Alberto Enciso

Bertrand's theorem revisited and superintegrable Hamiltonian systems

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 talk 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 
admitting a suitable, globally defined generalization of the Runge--Lenz 
vector.