New integrable systems: hydrogen atom in external fields

Abstract

We report the discovery of two new three-dimensional conservative Hamiltonian systems which are completely integrable. In both cases, two additional integrals of motion in involution are generators of purely dynamical symmetries, i.e., they do not follow from any apparent geometrical symmetry of the system and are fairly complicated functions of canonical coordinates. Physically, the obtained Hamiltonian functions are good approximations for the description of the hydrogen atom in a circularly polarized microwave field (for low frequencies) or in crossed magnetic and electric fields (for low magnetic field). More formally, they correspond to the hydrogen atom in the presence of both a circularly polarized laser field and a generalized Van der Waals potential.