Cayley-Type Conditions for Billiards within $k$ Quadrics in $R^d$

Details Cayley-Type Conditions for Billiards within $k$ Quadrics in $R^d$ Cayley-Type Conditions for Billiards within $k$ Quadrics in $R^d$

2005-03-22

arxiv.org/abs/math-ph/0503053v1

J. Phys. A: Math. Gen. 37 (2004), 1269-1276

Physics

Mathematical Physics

10 pages, some corractions are made in Section 4

Scientific paper

10.1088/0305-4470/37/4/014

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 $R^d$ and for the billiard ordered game within $k$ ellipsoids in $R^d$ are derived. In a limit, the condition describing periodic trajectories of billiard systems on a quadric in $R^d$ is obtained.

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