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 U r leading to H r = T r + U r . Secondly, we study the superintegrability of the four Hamiltonians 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.It is well known that the harmonic (isotropic) oscillator and the Kepler-Coulomb (KC) problem are integrable systems admitting additional constants of motion (Demkov-Fradkin tensor [1,2] and Laplace-Runge-Lenz vector, respectively). Systems endowed with this property are called superintegrable. It is also known that if a system is separable (Hamilton-Jacobi (HJ) separable in the classical case or Schrödinger separable in the quantum case), then it is integrable with integrals of motion of at most second-order in momenta. Thus, if a system admits multiseparability (separability in several different systems of coordinates) then it is endowed with 'quadratic superintegrability' (superintegrability with linear or quadratic integrals of motion).Fris et al. studied in [3] the two-dimensional (2D) Euclidean systems admitting separability in more than one coordinate system and they obtained four families of potentials V r , r = a, b, c, d, possessing three functionally independent integrals of motion (they were mainly interested in the quantum 2D Schrödinger equation but their results also hold at the classical level). Then other authors studied similar problems on higher-dimensional Euclidean spaces [4]-[6], on 2D spaces with a pseudo-Euclidean metric (Drach potentials) [7]-[10], and on curved spaces [11]-[21] (see [22] for a recent review on superintegrability that includes a long list of references).The superintegrability property is related with different formalisms and it can be studied by making use of different approaches, that is, proving that all bounded classical trajectories are closed, HJ separability, action-angle variables formalism, exact solvability, degenerate quantum energy levels, complex functions whose Poisson bracket with the Hamiltonian are proportional to themselves, etc. In this paper, we relate superintegrability with a geometric formalism introduced many years ago by Eisenhart [23].The theory of general relativity states that the motion of a particle under the action of gravitational forces is described by a geodesic in the 4D Riemannian spacetime. The Eisenhart formalism (also known as Eisenhart lift) associates to a sys...

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