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Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia, 2024, Volume 515, Pages 5–10
DOI: https://doi.org/10.31857/S2686954324010018 (Mi danma485) 		 
MATHEMATICS

Rotation functions of integrable billiards as orbital invariants

G. V. Belozerova, A. T. Fomenkoab

a Lomonosov Moscow State University, Moscow, Russia
b Moscow Center for Fundamental and Applied Mathematics
DOI: https://doi.org/10.31857/S2686954324010018
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 was studied, and edge orbital invariants (rotation vectors) were 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.
Keywords: integrable system, integrable billiard, rotation functions, orbital invariants.
Funding agency Grant number
Russian Science Foundation 22-71-00111
Foundation for the Advancement of Theoretical Physics and Mathematics BASIS
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”.
Received: 15.12.2023
Accepted: 20.01.2024
English version:
Doklady Mathematics, 2024, Volume 109, Issue 1, Pages 1–5
DOI: https://doi.org/10.1134/S1064562424701722
Bibliographic databases:
Document Type: Article
UDC: 517.938.5
Language: Russian
Citation: G. V. Belozerov, A. T. Fomenko, “Rotation functions of integrable billiards as orbital invariants”, Dokl. RAN. Math. Inf. Proc. Upr., 515 (2024), 5–10; Dokl. Math., 109:1 (2024), 1–5
Citation in format AMSBIB
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\by G.~V.~Belozerov, A.~T.~Fomenko
\paper Rotation functions of integrable billiards as orbital invariants
\jour Dokl. RAN. Math. Inf. Proc. Upr.
\yr 2024
\vol 515
\pages 5--10
\mathnet{http://mi.mathnet.ru/danma485}
\crossref{https://doi.org/10.31857/S2686954324010018}
\elib{https://elibrary.ru/item.asp?id=67973242}
\transl
\jour Dokl. Math.
\yr 2024
\vol 109
\issue 1
\pages 1--5
\crossref{https://doi.org/10.1134/S1064562424701722}
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