  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 $1 + Ar^{-\gamma}$ as $r\to\infty$, with $A\ne 0$ and
$\gamma > 1$. 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 $\rho$
with a slow angular speed $\varepsilon$. The values $\rho$ and $\varepsilon$
are suitably related to each other. This problem has some interest also in the
context of planar curves with prescribed curvature.
