Complex Structures in Electrodynamics

In this paper we show that the basic external (i.e. not determined by the equations) object in Classical electrodynamics equations is a complex structure. In the 3-dimensional standard form of Maxwell equations this complex structure $\mathcal{I}$ participates implicitly in the equations and its presence is responsible for the so called duality invariance. We give a new form of the equations showing explicitly the participation of $\mathcal{I}$. In the 4-dimensional formulation the complex structure is extracted directly from the equations, it appears as a linear map $\Phi$ in the space of 2-forms on $\mathbb{R}^4$. It is shown also that $\Phi$ may appear through the equivariance properties of the new formulation of the theory. Further we show how this complex structure $\Phi$ combines with the Poincare isomorphism $\mathfrak{P}$ between the 2-forms and 2-tensors to generate all well known and used in the theory (pseudo)metric constructions on $\mathbb{R}^4$, and to define the conformal symmetry properties. The equations of Extended Electrodynamics (EED) do not also need these pseudometrics as beforehand necessary structures. A new formulation of the EED equations in terms of a generalized Lie derivative is given.Comment: Latex2e, 19 page

A New Look on the Electromagnetic Duality. Suggestions and Developments

In this paper a new look on the electro-magnetic duality is presented and appropriately exploited. The duality analysis in the nonrelativistic and relativistic formulations is shown to lead to the idea the mathematical model field to be a differential form valued in the 2-dimensional vector space ${\cal R}^2$. A full ${\cal R}^2$ covariance is achieved through introducing explicitly the canonical complex structure ${\cal I}$ of ${\cal R}^2$ in the nonrelativistic equations. The connection of the relativistic Hodge * with ${\cal I}$ is shown and a complete coordinate free relativistic form of the equations and the conservative quantities is obtained. The duality symmetry is interpreted as invariance of the conservative quantities and conservation equations.Comment: Latex, 14 pages, no figure

Frobenius Curvature, Electromagnetic Strain and Description of Photon-like Objects

This paper aims to present a general idea for description of spatially finite physical objects with a consistent nontrivial translational-rotational dynamical structure and evolution as a whole, making use of the mathematical concepts and structures connected with the Frobenius integrability/nonintegrability theorems and electromagnetic strain quantities. The idea is based on consideration of {\it nonintegrable} subdistributions of some appropriate completely integrable distribution (differential system) on a manifold and then to make use of the corresponding curvatures and correspondingly directed strains as measures of interaction, i.e. of energy-momentum exchange among the physical subsystems mathematically represented by the nonintegrable subdistributions. The concept of photon-like object is introduced and description (including lagrangian) of such objects in these terms is given.Comment: 22 pages, no figures, comments and explanations added, submitted for publicatio

Electromagnetic field objects in terms of Balance of Geometric flows

This paper reviews our physical motivation for choosing appropriate formal presentation of electromagnetic field objects (EMFO). Our view is based on the understanding that EMFO are spatially finite entities carrying internal dynamical structure, so, their available integral time stability should be represented by appropriate adaptation of their internal dynamical structure to corresponding local stress-energy-momentum balance relations with other physical objects. This adaptation process has two aspects: internal and external. Clearly, finding adequate internal dynamical structure giving appropriate integral characteristics of the object, will bring also appropriate behavior of EMFO as a whole. Therefore, the internal local stress-energy-momentum balance among the subsystems of EMFO should formally be presented by appropriately defined tensor-field quantities, which are meant to suggest a dynamical understanding of the abilities of EMFO to successfully communicate with all the rest physical world.Comment: 20 pages, two figures. arXiv admin note: substantial text overlap with arXiv:1303.2808, arXiv:1210.832

Relativistic Strain and Electromagnetic Photon-Like Objects

This paper aims to relate some properties of photon-like objects, considered as spatially finite time-stable physical entities with dynamical structure, to well defined properties of the corresponding electromagnetic strains defined as Lie derivatives of the Minkowski (pseudo)metric with respect to the eigen vector fields of the Maxwell-Minkowski stress-energy-momentum tensor. First we recall the geometric sense of the concept of strain, then we introduce and discuss the notion for photon-like objects (PhLO). We compute then the strains along the eigen vectors of the stress-energy-momentum tensor $T_{\mu}^{\nu}$ and establish important correspondences with the divergence terms of $T_{\mu}^{\nu}$ and the terms determining some internal energy-momentum exchange between the two recognizable component-fields $F$ and $*F$ of a vacuum electromagnetic field. The role of appropriately defined Frobenius curvature is also discussed and emphasized. Finally, equations of motion and interesting PhLO-solutions are given.Comment: 9 pages, 2 figures, presented at Int.Workshop "Trends in Diff.Geometry, Complex analysis and Math.physics", August 25-29, 2008, Sofia, Bulgaria; published by World Scientific 2009; some changes in the introduction sectio

Extended Electrodynamics I. Basic Notions, Principles and Equations

This paper aims to present an elaborate view on the motivation and realization of the idea to extend Maxwell's electrodynamics to Extended Electrodynamics in a reasonable and appropriate way in order to make it possible to describe electromagnetic (3+1)-soliton-like objects in vacuum and in the presence of continuous media (external fields), exchanging energy-momentum with the electromagnetic field.Comment: 18 pages, LaTex, no figure

Coupling a nano-particle with isothermal fluctuating hydrodynamics: Coarse-graining from microscopic to mesoscopic dynamics

We derive a coarse-grained description of the dynamics of a nanoparticle immersed in an isothermal simple fluid by performing a systematic coarse graining of the underlying microscopic dynamics. As coarse-grained or relevant variables we select the position of the nanoparticle and the \emph{total} mass and momentum density field of the fluid, which are locally conserved slow variables because they are defined to include the contribution of the nanoparticle. The theory of coarse graining based on the Zwanzing projection operator leads us to a system of stochastic \emph{ordinary} differential equations (SODEs) that are closed in the relevant variables. We demonstrate that our discrete coarse-grained equations are consistent with a Petrov-Galerkin finite-element discretization of a system of formal stochastic \emph{partial} differential equations (SPDEs) which resemble previously-used phenomenological models based on fluctuating hydrodynamics. Under suitable approximations we obtain \emph{closed} approximations of the coarse-grained dynamics in a manner which gives them a clear physical interpretation, and provides \emph{explicit} microscopic expressions for all of the coefficients appearing in the closure. Our work leads to a model for dilute nanocolloidal suspensions that can be simulated effectively using feasibly short molecular dynamics simulations as input to a FEM fluctuating hydrodynamic solver.Comment: Submitted to J. Chem. Phy

How to Describe Photons as (3+1)-Solitons?

This paper aims to present the pure field part of the newly developed nonlinear {\it Extended Electrodynamics} [1]-[3] in non-relativistic terms, i.e. in terms of the electric and magnetic vector fields (${\mathbf E},{\mathbf B}$), and to give explicitly those (3+1)-soliton solutions of the new equations which have the integral properties of photons. The set of solutions to the new equations contains all solutions to Maxwell's equations as a subclass, as well as, new solutions, called nonlinear. The important characteristics {\it scale factor}, {\it amplitude function}, and {\it phase function} of a nonlinear solution are defined in a coordinate free way and effectively used. The nonlinear solutions are identified through the non-zero values of two appropriately defined vector fields $\vec{\cal F}$ and $\vec{\cal M}$, as well as, through the finite values of the corresponding scale factors. The intrinsic angular momentum (spin) is also defined. A limited superposition principle (interference of nonlinear solutions), yielding the well known classical {\it coherence} conditions, is found to exist.Comment: Latex, 15 pages (17 x 24cm text), no figure

On the Pre-metric Formulation and Nonlinearization of Charge-free Electrodynamics

This paper presents a coordinate free pre-metric formulation of charge free Maxwell-Minkowski electrodynamics, and of the developed by the authors non-linear Extended Electrodynamics. First we introduce some formal relations from multilinear algebra and differential geometry to be used further. Then we recall and appropriately modify the existing pre-metric formulation of linear charge free electrodynamics in pre-relativistic and relativistic forms as preparation to turn to corresponding pre-metric nonlinearization. After some preliminary examples and notes on nonlinearization, we motivate our view for existence and explicit formulation of time stable subsystems of the physical field objects considered. Section 5 presents the formal results of our approach on the pre-metric nonlinear formulations in static case, in time-dependent case, and in space-time formulation. In the Conclusion we give our general view on "why and how to nonlinearize". The Appendix gives a possible formal extension of our aproach to many subsystem field objects.Comment: 17 pages, no figures. arXiv admin note: text overlap with arXiv:1508.0482

A nonlinear prerelativistic approach to mathematical representation of vacuum electromagnetism

This paper presents an alternative prerelativistic approach to the vacuum case of classical electrodynamics represented by vacuum Maxwell equations. Our view is based on the understanding that the corresponding differential equations should be dynamical in nature and the physical relations represented by them should be directly verifiable at least in principle, so they must represent local energy-momentum balance relations.Comment: 22 pages, submitted for publication, minor text changes, reference 1 removed; some text amendments and typos corrected. arXiv admin note: text overlap with arXiv:1210.832