Motion of charged particles in magnetic fields created by symmetric configurations of wires

Abstract

In this paper we study the motion of a charged particle in the presence of a magnetic field created by three different systems of wires: an infinite rectilinear filament, a circular wire and the union of both. In the first case we prove that the equations of motion are Liouville integrable and we provide a complete description of the trajectories, which turn out to be of helicoidal type. In the case of the circular wire we study some restricted motions and we show that there is a trapping region similar to the Van Allen inner radiation belt in the Earth magnetosphere. We prove the existence of quasi-periodic orbits using Moser’s twist theorem, and the existence of scattering trajectories using differential inequalities. We also provide numerical evidence of Hamiltonian chaos and chaotic scattering by computing several Poincaré sections, Lyapunov exponents, fractal basins and their fractal dimensions. A similar study is done for the third system, although quasi-periodic orbits are proved to exist only under certain (perturbative) assumptions. From the viewpoint of the applications we propose a magnetic trap based on these configurations. Furthermore, the circular wire system can be interpreted as a simplified model of the levitated magnetic dipole–one of the recent proposals to confine a hot plasma for fusion power generation–and hence our work provides a verification of confinement and quasi-periodicity, beyond the adiabatic approximation, for this plasma system. Apart from contributing to the rigorous theory of the motion of charges in magnetic fields, this paper illustrates that very simple magnetic configurations can give rise to complicated, even chaotic trajectories, thus posing the question of how the complexity of magnetic lines affects the complexity of particle motions.