Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds

Abstract

The additive variable separation in the Hamilton–Jacobi equation is studied for a natural Hamiltonian with scalar and vector potentials on a Riemannian manifold with positive–definite metric. The separation of this Hamiltonian is related to the separation of a suitable geodesic Hamiltonian over an extended Riemannian manifold. Thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of Killing webs, Killing tensors, and Killing vectors. The results are applicable to the case of a nondegenarate separation on a manifold with indefinite metric, where no null essential separable coordinates occur.