Les systèmes super intégrables d’ordre trois séparables en coordonnées paraboliques
This thesis is a contribution to the study of classical and quantum superintegrability in a two-dimensional Euclidean space involving a third order integral of motion. It consists of an article. Because the classifications of all separable hamiltonians into Cartesian and polar coordinates are already complete, we bring to this picture the study of those systems in parabolic coordinates. First, we derive the determinating equations of a system into parabolic coordinates, after which we solve the obtained equations in order to find integrals of order three for potentials, which allow the separations of variables into the parabolic coordinates. Finally, we prove that all the third order integrals for separable potentials in parabolic coordinates in the Euclidean space of dimension two are reducible. In the conclusion of this article, we analyze the differences between the separable potentials in Cartesian and polar coordinates and the separable potentials in parabolic coordinates. Keywords: integrability, superintegrability, classical mechanics, quantum mechanics, Hamiltonian, separation of variables, commutation.
integrability superintegrability classical mechanics quantum mechanics Hamiltonian separation of variables commutation