Physics – Mathematical Physics
Scientific paper
2008-10-06
Commun.Math.Phys.290:1033-1049,2009
Physics
Mathematical Physics
17 pages
Scientific paper
10.1007/s00220-009-0793-5
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.
Ballesteros Angel
Enciso Alberto
Herranz Francisco J.
Ragnisco Orlando
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