Closed Geodesics and Billiards on Quadrics related to elliptic KdV solutions
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 Birkho® billiards in Rn. Namely, generic complex invariant manifolds are not Abelian v...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2004 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/929 |
| Online Access: | https://hdl.handle.net/2117/929 |
| Access Level: | Open access |
| Keyword: | Hamiltonian dynamical systems Lagrangian functions Hamiltonian systems Differentiable dynamical systems Curves elliptic KdV solutions Hamilton, Sistemes de Lagrange, Funcions de Sistemes dinàmics diferenciables Corbes Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::14 Algebraic geometry::14H Curves Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior Classificació AMS::70 Mechanics of particles and systems::70H Hamiltonian and Lagrangian mechanics |
| Summary: | 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 Birkho® billiards in Rn. 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 su±cient 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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