Integrability, Stäckel spaces, and rational potentials

Abstract

For a variety of classical mechanical systems embeddable into flat space with Cartesian coordinates {xi} and for which the Hamilton–Jacobi equation can be solved via separation of variables in a particular curvalinear system {uj}, we answer the following question. When is the separable potential function v expressible as a polynomial (or as a rational function) in the defining coordinates {xi}? Many examples are given.