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On fourth-degree polynomial integrals of the Birkhoff billiard. (English, Russian) Zbl 1371.37067

Proc. Steklov Inst. Math. 295, 27-32 (2016); translation in Tr. Mat. Inst. Steklova 295, 34-40 (2016).
Summary: We study the Birkhoff billiard in a convex domain with a smooth boundary \(\gamma\). We show that if this dynamical system has an integral which is polynomial in velocities of degree 4 and is independent with the velocity norm, then \(\gamma\) is an ellipse.

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
14H70 Relationships between algebraic curves and integrable systems
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