Concerning the full integrability of hamiltonian equations of charged particle motion in a weakly inhomogeneous magnetic field

The Hamiltonian of a charged particle in a weakly inhomogeneous magnetic field is calculated up to terms on the order of a small parameter. Fast phase-averaged equations of motion are derived. It is shown that these equations are intergrable in quadratures. Thus, the problem of particle motion in a weakly inhomogeneous field is solved in the first-order approximation. To calculate the Hamiltonian, the coordinates related to the field are used. Then, the canonical change of variables is done with the help of the generating function; in the case of a homogeneous field, this results in the action-angle variables. Such a procedure has been already used in [1]. However, the small parameter was not explicitly introduced and final expressions for small and large parts of the Hamiltonian were not calculated in that paper. It is shown that the small part of the Hamiltonian is a trigonometric polynomial of the fast phase (this can be important when analyzing the influence of additional perturbations). Besides, the averaged equations appear to be treatable and can be integrated in quadratures.