Periodic orbits, superintegrability, and Bertrand’s theorem
R. P. Martínez-y-Romero,
H. N. Núñez-Yépez,
A. L. Salas-Brito
Affiliations
R. P. Martínez-y-Romero
Facultad de Ciencias, Circuito Exterior, Ciudad Universitaria, UNAM, Coyoacan, CP 04510 CDMX, Mexico
H. N. Núñez-Yépez
Departamento de Física, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Apartado Postal 55-534, Iztapalapa, CP 09340 CDMX, Mexico
A. L. Salas-Brito
Laboratorio de Sistemas Dinámicos, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Apartado Postal 21-267, Coyoacan, CP 04000 CDMX, Mexico
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.
