On Differential Equations of Integrable Billiard Tables

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ℝ 3 with a first integra...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2024, Vol.40 (1), p.417-424
Main Authors: Dragović, Vladimir, Mironov, Andrey E.
Format: Article
Language:English
Subjects:
Billiards
Differential equations
Mathematical analysis
Mathematics
Mathematics and Statistics
Toruses
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Summary:We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ℝ 3 with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ℝ 3 homeomorphic to a torus, being a table of a billiard admitting two additional first integrals.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-024-2450-5