Examples Of Magnetic Fields Research Articles

On the semiclassical analysis of the ground state energy of the Dirichlet Pauli operator III: magnetic fields that change sign

We consider the semiclassical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm–Kovařik–Portmann and Helffer–Sundqvist for the asymptotics of the ground state energy in the semiclassical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semiclassical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disk are discussed. Various natural conjectures are disproved, and this leaves the research of an optimal result in the general case still open.

Integrability of magnetic fields created by current distributions

The existence of first integrals and periodic orbits of magnetic fields created by thin wires is investigated. When the current lines are planar we prove that magnetic orbits are closed near the wires and we provide two examples of magnetic fields without polynomial first integrals, thus contradicting Stefanescu's conjecture. When the current lines are non-planar we provide some examples of rectilinear configurations giving rise to helicoidal orbits near the wires and to chaotic portraits.

Alfvén waves in inhomogeneous magnetic fields

A perfectly conducting, nonviscous fluid is assumed to be permeated by an inhomogeneous, time-independent external magnetic field. Propagation of small-amplitude hydromagnetic waves with the velocity vector satisfying div v = 0 is investigated in a curvilinear coordinate system in which the lines of the external magnetic field are chosen as coordinate lines. It is shown that, if the lines have no torsion and if the external magnetic field has a certain symmetry, such that the field intensity is independent of one of the curvilinear coordinates, there exist rigorous solutions of the linearized hydromagnetic equations describing waves which travel along the lines of the external magnetic field, with the velocity of the fluid motion directed everywhere along the binormal to the local magnetic line of force. The examples of magnetic fields of plane and axial symmetry are discussed.