The Jacobi-Rosochatius problem on an ellipsoid: the Lax representations and billiards

Abstract

The Lax representations of the geodesic flow, the Jacobi-Rosochatius problem and its perturbations by means of separable polynomial potentials, on a ellipsoid are constructed. We prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces. Also, we consider billiards within an ellipsoid under the influence of the Hook and Rosochatius potentials between the impacts. A geometric interpretation of the integrability analogous to the classical Chasles and Poncelet theorems is given.

29 pages, 1 figure, to appear in Archive for Rational Mechanics and Analysis

References in corpus (3)

• Torus Actions and Integrable Systems
• Symmetries and Integrability
• Geodesic Flows and Neumann Systems on Stiefel Varieties. Geometry and Integrability

Cited by in corpus (5)

• Algebraic geometry and stability for integrable systems
• Heisenberg model in pseudo-Euclidean spaces
• Heisenberg model in pseudo-Euclidean spaces II
• Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere
• Self-focal points of ellipsoids of dimension $\ge 3$