Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature

Abstract

A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented.

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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• Twisted (2+1) $\kappa$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes
• The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature
• A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
• Classical Particle in Presence of Magnetic Field, Hyperbolic Lobachevsky and Spherical Riemann Models
• Clifford Algebras and Possible Kinematics
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
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• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• Clifford Fibrations and Possible Kinematics
• Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
• Harmonic Oscillator on the $\mathrm{SO}(2,2)$ Hyperboloid
• Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Curvature as an integrable deformation
• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum