Superintegrable potentials on 3D Riemannian and Lorentzian spaces with nonconstant curvature
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ELEMENTARY PARTICLES AND FIELDS Theory
Superintegrable Potentials on 3D Riemannian and Lorentzian Spaces with Nonconstant Curvature* A. Ballesteros1)** , A. Enciso2)*** , F. J. Herranz3)**** , and O. Ragnisco4)***** Received April 17, 2009
Abstract—A quantum sl(2, R) coalgebra (with deformation parameter z) is shown to underly the construction of a large class of superintegrable potentials on 3D curved spaces, that include the nonconstant curvature analog of the spherical, hyperbolic, and (anti-)de Sitter spaces. The connection and curvature tensors for these “deformed” spaces are fully studied by working on two different phase spaces. The former directly comes from a 3D symplectic realization of the deformed coalgebra, while the latter is obtained through a map leading to a spherical-type phase space. In this framework, the nondeformed limit z → 0 is identified with the flat contraction leading to the Euclidean and Minkowskian spaces/potentials. The resulting Hamiltonians always admit, at least, three functionally independent constants of motion coming from the coalgebra structure. Furthermore, the intrinsic oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of nonconstant curvature are identified, and several examples of them are explicitly presented. DOI: 10.1134/S1063778810020092
1. INTRODUCTION In the context of Hamiltonian systems with an arbitrary finite number of degrees of freedom, a deep connection between the coalgebra symmetry of a given system and its Liouville integrability was firmly established in [1]. Moreover, the intrinsic superintegrability properties of the coalgebra construction were further explored in [2]. Since then, this framework has lead to the coalgebra interpretation of the integrability properties of many well-known systems, as well as to the construction of many new superintegrable systems by using both Lie and q-Poisson coalgebras (see [1–4] and references therein). In particular, by making use of the Poisson coalgebra given by the nonstandard quantum deformation of sl(2, R), the construction of integrable 2D geodesic flows corresponding to 2D Riemannian and ∗
The text was submitted by the authors in English. Departamento de F ´ısica, Facultad de Ciencias, Universidad de Burgos, Spain. 2) ´ Departamento de F ´ısica Teorica II, Universidad Complutense, Madrid, Spain. 3) ´ Departamento de F ´ısica, Escuela Politecnica Superior, Universidad de Burgos, Spain. 4) Dipartimento di Fisica, Universita` di Roma Tre and Istituto Nazionale di Fisica Nucleare sezione di Roma Tre, Italy. ** E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] ***** E-mail: [email protected] 1)
Lorentzian spaces with nonconstant curvature was presented in [5]. Furthermore, these systems revealed a geometric interpretation of the quantum deformation, since the (in general, nonconstant) curvature of these spaces was just a smooth function of the deformation parameter. Later, integrable potentials on such 2D “quantum deformed” spaces were introduced by preservin
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