Closed geodesics and billiards on quadrics related to elliptic KdV solutions

Details Closed geodesics and billiards on quadrics related to elliptic KdV solutions Closed geodesics and billiards on quadrics related to elliptic KdV solutions

2004-12-14

arxiv.org/abs/nlin/0412034v1

Letters in Mathematical Physics (2006) 76:111-134

Nonlinear Sciences

Exactly Solvable and Integrable Systems

23 pages, Latex, 1 figure Postscript

Scientific paper

10.1007/s11005-006-0065-7

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero system

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