkowski geometries”, J. Geom. Phys., 40:3-4 (2002), 277–301
- Р. Хартсхорн, Алгебраическая геометрия, Мир, М., 1981, 600 с.
- V. Kaloshin, A. Sorrentino, “On the local Birkhoff conjecture for convex billiards”, Ann. of Math. (2), 188:1 (2018), 315–380
- В. В. Козлов, “Полиномиальные законы сохранения для газа Лоренца и газа Больцмана–Гиббса”, УМН, 71:2(428) (2016), 81–120
- В. В. Козлов, Д. В. Трещeв, Биллиарды. Генетическое введение в динамику систем с ударами, Изд-во МГУ, М., 1991, 168 с.
- M. Robnik, M. V. Berry, “Classical billiards in magnetic fields”, J. Phys. A, 18:9 (1985), 1361–1378
- V. Schastnyy, D. Treschev, “On local integrability in billiard dynamics”, Exp. Math. (to appear) , publ. online 2017
- С. Л. Табачников, “Внешние биллиарды”, УМН, 48:6(294) (1993), 75–102
- S. Tabachnikov, Billiards, Panor. Synth., 1, Soc. Math. France, Paris, 1995, vi+142 pp.
- S. Tabachnikov, “Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem”, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, 233–250
- S. Tabachnikov, “On algebraically integrable outer billiards”, Pacific J. Math., 235:1 (2008), 89–92
- D. Treschev, “Billiard map and rigid rotation”, Phys. D, 255 (2013), 31–34
- Д. В. Трещев, “Об одной задаче сопряжения в динамике бильярда”, Избранные вопросы математики и механики, Сборник статей. К 150-летию со дня рождения академика Владимира Андреевича Стеклова, Тр. МИАН, 289, МАИК “Наука/Интерпериодика”, М., 2015, 309–317
- D. Treschev, “A locally integrable multi-dimensional billiard system”, Discrete Contin. Dyn. Syst., 37:10 (2017), 5271–5284
- A. P. Veselov, “Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space”, J. Geom. Phys., 7:1 (1990), 81–107
- B. L. van der Waerden, Einführung in die algebraische Geometrie, Grundlehren Math. Wiss., 51, Springer, Berlin, 1939, vii+247 pp.
Supplementary files
