Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane

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Abstract

Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic field is considered in this paper. The question of the existence of an integral of motion which is a polynomial in the components of the velocity is investigated. It is shown that if such an integral exists, then the boundary of the domain defines a non-singular algebraic curve in $\mathbb{C}^3$. It is also shown that for a domain other than a geodesic disk, magnetic billiards does not admit a polynomial integral for all but perhaps finitely many values of the magnitude of the magnetic field. To prove our main theorems a new dynamical system, ‘outer magnetic billiards’, on a constant-curvature surface is introduced, a system ‘dual’ to magnetic billiards. By passing to this dynamical system one can apply methods of algebraic geometry to magnetic billiards.Bibliography: 30 titles.

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

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