Electrodynamics, Topsy-Turvy Special Relativity, and Generalized Minkowski Constitutive Relations for Linear and Nonlinear Systems

A Quest for Systematic Constitutive Formulations for General Field and Wave Systems Based on the Volterra Differential Operators

“…But abstract models that are convenient for mathematical manipulation, or empirically suggested functions are also used. The (almost trivial) example for the unmagnetized cold plasma, (see for example [10]), based on the single electron equation of motion, demonstrated how elimination of the extraneous physical quantities leads to operator constitutive parameters. In this case, two constitutive equations are used: It is assumed that charge and current densities (dependent fields in the present example) are related by j = ρv , with v being the velocity, and that the equation of motion of the electron is aptly represented by qE = m∂ t v , with q, m , denoting the electron charge, mass, respectively.…”

“…Both cases constitute an approximation when the series is truncated, keeping a finite number of leading terms. Therefor the Volterra series model is sometimes characterized as describing weak nonlinearity (see for example [10] for some earlier references). Volterra's series consist of functionals,…”

“…In each system the native spatiotemporal coordinates and the associated differential operators are used. Inasmuch as Einstein [7] addressed the free space (vacuum) case only, the question of constitutive relations did not arise, but as the theory evolved, it became clear that the covariance property above is actually attributed to the indeterminate set of equations [1,2,10].…”

Relativistic Electrodynamics: Various Postulate and Ratiocination Frameworks

“…The topsy-turvy model has been devised mainly for didactic reasons, in order to facilitate a straight-forward and compact analysis [5,7] for an application-oriented audience. Accordingly the Lorentz transformation (6) has been taken as a postulate, replacing the kinematical postulate of the constancy of c.…”

“…By applying the chain-rule of calculus to (7), the relations between derivatives in Γ and Γ are established…”