Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

Abstract

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian H = T +V into a geodesic Hamiltonian T with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians T_r (r = a, b, c, d) in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential Ur leading to H_r = T_r +U_r. Secondly, we study the superintegrability of the four Hamiltonians tilde{H}_r = H_r/μ_r, where μ_r is a certain position-dependent mass, that enjoys the same separability as the original system H_r. All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.