Periodic orbits, superintegrability, and Bertrand's theorem

Periodic orbits are the key for understanding classical Hamiltonian systems. As we show here, they are the clue for understanding Bertrand's result relating the boundedness, flatness, and periodicity of orbits with the functional form of the potentials producing them. This result, which is known as Bertrand's theorem, was proved in 1883 using classical 19th century techniques. In this paper, we prove such a result using the relationship between the bounded plane and periodic orbits, constants of motion, and continuous symmetries in the Hamiltonian system.