Abstract
Orbital invariants of integrable billiards on two-dimensional book tables are studied at constant energy values. These invariants are calculated from rotation functions defined on one-parameter families of Liouville 2-tori. For two-dimensional billiard books, a complete analogue of Liouville’s theorem is proved, action–angle variables are introduced, and rotation functions are defined. A general formula for the rotation functions of such systems is obtained. For a number of examples, the monotonicity of these functions is studied, and edge orbital invariants (rotation vectors) are calculated. It turned out that not all billiards have monotonic rotation functions, as was originally assumed by A. Fomenko’s hypothesis. However, for some series of billiards, this hypothesis is true.
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ACKNOWLEDGMENTS
The authors are grateful to V.V. Vedyushkina and E.A. Kudryavtseva for valuable comments and their interest in this work.
Funding
This work was performed at Lomonosov Moscow State University and was supported by the Russian Science Foundation, project no. 22-71-00111.
Belozerov also acknowledges the support of the Foundation for Advancement of Theoretical Physics and Mathematics “BASIS.”
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Translated by I. Ruzanova
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Belozerov, G.V., Fomenko, A.T. Rotation Functions of Integrable Billiards As Orbital Invariants. Dokl. Math. (2024). https://doi.org/10.1134/S1064562424701722
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DOI: https://doi.org/10.1134/S1064562424701722