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This article is cited in 12 scientific papers (total in 12 papers)
An elliptic billiard in a potential force field: classification of motions, topological analysis
I. F. Kobtsev Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
Abstract:
Given an ellipse ${\frac{x^2}{a}+\frac{y^2}{b}=1}$, $a>b>0$, we consider an absolutely elastic billiard in it with potential $\frac{k}{2}(x^2+y^2)+\frac{\alpha}{2x^2}+\frac{\beta}{2y^2}$, $a\geqslant0$, $\beta\geqslant0$. 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.
Received: 28.06.2019
Citation:
I. F. Kobtsev, “An elliptic billiard in a potential force field: classification of motions, topological analysis”, Sb. Math., 211:7 (2020), 987–1013
Linking options:
https://www.mathnet.ru/eng/sm9296https://doi.org/10.1070/SM9296 https://www.mathnet.ru/eng/sm/v211/i7/p93
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Abstract page: | 377 | Russian version PDF: | 67 | English version PDF: | 34 | References: | 42 | First page: | 8 |
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