Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds

TL;DR

This paper extends Bertrand's theorem to curved 3-manifolds, showing that certain spherically symmetric Hamiltonian systems are superintegrable and generalize classical oscillator and Kepler systems.

Contribution

It provides a curved-space generalization of Bertrand's theorem, introducing new superintegrable Hamiltonian systems and a global Runge-Lenz vector construction.

Findings

Hamiltonian systems on curved spaces are superintegrable

New classes of superintegrable systems are identified

Extension of classical oscillator and Kepler systems to curved manifolds

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.