Action at a distance as a full-value solution of Maxwell equations: The basis and application of the separated-potentials method

Abstract

The inadequacy of Liénard-Wiechert potentials is demonstrated as one of the examples related to the inconsistency of the conventional classical electrodynamics. The insufficiency of the Faraday-Maxwell concept to describe the whole electromagnetic phenomena and the incompleteness of a set of solutions of Maxwell equations are discussed and mathematically proved. Reasons of the introduction of the so-called "electrodynamics dualism concept" (simultaneous coexistence of instantaneous Newton long-range and Faraday-Maxwell short-range interactions) have been displayed. It is strictly shown that the new concept presents itself as the direct consequence of the complete set of Maxwell equations and makes it possible to consider classical electrodynamics as a self-consistent and complete theory, devoid of inward contradictions. In the framework of the new approach, all main concepts of classical electrodynamics are reconsidered. In particular, a limited class of motion is revealed when accelerated charges do not radiate electromagnetic field.

On Two Complementary Types of Total Time Derivative in Classical Field Theories and Maxwell’s Equations

“…In the classical definition of a partial time derivative (14), a function f does not explicitly possess mathematical characteristics of properly fluid quantity (recently, this fact was also critically pointed out in [6]− [9] but on different positions). Anyway, contrarily to what is explicitly assumed in (12), in the definition (14) there is no indication that f is submitted to non-zero geometrical transformation H t associated with non-zero flow velocity field v. Therefore, (14) is not directly applicable to treat time derivatives of fluid quantities.…”

“…Both definitions (12) and (13) are equivalent so that later on we shall refer only to the conventional expression (12). Important to note that in this classical definition of the total time derivative for Lagrangian observer, a function f is supposed to be submitted to non-zero geometrical transformation H t associated with non-zero flow velocity field v.…”

“…Therefore, the denial of the exclusive use of partial time derivatives to describe time variation for an observer at rest and a recognition of a deeper underlying meaning of the total time derivative in Eulerian description imply inevitable changes in the structure of mathematical solutions to Maxwell's equations (as regards this aspect, some alternative frameworks for classical electrodynamics were recently discussed in [6], [12]− [15] ). However, a wider analysis of the integral form of Maxwell's equations on basis of the local Convection Theorem does not look tractable at present stage.…”

“…As the next step, let us consider analytical expressions for (12) and (15) in Eulerian variables. The total time derivative is conceived as a linear part of the rate of change of f with respect to t. Thus, when t − t 0 tends to zero, only linear part of geometrical transformations H t makes sense for further discussions:…”

Instantaneous fields in classical electrodynamics

“…has recently been discussed [1][2][3][4][5]. The idea of this coexistence is partially motivated by the striking result that the scalar potential in the Coulomb gauge is an instantaneous quantity.…”