An elliptic billiard in a potential force field: classification of motions, topological analysis

I. F. Kobtsev

Sbornik. Mathematics, Tome 211 (2020) no. 7, pp. 987-1013

Résumé

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.

Détail

Keywords: integrable Hamiltonian system, potential
Mots-clés : billiard in an ellipse, Liouville foliation, bifurcations.

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