Superintegrability in classical mechanics: A contemporary approach to Bertrand's theorem

Superintegrable Hamiltonians in three degrees of freedom posses more than 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 Hamiltonian. Then, using basic algebraic ideas, we obtain a contemporary proof of Bertrand's theorem. That is, we are able to show that the harmonic oscillator and the Newtonian gravitational potentials are the only 3D potentials whose bounded orbits are all plane and periodic.