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.

Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

“…The study of superintegrability with magnetic fields was initiated in [1] and subsequently followed in both two spatial dimensions [2,3,4,5] and three spatial dimensions [6,7,8,9,10]. Also a relativistic version of the problem was recently considered, cf.…”

Cylindrical type integrable classical systems in a magnetic field

“…= 0 (8) using the coefficients in front of each individual combination of powers in momenta. Those equations in cartesian coordinates are listed in previous papers [36][37][38][39][40].…”

“…The augmented matrix of the system of linear equations (39) can be written in its reduced row echelon form as …”

“…. On the other hand, the solvability condition of the linear system (39), namely that the rank of M and of the corresponding augmented matrix coincide, imply that if either ρ(r) = 0 or σ(r) = 0, the function α must vanish. Thus in the two cases above we find constraints,…”

“…This means that the system is inconsistent and admits no solutions. The source of this inconsistency is that W r , W φ and W Z can be determined in a unique manner from the algebraic equation (39). They must however also be first derivatives of a smooth function W (r, φ, Z) and hence satisfy the Clairaut theorem on mixed derivatives.…”