The Gauss-Landau-Hall problem on Riemannian surfaces

We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field. 1 From a classical picture to a general setting Classically, the Landau-Hall problem consists of the motion study of a charged particle in the presence of a static magnetic field, H . In this setting, free of any electric field, a particle, of charge e and mass m, evolves with velocity v satisfying the Lorentz force law, [1], dP dt = e c v ×H, where c denotes the light speed, P = (ǫ/c) v stands for the momentum of the particle, and ǫ = mc [1− (‖ v ‖2 /c2)] is its energy. Since dP/dt is orthogonal