The Gauss-Landau-Hall problem on Riemannian surfaces

Abstract

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.Comment: 20 pages, LaTe

Magnetic vortex filament flows

“…In fact, we have shown in Ref. 4 that the above result also holds if the Riemannian manifold ͑M n , g͒ is geodesically complete.…”

“…Therefore we get in Ref. 4 the following result: Let ␥ be an inextensible magnetic curve of ͑M n , g , F͒ and such that ␥͑͑a , b͒͒ lies in a compact subset of M n for any finite interval ͑a , b͒ in its domain. Then, ␥ must be complete.…”

“…Furthermore, the whole families of magnetic curves of ͑M , g , F͒ and ͑M , g , F͒ coincide for any constant Ͼ0. Consequently, we have the following: 4 Let F be a nontrivial magnetic field on a Riemannian manifold, ͑M n , g͒. Then, there exists no affine connection on M n whose geodesics are the magnetic curves of ͑M n , g , F͒.…”

“…These are called uniform magnetic fields and they play an important role in the classical Landau-Hall problem. 4 Notice that they correspond with parallel Lorentz forces, ٌ =0.…”

“…4, the authors solved this problem for a certain class of magnetic fields in dimension 2, the so-called Gaussian magnetic fields. The associated Lorentz force equation of these magnetic fields corresponds with the field equations of a variational problem that describe the massive relativistic bosons, this is briefly recalled in Sec.…”