A solution of the motion of a charged particle in magnetic mirror systems

The paper solves the motion of a charged particle in an axially symmetrical, magnetostatic field, which forms magnetic mirror systems. The solution is based on the Hamiltonian for the motion of a charged particle in such a field. The whole problem is solved in orthogonal curvilinear coordinates, the coordinate curves being formed by the lines of force and the curves perpendicular to them. By applying the perturbation method of multi-periodical systems to the Hamiltonian in these coordinates, a Hamiltonian is obtained for the motion of the guiding centre which contains only the coordinate of motion along the lines of force, the others being cyclic. Thus, in addition to the energy two other first integrals of the equations of motion are obtained (since the impulses corresponding to the cyclic coordinates are constant), from which the conditions for confinement of motion in a magnetic field are immediately obtained. Since the Hamiltonian obtained in this way contains only one coordinate, the whole problem is solved by two quadratures, which define the dependence of the time and azimuthal angle on the line-of-force coordinate, while the other quantities are constant.In conclusion, the author thanks Dr. M. Seidl for valuable remarks and discussion on this work.