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

Abstract

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 $1+A{r}^{-\gamma }$ 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 $\epsilon$. The values $\rho$ and $\epsilon$ are suitably related to each other. This problem has some interest also in the context of planar curves with prescribed curvature.

19 pages, 1 figure

References in corpus (1)

• Nonlinear variational problems with lack of compactness

Cited by in corpus (1)

• Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space