Abstract:
Given an ellipse x2a+y2b=1, a>b>0, we consider an absolutely elastic billiard in it with potential k2(x2+y2)+α2x2+β2y2, a⩾0, β⩾0. This dynamical system is integrable and has two degrees of freedom. We obtain the iso-energy invariants of rough and fine Liouville equivalence, and conduct a comparative analysis of other systems known in rigid body mechanics. To obtain the results we apply the method of separation of variables and construct a new method, which is equivalent to the bifurcation diagram but does not require it to be constructed.
Bibliography: 17 titles.
Keywords:
integrable Hamiltonian system, billiard in an ellipse, potential, Liouville foliation, bifurcations.
This research was conducted in the framework of the Programme of the President of the Russian Federation for State Support of Leading Scientific Schools (grant no. НШ-6399.2018.1).
Citation:
I. F. Kobtsev, “An elliptic billiard in a potential force field: classification of motions, topological analysis”, Sb. Math., 211:7 (2020), 987–1013
\Bibitem{Kob20}
\by I.~F.~Kobtsev
\paper An elliptic billiard in a~potential force field: classification of motions, topological analysis
\jour Sb. Math.
\yr 2020
\vol 211
\issue 7
\pages 987--1013
\mathnet{http://mi.mathnet.ru/eng/sm9296}
\crossref{https://doi.org/10.1070/SM9296}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4133436}
\zmath{https://zbmath.org/?q=an:1448.37066}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2020SbMat.211..987K}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000573484500001}
\elib{https://elibrary.ru/item.asp?id=45280154}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85092096416}
Linking options:
https://www.mathnet.ru/eng/sm9296
https://doi.org/10.1070/SM9296
https://www.mathnet.ru/eng/sm/v211/i7/p93
This publication is cited in the following 13 articles:
Vivina L Barutello, Anna Maria Cherubini, Irene De Blasi, “Exploration of billiards with Keplerian potential”, Nonlinearity, 38:5 (2025), 055004
Airi Takeuchi, Lei Zhao, “Conformal transformations and integrable mechanical billiards”, Advances in Mathematics, 436 (2024), 109411
S. E. Pustovoitov, “Issledovanie struktury sloeniya Liuvillya integriruemogo ellipticheskogo billiarda s polinomialnym potentsialom”, Chebyshevskii sb., 25:1 (2024), 62–102
A. T. Fomenko, V. V. Vedyushkina, “Billiards and integrable systems”, Russian Math. Surveys, 78:5 (2023), 881–954
S. E. Pustovoitov, “Classification of singularities of the Liouville foliation of an integrable elliptical billiard with a potential of fourth degree”, Russ. J. Math. Phys., 30:4 (2023), 643
G. V. Belozerov, “Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space”, Sb. Math., 213:2 (2022), 129–160
A. T. Fomenko, V. V. Vedyushkina, “Evolutionary force billiards”, Izv. Math., 86:5 (2022), 943–979
G. V. Belozerov, “Topology of $5$-surfaces of a 3D billiard inside a triaxial ellipsoid with Hooke's potential”, Moscow University Mathematics Bulletin, 77:6 (2022), 277–289
S. E. Pustovoitov, “Topological Analysis of An Elliptic Billiard in a Fourth-Order Potential Field”, Mosc. Univ. Math. Bull., 76:5 (2021), 193–205
V. V. Vedyushkina, A. T. Fomenko, “Force evolutionary billiards and billiard equivalence of the Euler and Lagrange cases”, Dokl. Math., 103:1 (2021), 1–4
V. A. Kibkalo, A. T. Fomenko, I. S. Kharcheva, “Realizing integrable Hamiltonian systems by means of billiard books”, Trans. Moscow Math. Soc., 82 (2021), 37–64
S. E. Pustovoitov, “Topological analysis of a billiard bounded by confocal quadrics in a potential field”, Sb. Math., 212:2 (2021), 211
Anatoly T. Fomenko, Vladislav A. Kibkalo, Understanding Complex Systems, Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, 2021, 3
Citation in format AMSBIB
Contact us: