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

## TL;DR

This paper analyzes the complex motion of a charged particle in magnetic fields with cylindrical symmetry, revealing infinitely many bounded trajectories influenced by the field's decay rate and geometric properties.

## Contribution

It demonstrates the existence of infinitely many bounded trajectories for a charged particle in specific magnetic fields using perturbative-variational methods.

## Key findings

- Existence of infinitely many bounded trajectories.
- Trajectories involve superpositions of rotation and slow circumferential motion.
- Results relate to planar curves with prescribed curvature.

## 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 $1 + Ar^{-\gamma}$ 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.