Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds

Abstract

Bertrand's theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand's theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick's classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge-Lenz vector.

17 pages

References in corpus (14)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Superintegrable Systems in Darboux spaces
• Superintegrability in a two-dimensional space of non-constant curvature
• Nahm Transform For Periodic Monopoles And N=2 Super Yang-Mills Theory
• Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
• Maximal superintegrability on N-dimensional curved spaces
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Gravitomagnetism in the Kerr-Newman-Taub-NUT spacetime
• Bertrand spacetimes as Kepler/oscillator potentials
• Hidden symmetry of hyperbolic monopole motion
• Multi-center MICZ-Kepler system, supersymmetry and integrability
• Conformal Yano-Killing tensors for the Taub-NUT metric
• Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems
• Spectral curves and the mass of hyperbolic monopoles

Cited by in corpus (28)

• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• A new exactly solvable quantum model in N dimensions
• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• Curved manifolds with conserved Runge-Lenz vectors
• Generalization of Bertrand's theorem to surfaces of revolution
• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• Superintegrable systems with a position dependent mass : Kepler-related and Oscillator-related systems
• Superintegrable systems on spaces of constant curvature
• Classical and quantum higher order superintegrable systems from coalgebra symmetry
• Variational Contact Symmetries of Constraint Lagrangians
• Quantum Integrals from Coalgebra Structure
• The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
• On two superintegrable nonlinear oscillators in N dimensions
• Generalized Lenard Chains, Separation of Variables and Superintegrability
• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
• The ${\mathfrak{sl}}_2(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems
• Darboux inversions of the Kepler problem
• New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
• The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion
• Symmetries and integrals of motion of a superintegrable deformed oscillator
• Global versus local superintegrability of nonlinear oscillators
• Demkov-Fradkin tensor for curved harmonic oscillators
• A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system
• Curvature as an integrable deformation
• Higgs algebraic symmetry of screened system in a spherical geometry
• Exactly solvable deformations of the oscillator and Coulomb systems and their generalization
• Superintegrability, symmetry and point particle T-duality