Catastrophes and stable caustics in bound states of Hamiltonian systems

Abstract

Caustics—envelopes of families of classical trajectories, or boundaries between classically allowed and forbidden regions—correspond to singular points of a phase-space surface called a Lagrangian manifold. According to catastrophe theory, only a limited number of types of caustics are stable under general perturbations of the manifold. Most of the caustics that are found in calculations correspond to members of the canonical list of elementary catastrophes. However, there are some exceptions—examination of trajectories of typical Hamiltonian systems shows that stable structures exist which are not in accord with the stability theorem of catastrophe theory. These exceptional cases are discussed in this paper. They arise because of the special form of the typical Hamiltonian of physical systems.