The classical superintegrable generalizations of the Kepler problem

Seminario de Ecuaciones Diferenciales

Ponente: Alain Albouy (Observatorio de Paris/CNRS, Francia)

Abstract: An open set in the phase space of the Kepler problem is filled up with periodic orbits (the elliptic orbits). This property may be expressed as a "superintegrability": there are as many independent first integrals as possible, which happen to be algebraic.
We will discuss two generalizations discovered in the 19th century: one discovered by Darboux by replacing the plane of motion by a surface of revolution, the other one, non Hamiltonian, due to Jacobi. We will address the same questions in each case: Is there a "hidden symmetry"?
What are the possible dimensions of the configuration space? Is the force field divergence free in dimension 3? Are there transformations from a generalization to another one? Is there a Lambert theorem? The results on a generalization were obtained with Lei Zhao. The results on the other one, with Antonio Ureña..