Integrable potentials on spaces with curvature from quantum groups

Abstract

A family of classical integrable systems defined on a deformation of the twodimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed through Hamiltonians defined on the non-standard quantum deformation of a sl(2) Poisson coalgebra. All these spaces have a non-constant curvature that depends on the deformation parameter z. As particular cases, the analogues of the harmonic oscillator and Kepler-Coulomb potentials on such spaces are proposed. Another deformed Hamiltonian is also shown to provide superintegrable systems on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant curvature that exactly coincides with z. According to each specific space, the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers. The non-deformed limit z → 0 of all these Hamiltonians can then be regarded as the zero-curvature limit (contraction) which leads to the corresponding (super)integrable systems on the flat Euclidean and Minkowskian spaces. PACS: 02.30.lk 02.20.UwRecently, this integrability-preserving deformation procedure has been used to introduce both superintegrable and integrable free motions on two-dimensional (2D) spaces with curvature, either constant or variable, respectively [12]. Therefore one could expect that potential terms can also be considered, in such a way that the coalgebra approach should provide (super)integrable potentials on curved spaces. The aim of the present paper is to prove this assertion through the construction of some relevant Hamiltonians.In order to make these ideas more explicit, let us consider the non-standard quantum deformation of sl (2) [13] written as a Poisson coalgebra (sl z (2), ∆ z ) with (deformed) Poisson brackets, coproduct and Casimir given by {J 3 , J + } = 2J + cosh zJ − {J 3 , J − } = −2 sinh zJ − z {J − , J + } = 4J 3 (1.1) ∆ z (J − ) = J − ⊗ 1 + 1 ⊗ J − ∆ z (J i ) = J i ⊗ e zJ − + e −zJ − ⊗ J i i = +, 3 (1.2)

Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations

“…3 ) determines a family of integrable systems as the three functionally independent functions {H z , C (2) z , C (3) z } are mutually in involution.…”

“…We now consider the Hamiltonian H S z = 1 2 J (3) + e zJ (3) − which has four (functionally independent) constants of motion, namely C (2) z (6), C (3) z (13), I (2) z (9) and [4] I ( …”

“…3 ) gives rise to an integrable system, for which C (2) z is the constant of the motion. In particular, we find a large family of integrable deformations of the free motion of a particle on the 2D Euclidean space defined by…”

“…Likewise, their "deformed" counterpart (understood as spaces with non-constant curvature) have been deduced from the integrable metric (8). The explicit solution of the geodesic flows for all these spaces has been studied in [2], as well as a method to introduce (super)integrable potentials on them by adding a potential term of the form U (zJ (2) − ); in this way, some known potentials are recovered (appearing in the classifications [6,7]) and also new ones are obtained. We recall that another approach to superintegrability on 2D spaces of variable curvature can be found in [8,9].…”

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

“…Some of these variable curvature systems in 2D and 3D have been already studied (see [31,32,33]), and we present here the most significant elements for their N D generalizations. We will show that this scheme is quite efficient in order to get explicitly a large family of QMS systems.…”