On the dynamics of a charged particle in magnetic fields with cylindrical symmetry

We study the motion of a charged particle under the action of a magnetic field with cylindrical symmetry. In particular we consider magnetic fields with constant direction and with magnitude depending on the distance r from the symmetry axis of the form 1+Ar−γ as r→∞, with A≠0 and γ>1. With perturbative-variational techniques, we can prove the existence of infinitely many trajectories whose projection on a plane orthogonal to the direction of the field describe bounded curves given by the superposition of two motions: a rotation with constant angular speed at a unit distance about a point which moves along a circumference of large radius ρ with a slow angular speed ε. The values ρ and ε are suitably related to each other. This problem has some interest also in the context of planar curves with prescribed curvature.