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Implementation of Integrable Systems by Topological, Geodesic Billiards with Potential and Magnetic Field

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Abstract

In the paper, eight classes of integrable billiards are studied; in particular, classes introduced by the authors: elementary, topological, billiard books, billiards on the Minkowski plane, geodesic billiards on quadrics in three-dimensional Euclidean space, billiards in a magnetic field, and also a class containing all of the ones above. It turns out that, in the class of billiard books, topological obstacles to implementation occurred, for example, for the “twisted” Lagrange top (as we conventionally call a modification of the usual Lagrange top which we had discovered) for one of energy zones. We indicate this obstacle explicitly. It turns out further that this system can still be implemented in the class of magnetic billiards.

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Acknowledgment

The research was carried out as a part of the Program of the President of the Russian Federation for State Support of Leading Scientific Schools of the Russian Federation (grant NSh-6399.2018.1, agreement no. 075-02-2018-867) and was also supported by the Russian Foundation for Basic Research (grant 19-01-00775-a).

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991, Moscow, Russia

    A. T. Fomenko & V. V. Vedyushkina

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  1. A. T. Fomenko
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  2. V. V. Vedyushkina
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Correspondence to V. V. Vedyushkina.

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To the memory of Mikhail Karasev

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Fomenko, A.T., Vedyushkina, V.V. Implementation of Integrable Systems by Topological, Geodesic Billiards with Potential and Magnetic Field. Russ. J. Math. Phys. 26, 320–333 (2019). https://doi.org/10.1134/S1061920819030075

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