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, A., Enciso, A., Herranz, F.j., Ragnisco, O. (2009). Hamiltonian Systems Admitting a Runge-Lenz Vector and an Optimal Extension of Bertrand's Theorem to Curved Manifolds RID B-5702-2011 RID F-2453-2010. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 290(3), 1033-1049 [10.1007/s00220-009-0793-5].

Hamiltonian Systems Admitting a Runge-Lenz Vector and an Optimal Extension of Bertrand's Theorem to Curved Manifolds RID B-5702-2011 RID F-2453-2010

RAGNISCO, Orlando
2009-01-01

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.
2009
Ballesteros, A., Enciso, A., Herranz, F.j., Ragnisco, O. (2009). Hamiltonian Systems Admitting a Runge-Lenz Vector and an Optimal Extension of Bertrand's Theorem to Curved Manifolds RID B-5702-2011 RID F-2453-2010. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 290(3), 1033-1049 [10.1007/s00220-009-0793-5].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/124589
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