Group Analysis of Differential Equations and Integrable Systems − 2018

Libor Šnobl (Czech Technical University in Prague, Czech Republic)

Spherical type integrable classical systems in a magnetic field

We show that 4 classes of second order spherical type integrable classical systems in a magnetic field exist in the Euclidean space $\mathbb{E}_3$ 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 4 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 4 or 6 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.

Joint work together with A. Marchesiello and P. Winternitz.