Eur. Phys. J. Plus (2014) 129: 164
https://doi.org/10.1140/epjp/i2014-14164-5

Regular Article

Superintegrable Lissajous systems on the sphere

¹ Departamento Matemática Aplicada, Universidad de Valladolid, 47011, Valladolid, Spain
² Department of Physics, Faculty of Sciences, Ankara University, 06100, Ankara, Turkey
³ Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47071, Valladolid, Spain

^* e-mail: jnegro@fta.uva.es

Received: 24 June 2014
Accepted: 6 July 2014
Published online: 5 August 2014

Abstract

The kind of systems on the sphere, whose trajectories are similar to the Lissajous curves, is studied by means of one example. The symmetries are constructed following a unified and straightforward procedure for both quantum and classical versions of the model. In the quantum case it is stressed how the symmetries give the degeneracy of each energy level. In the classical case it is shown how the constants of motion supply the orbits, the motion and the frequencies in a natural way. The associated sets of polynomial constants of motion and symmetries as well as their algebraic relations are also obtained and compared for the classical and quantum cases.