Sta¨kel equivalent integrable Hamiltonian systems

The Stackel transform is a mapping of the commuting constants of the motion (corresponding to a separable coordinate system) for one completely integrable classical or quantum Hamiltonian system to the constants of the motion for another such system. Here the transform is defined and given an intrinsic characterization, and a large family of nontrivial examples is worked out of systems which are “Stackel equivalent”. Among the simplest examples are geodesic flow on an n-dimensional ellipsoid with distinct axes, which is equivalent to the motion of a mass point on the unit sphere in $R^{n + 1} $ under the influence of a quadratic potential with distinct eigenvalues, and the Kepler (Coulomb) problem in three dimensions which is equivalent to the pseudo-Coulomb problem.