Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems

Abstract

We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painlevé transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.