Maxwell Faraday and Maxwell Ampere Equations
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)
INTRODUCTION
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 [1] along with complete details of the constructions on a four dimensional variety. The method will demonstrate that the laws of
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)
INTRODUCTION
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 [1] along with complete details of the constructions on a four dimensional variety. The method will demonstrate that the laws of
References: [1] Bateman [2] Whitaker [3] Whitaker [4] Kiehn [5] Bateman [6] Fock Luneberg [7] Osserman [8] Post [9] [10] (UNFINISHED) page 24