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Dynamic Model of Elementary Particles and Fundamental Interactions
George P. Shpenkov and Leonid G. Kreidik Institute of Mathematics & Physics, University of Technology & Agriculture Al. Kaliskiego 7, Bydgoszcz 85-796, POLAND; e-mail: shpenkov@janmax.com
This paper describes a physical model of elementary particles based on the wave features of their beha v ior. Elementary particles are regarded as dynamical structures of the micro-world, interrelated with all levels of the Universe; i.e., inseparable from the structure of the Universe as a whole. Between any elementary particles and the ambient field of matter-space-time, as well as between elementary particles themselves, there exists an interchange of matter-space-time occurring both in horizontal (within the same level) and vertical (between di f ferent levels) directions. This model reveals the nature of mass and charge of elementary particles, which in turn leads to the unified description of fundamental (electromagnetic, gravitational, and nuclear) interactions, and other important results considered concisely here. PACS Numbers: 01.58.+b, 11.90.+t, 12.90.+b

1. Introduction
Our understanding of fundamental interactions depends upon our knowledge about the nature, features, and behavior of the micro-objects of the Universe, called òelementary particlesó (although no one doubts that these particles too are complex). The elementary-particle model accepted in modern physics is the Standard Model (SM). It just attempts to describe behavior; i.e., it focuses on answering questions of òhowó; but it encounters diffi culties when questions of òwhyó or òwható arise. One unresolved mystery is the nature of mass and charge of elementary particles. Another is the relation between bare elementary particles and ambient space, etc. Following the as-yet unquestioned modern nuclear model of atoms, it is assumed that the dimensions of elementary particles do not exceed the size of atomic nuclei. But we do not know whether this is true. Despite not knowing primordial features of matter, physicists created not only the abstract SM, but also an abstract quantummechanical model of atoms, and models of more complicated systems, including the whole Universe and its origin. In the course of time, many began to realize that some widely-accepted basic concepts were doubtful, and they noted that: ðèThe ideas that were put in place by our intellectual ancestors in the early 1900ós are insufficient to deal with the deep issues that are now being explored. The neat and tidy view of the 1970ós has given way to confusing colle c tions of paradoxes, puzzles, enigmas, and contradictionsè [1]ñ. This comment refers, mostly, to the problems of elementary particles, gravity, and relativity. It is also widely recognized that the SM ð will not be the final theoryñ and ðany efforts should be undertaken to finds hints for new physicsñ [2]. Knowing that our ideas concerning the fundamentals of physics are poor, but not knowing better ways, the overwhel ming majority of physicists continue research in a traditional way, creating more and more complicated, abstract theories based on sophisticated mathematics. But a new approach to the aforementioned problems is developed in works of the present authors [3]. According to our analysis of the fundamentals of physics, cogn i tion of Nature is impossible without resolving the primordial problems of natural science mentioned above.

In the new approach developed in our works, atoms and elementary particles are regarded as structures of distinct levels of the Universe, which has many such levels ( e.g., molecular, atomic, subatomic, etc.). Perfect harmony and correlation inside and between all levels takes place in this Universe. From this viewpoint, the physical field-space of the Universe represents by itself an infinite series of spaces embedded in each other [recall ing a set of nesting dolls, or an infinite functional series ]. This series of spaces expresses the funda mental concept of natural philosophy concerning the infinite d i visibility of matter. Every level of space is the basis level for the nearest above-situated level and, simultaneously, it is the level of superstructure for the nearest below-situated level. This means that above-situated field-spaces are formed on the basis of belowlying fields-spaces. Accordingly, there is no meaning to the concept of òvery last elementary particleó in the common classical sense of this phrase. Therefore, it is clear; we should not consider atoms and elementary particles separately from the total structure of the Un i verse. This means that in a consideration of the problem of structure of any material objects, one should begin from the precise definition of the principal axioms on the structure of the Universe on the whole. As follows from the first of the axioms of the general structure of the Universe [3] (see also the web site [4]), mutual transformations of fields with opposite properties cause the wave nature of the world (e.g., transformations of the potential field into the kinetic field and vice versa). Waves appearing at one level generate other waves going deep into an infinite series of embedded field-spaces. Based on these and other relevant axioms, the wave equation of matter-space-time was solved. As a result, we found the spatial distribution of characteristic points (called conditionally nodes) of wave fields, where the wave function of the wave equation reaches maximal and zero values. In particular, these solutions revealed the distribution of nucleons in atoms and, accordingly, the nature of Mendeleevós periodic law and symmetry [5] (including ðforbidden to ordinary crystalsñ [6]).


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where relative The rate of we call

Vol. 15, SI No. 2

According to the solutions obtained, nucleons in atoms are in the primary potential polar-azimuth nodes (maximum of two per a node) located along characteristic meridians and parallels of spherical shells, corresponding to radial solutions of the wave function (details are in [3] and partly in the web site [7]). This atomic model accounts for the known physical properties and phenomena already considered in [3]. It predicts and yields the structure and mass of all possible isotopes. In essence, it reveals the ògenetic codeó of the structural variety in Na ture. Based on the solution of the wave equation, the new atomic model also allows an understanding of the physics of atomic reactions caused by an inelastic collision of high-energy particles with substance. Calculated binding energies and the proper energy of nucleons in the nodes conform to the experimental data of nuclear physics. A deeper understanding of atomic properties and atomic structure cannot be achieved without understanding the nature of atomic components ï òelementary particlesó, i.e., micro-objects of atomic and subatomic levels of the Universe: protons, electrons, neutrons, etc. In this way, the consideration of the wave nature of the particles led us to an understanding of the nature of their mass and charge . Using the framework of our approach and the results obtained, a unified description of fundamental interactions (electromagnetic, gravitational, and nuclear) became possible [3], and is demonstrated here. The goal of this paper is to show all major stages of this research.

is the absolute unit density and is the density. ratio of mass and time expresses the volumetric mass exchange of the particles with environment, which the exchange charge, or merely the charge (2.2)

where is the area of a closed surface separating the space of an elementary particle from the surrounding field of matter-spacetime, is some speed of wave exchange (interaction) at the sep a rating surface. It is natural to present the speed of wave e x change (interaction) in the form (2.3) is the wave number corresponding to where the definite fundamental fr equency of the exchange field (which is characteristic of the subatomic level of the Universe).

2. Model of Elementary Particles; Definitions
Let us imagine an elementary particle as a dynamic spherical formation of a complicated structure being in a dynamic equilibrium with environment through the wave process of the definite frequency . Longitudinal oscillations of its wave shell in the radial direction provide an interaction of the particle with other objects and the ambient field of matter-space-time (Fig. 2.1). In the approach presented, the logical triad, matter-space-time, ex presses an indissoluble bond of matter, space, and time. The logical pair: motion-rest presents indissoluble bond of motion and rest, etc. We assume that a spherical wave shell bounds the space of an elementary particle, separating it from the ambient wave field. We call this sphere the characteristic sphere of a micro-particle. The characteristic sphere restricts the main part of the micro-particle from its field part that merge gradually with the ambient field of matter-space-time. The main part (core) is the basis of a micro-particle, whereas the field part represents its superstructure. Thus, the basis space of a micro-particle is restricted by the characteristic sphere , beyond which there is the space of its s uperstructure. Such a model inter prets a micro-particle as a particular discrete physical point of an arbitrary level of matter-space-time, restricted by the cha r acteristic sphere and being in rest in the field-space. The ratio of mass and volume of elementary part i cles defines their absolute-relative density : (2.1)

Figure 2.1. An element of the volume (a) of the wave shell in a spherical field of exchange: a particle - ambient field of matter-space-time; and are pow ers of exchange of the field with the element of shell of the particle, is the two-dimensional density of exchange, or the pressure of the field of exchange;. (b) the internal and external parts of an elementary particle.

Strictly speaking, the exchange charge is the measure of the rate of e x change of matter-space-time , or briefly the power of mass exchange. In this wider sense, the area of exchange does not necessarily concern the closed sur face. In a case of a micro-object of spher i cal structure, the measure of exchange charge (2.2) is (2.4) where is the radius of the wave shell of the micro-object. The Universe is an infinite series of material and ideal spaces. Between objects of the spaces, there occur complicated interactions that represent the ex change of matter-space-motion-rest (matterspace-time for brevity). The exchange of matter-space-time occurs simultaneously in many levels, which are represented by corresponding subspaces of matter of the Universe. These subspaces should be regarded as embedded into each other; they form the space of the Universe. The embedding is one of the aspects of the physical multi-dimensionality of fields of matterspace-time of the Universe.


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(3.4)

As a measure of exchange intensity in matter-space-time, it is possible to take any parameter of exchange. If it is momentum, then we deal with the rate of exchange of momentum, etc . In such a broad sense of the word, the expression , known as Newtonós second law, is a simple writing of the fo r mula of the vector power of exchange of momentum. By virtue of this, we will also call the vector the power of exchange of momentum . Of course, this power of exchange cannot be identified with the scalar power of exchange of energy : . However, in spite of their difference, both and are powers of exchange, e x pressed by the language of the concrete measures of exchange, and nothing more . This is why the same term the power of exchange is the rightful one as the measure of the rate of exchange. The dynamical model presented here uses terminology defined in [8]. The geometrical space (spherical volume) delimited by the spherical wave shell of an elementary particle is òexternaló to the Universe. As the òexternal worldó of the Universe (Fig. 2.1b), this space inside the spherical volume can naturally be called the òAnti-Universeó. In this sense, the World (Being and Non-Being) is presented here through the Universe and AntiUniverse. Obviously, the spaces of the Universe and AntiUniverse are closed on each other. Most probably, the main essence of life, its mystery, is hidden in the Anti-Universe. Resting on the aforementioned definitions, we can start the consideration of wave exchange (interaction) of a particle with the ambient field of matter-space-time.

where is the amplitude of the exchange density at the . boundary of the wave zone defined by the condition Joining the equalities (3.3) and (3.4), we have (3.5) On the basis of Eqs. (3.3) and (3.5), we find the power of exwith the ambient field at the boundary of the spher i change cal shell of a particle with the area and radius : (3.6) or (3.7) The expression in brackets can be regarded as a resulting mass of particle - environment exchange. It is an associated field mass of the particle . (3.8)

We can present the expression (3.7) in another form. Because , the right part of this expression can be rewritten as (3.9) In such a case, the equation of radial exchange of a particle of through the spherical surface, within which the particle mass is localized, can be presented in the form of the following equation of exchange powers: (3.10) where is the exchange power of the particle with an object in , takes into acterm, with the a mbient field of matequation of exchange powers degree of freedom can be pre-

3. Derivation of Elementary Particle Mass
In a spherical field (Fig. 2.1), an equation of powers of ex change of momentum for an elementary volume of a characteristic spherical shell of a particle of area and thickness is defined by the equality (3.1) are described where the speed and the power of exchange by the field of binary numbers [9, 10], expressing the potentialkinetic character of e x change. The resulting action is

Because the following form

, the equation of exchange (3.1) will take

the ambient space, and the second count the exchange of the particle ter-space-time. Taking into account (3.9), the for the particle with the one radial sented as

or On the basis of Eq. (3.2) and because arrive at

(3.2) [see (2.3)], we (3.3) where

(3.11)

(3.12) is the coefficient of resistance, or the dispersion of rest-motion at exchange. The exchange-power equation (3.11) is presented in the classical form of Newtonós second law, describing the motion in the field-space with resi stance . In such a description of motion-

In a spherical field, elementary cone is versely to a distance sequently, a wave of

the flow constant. from the exchange

of oscillatory energy through an Hence, the speed decreases incenter of the spherical field. Condensity has the form


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Shpenkov and Kreidik: Dynamic Model
of (3.13) , or

Vol. 15, SI No. 2
(4.4)

rest, the expression in brackets represents the ef fective mass the particle:

The second term is abbreviated as (3.14) called the assoc i ated potential mass of the particle, or merely the associated mass of the particle, or briefly the mass of the particle: This is the mass of the particle in longitudinal (central) exchange. If the rest mass (own mass) of the particle is significantly less than the associated mass , then the mass of the parti, and it is the field cle is defined only by its associated mass mass in the central exchange . Obviously, the rest mass of the particle is the associated mass with respect to the deeper level of the field of matter-space-time. Therefore, we can assert that all masses of micro-particles in the Universe have an associated field char acter, and that their own (proper, rest) masses do not exist . If situations are possible where exchanges of particles with the ambi ent field of matter-space-time of the subatomic level do not occur, then masses of particles with respect to this level are equal to zero, and no exper i ment will find such a world of micro-particles. Accordingly, this world will be unknowable to physics.

Eq. (4.4) determines the fundamental frequency of the field of e x change, which is the distinctive òtimeó frequency of exchange at the atomic and subatomic levels [3]. Using (4.4), the active charge can be presented as (4.4a) The active mass of dispersion at exchange, corresponding to the active charge, is (4.4b) In such a case, the associated mass m should be regarded as the reactive mass.

5. Elementary Law of Central Exchange
The simplest potential of exchange speed in a spherical field has the following form (5.1) Let the radial speed of exchange (at the wave spherical surface of a particle) follow the law the boundary condi tions . Then, taking into account , we obtain

4. The Charge of Exchange
Equation (3.10) describes the exchange of motion, whereas the mass exchange is defined by charges (2.2). In this case, we present the field component of mass ex change in the form (4.1) and taking into account the equality (3.10), we Assuming obtain the following equation of exchange powers: (4.2) The exchange charge has an active-reactive character; it fo l lows from Eqs. (3.6), (4.1) and (4.2) that (4.3) where (4.3a) is the active charge, and (4.3b)

On this basis, the potential of the spherical field of exchange can be pr esented as (5.2) is the active potential of dispersion, and where active potential of exchange is the re-

(5.3) The potential of radial exchange wave of exchange is determined by the charge

(5.4) is the amplitude of charge, determined by the expre swhere sion (4.3b). The potential (5.3) will not be changed if we will assume that at the field level . Then, the amplitude (and also the mean value) of the potential will be determined by the equality (5.5) where

is the reactive charge. The active component defines the dispersion during ex change, which in a steady-state process of exchange is compensated by the inflow of motion and matter from the deeper levels of space. The reactive component of charge (further for brevity, the charge of exchange ) is connected with the associated mass by the relation

(5.6) is the amplitude (or mean value) of the charge. The gradient of the exchange potential defines the intensity (or strength, or the rate, or the vector) of central exchange (its amplitude and mean value):


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The vector of central exchange by definition equal to or

defines the dynamic vector

,

Assuming, naturally, that , we arrive at the formula of correspondence between exchange charge and Coulomb charge : . (6.3)

(5.8)

This vector represents the density of exchange momentum of rest-motion. In accord with (4.1), the following power of exchange co r responds to the strength-rate of exchange : (5.9) is the absolute unit density. This expreswhere sion represents the law of central exchange of the Coulomb kind. Its general form is (5.10) The speed of exchange at the basis level is equal to . In this case, the equation of the power of exchange (4.2) takes the form (5.11) Hence, the (carrier) energy of mass exchange (interaction) on the basis level or, in other words, the dynamic energy of the subatomic level, will be equal to (5.12) where and is the displacement of the wave front of exchange (at the separating surface of a particle, see Fig. 2.1). In the case of the differential exchange, we have (5.12a) We arrive at the so-called rest energy of particles, well known , which appeared by chance in Einsteinós in the form manipulation with the fictitious mathematical empty spaces [3]. The sense (nature) of this energy is not (and cannot be) properly understood within the framework of generally accepted modern physical theories.

Hence, the exchange reactive charge of an electron at the level of the fundamental frequency is g/s where (6.4)

is the (Coulomb) elec-

tron charge. Thus, the physical quantity (6.4) is the exchange charge of an electron obtained on the basis of the experimental value of the electronós electric charge. On the basis of (3.4), knowing the exchange charge of an electron (6.4), we find the fundamental frequency of the wave field of exchange (interaction) at the subatomic level (the òfrequency of electrostatic fieldó) (6.5) and the fundamental wave radius , corresponding to this frequency, cm where g is the mass of the electron. (6.6)

nm correThe fundamental wave diameter lates with the average value of lattice parameters in crystals, defining an average discreteness of space at the subatomic level of exchange (intera ction).

7. Unified Approach to Electromagnetic, Gravitational, and Nuclear Interactions
Taking into account the relation (4.4), charge of exchange and the associated mass tral exchange (5.10) can be presented as . , between the , the law of cen-

(7.1)

This law lies at the foundation of Nature . Its particular case is the law of universal gravitation . Discovered by Newton in 1687, its original form is (7.2) Following the general form of the law of central exchange (inte r action), (5.10) or (7.1), we should present the law of universal gravitation in its correct form. For this aim, obviously, the for mula (7.2) must contain in the denominator the coefficient , which expresses the spherical isotropic character of exchange, and the absolute unite density , which expresses the interrelation of matter and space (mass and volume, or contents and form [3]). Introducing these multipliers in numerator and denomin a tor of (7.2), we arrive at (7.2a)

6. Fundamental Frequency and Wave Radius of the Electrostatic Field
The energy of exchange between a particle and the surrounding field, taking into account Eq. (5.5), is equal to (6.1) In an electrostatic field theory, the Coulomb energy (in CGSE units) corresponds to the energy of exchange (6.1): (6.2) where is the Coulomb òelectric chargeó.


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and ,

Vol. 15, SI No. 2
. The average value of

where . Comparing now the central exchange presented in the two forms, (7.1) and (7.2a), we arrive at the interrelation between the fundamental frequency of the field of exchange at the gravitational level with the gravitational co nstant : (7.3) The important effects originating from this equality can be found in [11]. The fundamental gravitational frequency , obtained from the equality (7.3), is (7.3a) where 2. Knowing and assuming

shell, defined from the expression (3.14) under the condition MeV. This value the energy in this case is correlates with the height of the potential barrier of division for a series of atoms [13].

8. Conclusion
Recognition of the wave nature of all phenomena in the Uni verse has required development of a physical (dynamical) model of elementary particles corresponding to such a wave nature, which has been carried out by the present authors. The existence and interactions of the particles are, in essence, a continuous process of wave exchange of matter-space and motion-rest, or, for brevity, exchange of matter-space-time. The wider (and, hence, truer) notion exchange is thus more correct because it reflects behavior of elementary particles in their dynamic equilibrium with the ambient field, at rest and motion, and interactions with other objects (and particles themselves). In other words, the notion exchange is more appropriate from the point of view of the phy sics of the complex behavior of elementary particles viewed as dynamic micro-objects belonging to one of the interrelated levels of the many-level Universe. (This notion was first introduced in [11].) It follows that the notion rest mass of elementary particles is, in principle, not valid for such a model. Accordingly, one could conclude that the rest mass of elementary particles does not exist. The associated nature of mass, as the field mass of the central wave exchange, naturally originates from this model. The power of mass exchange, i.e., the rate of exchange of mass, defines the exchange charge or simply the charge of elementary particles, which in contemporary physics is called the òelectric chargeó. The correctness of the dynamical model is reinforced by the fact that from this model it naturally (and logically) originates that: 1) The fundamental law of central exchange (5.10) (of the Coulomb kind), which unifies the fundamental interactions, distinguished in contemporary physics as electromagnetic, gravitational and nuclear; 2) The formula of dynamic energy of mass exchange of the subatomic level [see (5.12) or (5.12a)]; 3) The fundamental frequency of exchange (6.5), i.e., the frequency of the so-called òelectrostatic fieldó, that reveals its essence, in particular, non-stationary nature; 4) The fundamental wave radius [see (6.6)], defining the aver (lattice age atomic diameter and, hence, the average distance parameter) in ordered material structures ( e.g., crystals); 5) The fundamental gravitational frequency (7.3a) and the wave gravitational radius of elementary particles (7.4); 6) The energy of interchange in atoms (ònuclear forcesó), etc. Interested readers in can find details and other parameters, which have not been presented above, in other works [3, 8, 11]. The plentiful results obtained have universal meaning because they touch upon many fundamental interactions described on the basis of one theoretical concept. We believe that these data will stimulate corresponding theoretical research in all domains of physics, including High Energy and Elementary Particles Phy sics, etc.

that the gravitational interaction relates to the subatomic level with the basis speed wave gravitational radius of a particle: cm/s, we find the

Mkm

(7.4)

This radius determines the wave gravitational sphere with the transient wave zone, which divides the spherical space-field of a particle into the near oscillatory domain (domain of basis) and the far wave domain (domain of superstructure). If the particles form cosmic objects, for example stars, then the domain of the gravitational radius (as the transient zone, separating the basis and the s uperstructure of the field of the star) must be presented by a series of rings-shells. In the solar system, these are represented by the rings of asteroids of the Sun, a djoined to the shell of the gravitational radius. In this domain, big planets cannot exist because, in the process of formation of the Solar system, this transient domain was the place of the most intense m otion. It should be noted that the existence of the gravitational fre quency and the gravitational radius of elementary particles shows the indissoluble bond of micro- and mega-objects of the Universe in the unit complex of the Infinitely Small and Infinitely Big, as the coexisting polar oppositions Yes and No. The gravitational spectrum of H-atomic wave shells, coinciding with the spectrum of planetary orbits, is presented in [3] (see there a Section ðThe wave field of H-atom at the micro- and mega-levelsñ, pages 495 ï 503). The nucleon exchange charge defines, at the fundamental frequency , the high steadiness of atomic structures, i.e., the interbond of nucleons (ònuclear forcesó) in an atom. According to the new atomic model [3, 8, 11, 12] mentioned in the Introduction, the amplitude energy of interchange [see (6.1)], in the case of two nucleons touching on their outer shells, is MeV where g/s is the exchange (associated) charge of a proton, cm is the radius of the protonós characteristic (7.6) (7.5)


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[7] [8] [9]

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References
[1] [2] S . Shenker, ðAbout ITP of Stanford Universityñ, http://

L . Kreidik and G. Shpenkov, ðThe Shell Structure of Matter Spacesñ, in http://shpenkov.janmax.com/ShellStr.pdf L. Kreidik and G. Shpenkov, pp. 125-127 in Foundations of Phy s ics; 13.644èCollected Papers (272 p, Bydgoszcz, 1998). L. Kreidik and G. Shpenkov, ðMaterial-Ideal Numerical Fieldñ, pp. 34-39 in Contact ó95, Proceedings of the General ScientificTechnological Session Contact ó95, v. II (Bulgaria, Sofia, 1995),. L. Kreidik and G. Shpenkov, ðMysteries of the Numerical Field and Errors of Natural Science Related to ðImaginary Numbersññ; ðThe Binary Numerical Field and Potential-Kinetic Oscillationsñ; ðThe Binary Numerical Field and Longitudinal Transversal Motionñ; in

itp.stanford.edu/about.htm
L. Simons, Sect. ðFundamental Interactionsñ, in Chapt. ðAtomic and Condensed Matter Physicsñ of the NuPECC report devoted to ðImpact and Applications of Nuclear Science in Europeñ, ( Dourdan , France, 21-23 November, 2001). L. Kreidik and G. Shpenkov, Atomic Structure of Matter-Space, (584 p., Bydgoszcz, 2001). L. Kreidik and G. Shpenkov, ðThe Base of Dialectical Physics (Grand Survey)ñ, in http://shpenkov.janmax.com/

[10]

[3] [4]

TheBaseGrandSurvey.pdf
[5] L.G. Kreidik, G.P. Shpenkov, ðA Wave Field of Probability and the Form of Crystalsñ, p. 258 in Abstracts of the XVIIIth IUCr (Intern ational Union of Crystallography) Congress & General Assembly (Glasgow, Scot land, UK, 4 ï 13 August 1999). D. Shechtman, I. Blech, D. Gratias, and J.W. Chan, ðMetallic Phase with Long-Range Orientation Order and no Translation Symmetryñ, Phys. Rev. Lett. 53, (20) 1951-1953 (1984).

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[6]

[13]

L. Kreidik and G. Shpenkov, Alternative Picture of the World, Vol. 1-3 (148, 156, & 178 p., Bydgoszcz, 1996). L. Kreidik and G. Shpenkov, ðImportant Results of Analysing Foundations of Quantum Mechanicsñ, Galilean Electrodynamics & GED-East, Speical Issues 2, 13, 23-30, (2002). A.B. Ignatyuk, et al , Fizika Elementarnyh Chastits i Atomnogo Yadra 16, 709-772 (1985),