Applications of Differential Forms
Maxwell Faraday and Maxwell Ampere Equations
R. M. Kiehn (in preparation - last update 10/31/97) Physics Department, University of Houston, Houston, Texas Abstract: The topological universality of the Maxwell Faraday and Maxwell Ampere equations is an artifact of C2 differential forms on a domain of dimension n * 4. Starting with a 1-form of (electromagnetic) Action, the Maxwell Faraday equations become a consequence of the Poincare lemma. Starting from an N-1 form density, the Maxwell Ampere equations become a consequence of the topological constraint that the N-1 form density is exact. The conservation of charge current is a consequence of the Poincare lemma. Geometrical structure constraining the deduced 2-form and the induced N-2 form establish equivalence classes of constitutive equations. Evolutionary processes acting on Maxwell-Faraday systems can be classified into reversible and irreversible categories, depending upon the Pfaff dimension of the Action 1-form. The perfect plasma equations are equivalent to the unique Hamiltonian dynamical systems on spaces of Pfaff dimension 3, and the Master equations describe reversible processes on the symplectic manifold of Pfaff dimension 4. Irreversible processes generate dynamical systems proportional to vector fields ÝExA + Bd, A 6 BÞ, on symplectic domains of Pfaff dimension 4.
THIS ARTICLE IS UNDER RE-CONSTRUCTION (10/18/97) (Suggestions are appreciated)
In this article, Classical Electromagnetism will be defined in terms of two topological statements or postulates: the existence of a non-exact global 1-form of potentials, A, and the existence of a global exact N-1 form of charge currents, J. Then, the ideas implied by these topological postulates will be expressed in terms of Cartan’s theory of differential forms  along with complete details of the constructions on a four dimensional variety. The method will demonstrate that the laws of electromagnetism, as defined by a set of partial differential equations, or equivalently, an exterior differential system , are concepts independent from a choice of metric, or a choice of a group structure that is often used to define a connection . Following this expose, the various topological components will be subjected first to geometrical constraints that will lead to a constitutive theory of signals as propagating discontinuities, and then to evolutionary processes by means of Cartan’s Magic formula , demonstrating how equivalence classes of electromagnetic problems can be formulated in a topological manner. The constitutive technique demonstrates that there exist characteristic wave solutions to Maxwell’s equations for which the propagation velocity is not four-fold degenerate. Each of two states of polarization propagate in opposite directions with 4 (four) distinct speed. It is a characteristic of the Lorentz vacuum that these 4 speeds are degenerate. The method also indicates that there exists another type of vacuum, the chiral vacuum, that preserves the Lorentz symmetries. The only difference between the Lorentz vacuum and the chiral vacuum is in the conformal factor that represents the radiation impedance. In the language of differential forms, the exterior derivative is related to the Kuratowski closure operator, which in simple terms implies that the exterior derivative is a limit set generator. This idea has explicit exhibition in terms of the elementary concept that the divergence of the D field implies that the field lines terminate on their limit points, the page 1
charges. However, the concept is much more general. For example, it will become apparent below that the E and B fields are the ”limit” points of the electromagnetic potentials. Of particular interest are those topological sub-systems whose limit sets are ”empty”. For then the system is closed, but not necessarily isolated. Such configurations can have harmonic components whose integral...
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