Topology and Topological Evolution of Clasic ic Fields and Curr (Kiehn RM 1998).pdf

TOPOLOGY AND TOPOLOGICAL EVOLUTION OF
CLASSICAL IC FIELDS AND
CURRENTS.
R. M. Kiehn
University of Houston
Houston, Texas, 77036
updated 9/11/98
Abstract The theory of classical ism can be put into correspondence
with two topological constraints placed on the variety of independent variables
{x,y,z,t}. The topological constraints are formulated in terms of the exterior
differential systems, F ­ dA = 0, and J ­ dG = 0. These topological constraints
imply that the domains of support for finite non-zero ic field
intensities, and finite non-zero ic currents, in general, cannot be
compact without boundary. The method emphasizes the physical importance of the
potentials, for the two fundamental constraints lead to the independent concepts of
topological torsion, A^F, and topological spin, A^G, with topological features that
are explicitly dependent on the potentials, A. The exterior derivatives of these two
3-forms create the familiar Poincare invariants as 4-forms, whose closed integrals are
evolutionary deformation invariants. The zero sets of each 3-form can be used to
define the concepts of transverse ic and transverse electric waves on
topological grounds. The direction fields of the 3-forms A^F and A^G can exhibit
linking and separation ponent domains. The possible evolution of these
topological properties is studied with respect to classes of processes that can be
defined in terms of singly parameterized vector fields. Non-zero values of the
Poincare 4-forms are the source of topological change and non-equilibrium
thermodynamics.
1. Introduction
In the language of exterior differential systems [1] it es evident that classical
ism is equivalent to a set of topological constraints on a variety of independent
variables. Certain integral properties of such an ic system are deformation invariants
with respect to all continuous evolutionary processes that can be described by a singly parameterized
vector field. Thes