• DOI:10.1007/s00220-009-0793-5
• Corpus ID: 14862729

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

@article{Ballesteros2008HamiltonianSA,
  title={Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds},
  author={{\'A}ngel Ballesteros and Alberto Enciso and Francisco J. Herranz and Orlando Ragnisco},
  journal={Communications in Mathematical Physics},
  year={2008},
  volume={290},
  pages={1033-1049}
}

• Á. Ballesteros, A. Enciso, +1 author O. Ragnisco
• Published 6 October 2008
• Mathematics
• Communications in Mathematical Physics

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.