Superintegrability in a two-dimensional space of non-constant curvature

Abstract

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton-Jacobi and Schroedinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.

27 pages

References in corpus (3)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Exact Solvability of Superintegrable Systems

Cited by in corpus (75)

• Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
• Superintegrability with third order invariants in quantum and classical mechanics
• A non-linear Oscillator with quasi-Harmonic behaviour: two- and $n$-dimensional Oscillators
• Superintegrable Systems in Darboux spaces
• Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability
• Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Quantum superintegrability and exact solvability in N dimensions
• Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• $f(R)$-gravity from Killing Tensors
• Invariant characterization of Liouville metrics and polynomial integrals
• Bertrand spacetimes as Kepler/oscillator potentials
• Deformation Quantization of Superintegrable Systems and Nambu Mechanics
• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
• Spacetime Encodings II - Pictures of Integrability
• Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties
• The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach
• Invariant classification of second-order conformally flat superintegrable systems
• (Super)integrability from coalgebra symmetry: formalism and applications
• The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach II
• Integrable potentials on spaces with curvature from quantum groups
• N-dimensional sl(2)-coalgebra spaces with non-constant curvature
• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
• Superintegrable systems with spin and second-order integrals of motion
• Superintegrable systems with a position dependent mass : Kepler-related and Oscillator-related systems
• Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials
• Spacetime Encodings III - Second Order Killing Tensors
• Classical and quantum higher order superintegrable systems from coalgebra symmetry
• Maximal superintegrability of Benenti systems
• Superintegrable Systems on 3 Dimensional Conformally Flat Spaces
• A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
• The Variety of Integrable Killing Tensors on the 3-Sphere
• Group classification of Schrödinger equations with position dependent mass
• First Integrals from Conformal Symmetries: Darboux-Koenigs Metrics and Beyond
• Eisenhart Lift of $2$--Dimensional Mechanics
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII
• Superintegrable deformations of superintegrable systems : Quadratic superintegrability and higher-order superintegrability
• Photonic realization of the deformed Dirac equation via the segmented graphene nanoribbons under inhomogeneous strain
• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Exact solvability of superintegrable Benenti systems
• Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
• Are all classical superintegrable systems in two-dimensional space linearizable?
• Integrable geodesic flows and Multi-Centre versus Bianchi A metrics
• Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
• Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV
• Invariant prolongation of the Killing tensor equation
• Second Order Superintegrable Systems in Three Dimensions
• Superintegrable systems in non-Euclidean plane: hidden symmetries leading to linearity
• Global structure and geodesics for Koenigs superintegrable systems
• Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
• Superintegrability on sl(2)-coalgebra spaces
• Path Integration on Darboux Spaces
• Global versus local superintegrability of nonlinear oscillators
• General Nth order integrals of the motion
• Adding Potentials to Superintegrable Systems with Symmetry
• Projectively equivalent 2-dimensional superintegrable systems with projective symmetries
• Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions
• Infinite dimensional representations of cubic and quintic algebras and special functions
• Doubly Exotic $N$th-order Order Superintegrable Classical Systems Separating in Cartesian Coordinates
• Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations
• Exact and quasi-exact solvability of two-dimensional superintegrable quantum systems. I. Euclidean space
• Koenigs Theorem and Superintegrable Liouville Metrics
• Curvature as an integrable deformation
• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras
• Path Integral Approach for Quantum Motion on Spaces of Non-constant Curvature According to Koenigs
• On polynomial symmetry algebras underlying superintegrable systems in Darboux spaces
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$
• Path Integral Approach for Spaces of Non-constant Curvature in Three Dimensions
• Superintegrability in the interaction of two particles with spin