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

TL;DR

This paper constructs Lax representations for the Jacobi-Rosochatius problem on ellipsoids, proves integrability for symmetric cases, and explores billiard systems with potentials, providing geometric interpretations akin to classical theorems.

Contribution

It introduces Lax representations for the problem, establishes integrability for symmetric ellipsoids, and extends the analysis to billiards with potentials, offering new geometric insights.

Findings

Lax representations for geodesic flow and Jacobi-Rosochatius problem.

Complete integrability on generic symmetric ellipsoids.

Geometric interpretation similar to classical theorems.

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.