SUPERINTEGRABILITY IN CLASSICAL MECHANICS - A CONTEMPORARY APPROACH TO BERTRANDS THEOREM

Superintegrable Hamiltonians in three degrees of freedom posses more t han three functionally independent globally defined and single-valued constants of motion. In this contribution and under the assumption of the existence of only periodic and plane bounded orbits in a classical system we are able to establish the superintegrability of the Hamilto nian. Then, using basic algebraic ideas, we obtain a contemporary proo f of Bertrand's theorem. That is, we are able to show that the harmoni c oscillator and the Newtonian gravitational potentials are the only 3 D potentials whose bounded orbits are all plane and periodic.