Orthogonal Separation of the Hamilton–Jacobi Equation on Spaces of Constant Curvature

We review the theory of orthogonal separation of variables of the Hamilton– Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins–Eisenhart–Miller coordinates. Then we present the Benenti–Eisenhart–Kalnins–Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins–Eisenhart–Miller coordinates which separate a given natural Hamilton–Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti–Eisenhart–Kalnins–Miller separation algorithm to study the separability of the three dimensional Calogero–Moser and Morosi–Tondo systems.