Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature

Abstract

An infinite family of classical superintegrable Hamiltonians defined on the Ndimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N − 3) functionally independent constants of the motion. Among them, two different subsets of N integrals in involution (including the Hamiltonian) can always be explicitly identified. As particular cases, we recover in a straightforward way most of the superintegrability properties of the Smorodinsky-Winternitz and generalized Kepler-Coulomb systems on spaces of constant curvature and we introduce as well new classes of (quasi-maximally) superintegrable potentials on these spaces. Results here presented are a consequence of the sl(2, R) Poisson coalgebra symmetry of all the Hamiltonians, together with an appropriate use of the phase spaces associated to Poincaré and Beltrami coordinates.

Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

“…The results above summarized allows for a straightforward superintegrable even-order anharmonic oscillator perturbation given by [1]:…”

“…Therefore, the remaining functionally independent constant of the motion does exist and, therefore, it has to be found by direct methods. Such an additional integral can be shown to be any of the following N functions [1]:…”

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

“…Let us briefly recall the main result of [30] that provides an infinite family of QMS Hamiltonians. We stress that, although some of these Hamiltonians can be interpreted as motions on spaces with constant curvature, this approach to QMS systems is quite general, and also nonnatural Hamiltonian systems (for instance, those describing static electromagnetic fields) can be obtained.…”

“…tan( √ κ r) (see [30] for the expression of this quantity in terms of Poincaré and Beltrami coordinates). Also, for the sake of simplicity, the centrifugal terms coming from the symplectic realization with arbitrary b i will be expressed in ambient coordinates x i [30]:…”

“…In the the next Section we will briefly summarize the N D constant curvature construction given in [30], that makes use of an sl(2, R) Poisson coalgebra symmetry. The generic variable curvature approach will be obtained in Section 3 through a non-standard quantum deformation of an sl(2, R) Poisson coalgebra.…”