Line-integral Lagrangians for the electromagnetic fields interacting with charged particles

Abstract

An examination is made of the relation between line-integral representations of the scalar and vector potentials for the electromagnetic fields and line-integral representations of the polarization and magnetization fields due to an aggregate of charged point particles. The connection between the two types of representation is to be found in the Lagrangian that describes the interaction of the fields with the particles. This can be expressed completely in terms of the charge and current densities coupled to the scalar and vector potentials or partially (though sometimes completely) in terms of the polarization and magnetization fields coupled to the electric and magnetic induction fields. It is shown that the integration paths for the polarization and magnetization fields are determined by those for the scalar and vector potentials and by the motion of the charged particles; the integration paths are in general curvilinear and emanate from a moving reference point. It is also shown that altering the reference point or the integration paths amounts to a transformation of the potentials and causes a similar gauge transformation of the polarization and magnetization fields. The results obtained here reduce in special cases to some given previously by other authors.