Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
arXiv:0810.0999 · doi:10.1007/s00220-009-0793-5
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 (5)
- Superintegrable Systems in Darboux spaces
- A maximally superintegrable system on an n-dimensional space of nonconstant curvature
- Hidden symmetry of hyperbolic monopole motion
- Multi-center MICZ-Kepler system, supersymmetry and integrability
- Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems