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Cayley-type conditions for billiards within \(k\) quadrics in \(\mathbb R^d\). (English) Zbl 1108.37041

J. Phys. A, Math. Gen. 37, No. 4, 1269-1276 (2004); corrigendum 38, No. 36, 7927 (2005).
Summary: The notions of reflection from outside, reflection from inside and signature of a billiard trajectory within a quadric are introduced. Cayley-type conditions for periodical trajectories for the billiard in the region bounded by \(k\) quadrics in \(\mathbb R^d\) and for the billiard ordered game within \(k\) ellipsoids in \(\mathbb R^d\) are derived. In a limit, the condition describing periodic trajectories of billiard systems on a quadric in \(\mathbb R^d\) is obtained.
In their corrigendum the authors correct the statements of Theorems 3 and 4.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
70G55 Algebraic geometry methods for problems in mechanics
70J10 Modal analysis in linear vibration theory
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