INTRODUCTION

** **

*1.** **The choice of initial position and
statement of the problem*

* *

At the turn of the 19^{th} and 20^{th}
centuries the physicists were inclined to think that dynamic
electromagnetic processes take part in the formation of energy bundles, which
exist in fully open space, and approve themselves as a special family of objects,
which are nowadays referred to as fundamental particles. In 1908 Henry Poincare
generalized this idea: «Abraham’s calculations and Kaufman’s experiments have
shown that mechanical mass is equal to zero and the mass of electrons, at least
the mass of negative electrons, is of an exclusively electrodynamics origin»
[1]. In
essence, it is not a hypothesis, but quite a well-founded, logically unerring
statement, which has capsuled certain discussions. Moreover, earlier, in 1900 the same author has already derived a
formula connecting mass (*m*) with energy (*E** _{n}*) and propagation velocity (

_{}.
(1)

However,
as it has turned out later, Poincare’s statement all the same requires additional
perfect check-up. Therefore, we shall consider it
as our main starting position and will make it a general working hypothesis
only as a tribute to the fashion. At that, we consider
that the point
of view of Poincare and his adherents on objective reality
has not yet gained any consecutive development from the point of view of
physics, a science, in which conceptually electrodynamics, formulated by James Clerk Maxwell [3] according to many conclusive experiments,
should not be the last one. We mean the
electrodynamics, which together with classical mechanics forms the basis of
modern physics. As far as we know, after Poincare there were only new hypotheses, which
were sometimes playful combinations of hypotheses, for some reason called
theories or models [4], and which in form do not go beyond Poincare’s
statement, but in essence only divert the problem into the area of ideas that
are even less connected with the reality. As a result, these ideas have turned
into a system of afflation, which has absolutely left the territory of natural
sciences (see the Preface).

The reader will not find new hypotheses in this work.
In order not to confuse quite a well-formed and precise conception of a
mathematical or computer-based model with the free and purely speculative
creation, which is now called a physical model, we are forced to refrain from
using the word "model" further.

Comparison of our
working hypothesis to already known and well checked up physical facts allows
to define conditions, which are obviously necessary to work with it in perfect
union with the electromagnetic field theory and concrete results of its
application, and which in case of their nonobservance lead to obvious denying
of the very opportunity to decide a problem. It is possible to regard such
conditions as consequences of a working hypothesis. For
us the first consequence is the following statement: or the open space should
be nonlinear, or there should be nonlinear the equations describing of energy distribution to let stable bundles
of electromagnetic energy exist in it. It follows from that fact, that any
localization of energy should be connected with the local changes of
electromagnetic properties of the medium. Otherwise, the linear equations of
Maxwell present blurring-in-space decisions, and the amplitudes of the maximum
possible fluctuations of fields aspire to an average background according to
the laws of thermodynamics. But the reason of spatial inhomogeneity of the open
medium can be only an electromagnetic field itself under condition of
nonlinearity of the medium (or the equations) in
which it exists.

Let us mark that the idea on nonlinearity is not new. Nonlinearity is in
keeping with the theory of vacuum polarization, which was developed by P.A.M. Dirac (he has given a detailed account of it in [5]). However, in this theory P.A.M. Dirac employed artificial
mathematical devices, which allowed abstracting from the structure of
fundamental particles. Our work, on the
contrary, is aimed at the research of the field structure inside the local
process, which is called a particle. Therefore, we
are compelled to separate a problem of the nature of physical nonlinearity and
the mechanism of its occurrence from electrodynamics calculations and whenever
possible to discuss it independently, though at the same time keeping in mind the logic of Dirac.

Thus, recognizing
the hypothesis of Poincare’s statement, we should recognize that nonlinearity of media, in which we are going to
investigate local electromagnetic concentration of energy, or initial equations
obviously exists, and it should be mathematically expressed with
the minimal restrictions in the form of equations. As
these equations are so far unknown to us, at the initial stage of computation
we are compelled to design them on the ground of physics, and to gradually
build formulas, which should be specified in the course of comparison of the
facts that are known already (charges, moments, etc.) with the results of
numerical experiments. Maxwell equations will not avoid changes either.

The binding presence of
a circumrotate component, the so-called particle, in the direction of
electromagnetic-field distribution inside the field disposition; can be
considered the second consequence of a working hypothesis. I.e. in a medium or
space there should be an electromagnetic vortex. It follows from the general
laws of distribution of electromagnetic waves in non-uniform media and the laws
of Snel
van Royen (1620) in classical optics [6]. Self-concentration of
electromagnetic energy assumes that the system involves the factors, returning
the waves, i.e. their reflection by local inhomogeneity of the medium (It is
not dependent from a inhomogeneity nature). And it is known that full internal
reflection without attenuation (we do not consider mirrors of superconductors)
can take place only in case of observance of one important condition, in
addition to other conditions: waves should have such a direction of
distribution, which is mainly perpendicular to the gradient of optical density.
Waves, the direction of which coincides with the gradient, are not reflected.
Naturally, in any spatial localization, not grounded to itself, the direction
of a gradient of any properties is approximately parallel to the radius coming
from its center. Initial data on rotation of an electromagnetic field are
stated in p.2 of the present Introduction.

Researches of the
electromagnetic phenomena of such complexity as the structure and properties of
an electromagnetic vortex, should at least rely on approximate solutions of
Maxwell’ equations for the elemental rotating field configurations, for
example, to answer such a question of principle as: “What is a circumrotate
degree of freedom of electromagnetic field?” However, in this case neither the
equations of Maxwell, nor the wave equations derived from them, cannot be
solved by means of usual methods of mathematical physics, because in them the
necessary variables are not differentiated [7]. Besides any displays of
nonlinearity of medium present an additional barrier to the use of usual
differential and integral calculus for the solution of our problem. Naturally,
modern computing means can help to overcome the deadlock. But in this case one
will come across some other difficulties. One has to tackle a problem, which is
initially under specified, and has to resort to a stepwise manner, accumulating
the findings step by step. The best way to characterize the described technique
is to use the concept of a numerical experiment. All the more so as correct
solutions of Maxwell’s equations are characterized by extreme reliability,
which is proved by more than a century’s practice of their check by a vast
number of real experiments, the results of which invariably confirmed the
theory (see, for example, [8]). If, for example, a radio-engineer learned from
the solution of the equations that phase velocity of a wave in a wave-guide in
front of him is equal to three velocities of light, he was always sure that if
he measured the velocity by means of a ruler and a tester, the velocity would be the same. The error will be equal
only to the error of the tester and the ruler. The
same concerns both internal electrodynamics – an electrodynamics of the
cavities limited by walls, and to electrodynamics of open space, which is
usually used in physical optics and in designing microwave antennas
or in radio-range. Therefore Maxwell’s theory of electromagnetic field (the
author gave his work the name of the theory [3]) and his equations should have
an absolute priority for us if compared to any other theories dealing with
electromagnetic fields and waves.

It is obvious, that
fundamental particles should not be bindingly a unique form of appearance of an
electromagnetic vortex. There can also be such spatial configurations of a
macroscopic size, for example local formation of a globular lightning. This
phenomenon as an object of
research is especially attractive as it can be gained
experimentally with sufficient information on its structure. The more so as the property of
self-concentration of streams of electromagnetic energy in a nonlinear media,
including gases in a stage of development of an electrical breakdown, is an
experimentally established fact, which was registered in 1961 by Gurgen A.
Askarjan [9], when observing self-focusing of beams of powerful laser
radiation. Therefore, a vortex of a macroscopic calibration, similar to a globular lightning, is used further on as an object for preliminary
adjustment of a numerical-experiment technique.

Let us note one
more feature of the object of our researches, which is important for the
statement of our problem. The thing is that a self-organized electromagnetic
vortex itself, if it exists at all, represents an extremely unique phenomenon,
and we cannot investigate it, relying exclusively on the known properties of
the physical media, which have already been described. Here by necessity the
problem can be set only in reverse order: we search for such initial field
configuration and such parameters of a physical medium, at which this
phenomenon can exist in general, if it possesses even short-term stability.
Thus, we consider the fact, that a question on how there can be rare
combinations of existence conditions for a vortex in nature or how they can be
created artificially in experimental conditions, can be correctly raised only
after thorough investigation of the electromagnetic process itself.

The
general solution of our problem involves a procedure of calculations based on
the solution of Maxwell’s equations by a well-known relaxation method [7, 10,
11], which is presented in p.3 of the present
Introduction. Practically any programming language, accessible to a modern
student or a schoolboy, can be used to do any calculations. However, the
reader, who is familiar with the system of computer mathematics “Matlab”
(“MathWorks”, Inc), has a decided advantage.

All our calculations were carried out by means of (2005, 2006) a modern
personal computer, the resources of which are rather limited, especially
resources of operative memory. Therefore, they are suitable only for an
approximate quantitative or semi-qualitative solution of the
problem in terms of phenomenology. Certainly, it is not too little. However, precision of such a
technique is obviously insufficient to determine the important constants. We believe, that the involvement of more powerful computing means will allows us in the long
view to specify quite purposefully the concrete properties of investigated
fundamental processes, which we have already discovered, as well as to find
absolutely new ones. However, certainly, we recognize that it is not an
abstract numerical experiment, but a real physical one, that will have the last
word.