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arXiv · 2608.26412

A closure criterion for electromagnetic curves on the sphere in a uniform ambient field

C S López-Monsalvo
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Abstract

We consider the motion of a charged test particle confined to the round unit sphere in a uniform ambient magnetic field. We use an extension of Noether's theorem for systems with magnetic forces to reduce the problem to a quadrature. The rotation number of a trajectory, the azimuth it gains over one oscillation in latitude, is a complete elliptic integral of the third kind in Legendre form. A trajectory closes if and only if that advance is a rational multiple of a full turn. The criterion holds on every level set of the two first integrals. Then, we differentiate the rotation number with respect to the half-cyclotron frequency and sign that derivative on each of the two branches into which a level set divides. The rotation number sees the charge, the mass, the field and the speed through a single dimensionless ratio, that of the half-cyclotron frequency to the speed. A closed trajectory therefore fixes a value of that ratio. We say how many values carry a given rational winding. The poles are attainable on a single level set, where the motion reduces to a pendulum. The field has no flux through the sphere, so it is globally exact. We compute the Ma\~n\'e strict critical value of the resulting exact magnetic flow. Above that value the closed trajectories are closed Reeb orbits of an explicit contact form.

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C S López-Monsalvo. 2026-08-26. A closure criterion for electromagnetic curves on the sphere in a uniform ambient field. https://arxiv.org/abs/2608.26412

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