1. The choice of initial position and statement of the problem
At the turn of the 19th and 20th 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» . 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 (En) and propagation velocity (c) of a plane wave :
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  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 , 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 ). 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 . 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 . 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, ). 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 ) 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 , 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.