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Liouville Foliations of Topological Billiards with Slipping

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Abstract

In the paper, a new class of integrable billiards, namely, billiards with slipping, is studied. At the reflection from the boundary, a billiard particle of such a system may not only change its velocity, but also move some distance along the border. Some laws of slipping preserve the integrability of flat confocal and circular billiards and billiard books glued from them, i.e., billiards on cell complexes. In the paper, the topology of the Liouville foliations for several integrable billiards with slipping, both flat and locally flat, is studied. Two such systems are Liouville equivalent to integrable geodesics flows of small degrees on nonorientable two-dimensional surfaces, namely, the projective plane and the Klein bottle. This shows that the nonorientability of a two-dimensional surface, in itself, is not an obstacle to its implementability by an appropriate integrable billiard.

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Acknowledgments

The authors are grateful to Vladislav Kibkalo for valuable comments and remarks.

Funding

The research in Sec. 1 of this work was funded by Russian Foundation for Basic Research (project 19-01-00775a) and done at Moscow Center for Fundamental and Applied Mathematics. The research in Sections 2-5 was supported by the Russian Science Foundation (project 20-71-00155) and done at Lomonosov Moscow State University.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    A. T. Fomenko, V. V. Vedyushkina & V. N. Zav’yalov

  2. Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

    A. T. Fomenko, V. V. Vedyushkina & V. N. Zav’yalov

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

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Fomenko, A.T., Vedyushkina, V.V. & Zav’yalov, V.N. Liouville Foliations of Topological Billiards with Slipping. Russ. J. Math. Phys. 28, 37–55 (2021). https://doi.org/10.1134/S1061920821010052

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