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

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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.

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Authors and Affiliations

  1. Depto. de Física, Universidad de Burgos, 09001, Burgos, Spain

    Ángel Ballesteros & Francisco J. Herranz

  2. Depto. de Física Teórica II, Universidad Complutense, 28040, Madrid, Spain

    Alberto Enciso

  3. Dip. di Fisica, Università di Roma 3, and Istituto Nazionale di Fisica Nucleare, 00146, Rome, Italy

    Orlando Ragnisco

Authors
  1. Ángel Ballesteros
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  2. Alberto Enciso
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  3. Francisco J. Herranz
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  4. Orlando Ragnisco
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Correspondence to Alberto Enciso.

Additional information

Communicated by G.W. Gibbons

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Ballesteros, Á., Enciso, A., Herranz, F.J. et al. Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds. Commun. Math. Phys. 290, 1033–1049 (2009). https://doi.org/10.1007/s00220-009-0793-5

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  • DOI: https://doi.org/10.1007/s00220-009-0793-5

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