Integrable systems on the sphere, ellipsoid and hyperboloid

TL;DR

This paper explores the relationships between integrable systems on different quadratic surfaces in Euclidean space, demonstrating how affine transformations and coupling constant changes can generate new integrable systems with polynomial invariants.

Contribution

It introduces a method to derive integrable systems on hyperboloids from those on spheres and ellipsoids using affine transformations and coupling constant transformations.

Findings

Real integrals of motion on hyperboloids can be obtained from sphere systems.

Examples include systems with cubic, quartic, and sextic polynomial invariants.

The approach unifies integrable systems across different quadratic surfaces.

Abstract

Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant transformations we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.