The contact magnetic flow in 3D Sasakian manifolds

We first present a geometrical approach to magnetic fields in three-dimensional Riemannian manifolds, because this particular dimension allows one to easily tie vector fields and 2-forms. When the vector field is divergence free, it defines a magnetic field on the manifold whose Lorentz force equation presents a simple and useful form. In particular, for any three-dimensional Sasakian manifold the contact magnetic field is studied, and the normal magnetics trajectories are determined. As an application, we consider the three-dimensional unit sphere, where we prove the existence of closed magnetic trajectories of the contact magnetic field, and that this magnetic flow is quantized in the set of rational numbers.Oficina de Gestión de la Investigación Científica y TécnicaJunta de Andalucí