Spherically restricted motion of a charge in the field of a magnetic dipole

TL;DR

This paper analyzes the classical non-relativistic motion of a charge restricted to a spherical surface in a magnetic dipole field, revealing integrable dynamics, effective potentials, and various trajectory types through analytical and numerical methods.

Contribution

It provides a detailed analytical reduction of the restricted Stoermer problem to a quadrature and effective potential analysis, enhancing understanding of its complex dynamics.

Findings

Closed horizontal trajectories identified.

Effective double-well potential derived.

Conditions for periodic and non-periodic trajectories established.

Abstract

We study the restricted motion of an electric charge in a spherical surface in the field of a magnetic dipole. This is the classical non-relativistic St\"oermer problem within a sphere, with the dipole in its centre. We start from a Lagrangian approach which allows us to analyze the dynamical properties of the system, such as the role of a velocity dependent potential, the symmetries and the conservation properties. We derive the Hamilton equations of motion and observe that in this restricted case the equations can be reduced to a quadrature. From the Hamiltonian function we find for the polar angle an equivalent one-dimensional system of a particle in the presence of an effective potential. This equivalent potential function, which is a double well potential, allows us to get a clear description of this dynamical problem. We are able to find closed horizontal trajectories, as well as their period. Depending on initial conditions, we can find also some bands covered by non-periodic trajectories, as well as the conditions for the presence of loops. Then we obtain by means of numerical integration different plots of the trajectories in three dimensional graphs in the sphere. This restricted case of the Stoermer problem, which is formally integrable, is still a nonlinear problem with a complex and interesting dynamics and we believe that it can offer the student a better grasp of the subject than the general three dimensional case.