Spherical type integrable classical systems in a magnetic field

Abstract

We show that four classes of second order spherical type integrable classical systems in a magnetic field exist in the Euclidean space , and construct the Hamiltonian and two second order integrals of motion in involution for each of them. For one of the classes the Hamiltonian depends on four arbitrary functions of one variable. This class contains the magnetic monopole as a special case. Two further classes have Hamiltonians depending on one arbitrary function of one variable and four or six constants, respectively. The magnetic field in these cases is radial. The remaining system corresponds to a constant magnetic field and the Hamiltonian depends on two constants. Questions of superintegrability—i.e. the existence of further integrals—are discussed.