Superintegrabilite avec separation de variables en coordonnees polaires et integrales du mouvement d'ordre superieur a deux.

摘要

In this thesis, we propose new superintegrable systems separable in polar coordinates. After the introduction, in chapter 2, we present a complete classification of all separable systems in polar coordinates which admit a third order integral in addtion to the second order one responsible for the separation of variables. New potentials expressed in terms of the sixth Painleve transcendent and of the Weierstrass elliptic function are obtained. In chapter 3 we introduce an infinite family of integrable and exactly sovable classical and quantum systems separable in polar coordinates. This family is described in term of a parameter k. The energy spectrum and the wave functions of the quantum systems are obtained. A conjecture postulating the superintegrability of these systems is formulated and is verified for the first cases k = 1, 2, 3, 4. The order of the integrals is 2k where k ∈ N . The algebraic structure of the family of quantum systems is formulated in term of a hidden algebra where the number of generators depends on the parameter k. A quasi-exactly solvable and integrable generalization of the family of potentials is proposed. Finally in chapter 4, the classical trajectories of the family of systems are calculated for all the rational cases k ∈ Q . Those are expressed in term of Chebyshev polynomials. We plot the curves associated with the trajectories for k = 1, 2, 3, 4, 1/2, 1/3 and 3/2. The bounded curves are closed and periodic in the two dimensional phase space. Those results obtained reinforce the possible veracity of the conjecture.;Keywords: integrability, superintegrability, exactly solvable, Painleve property, Painleve transcendents, separation of variables.