Abstract-The relativistic-invariance representation of generalized momentum of a particle in the external field is proposed. In this representation the dependence ...
ISSN 1068–3372, Journal of Contemporary Physics (Armenian Academy of Sciences), 2012, Vol. 47, No. 6, pp. 249–256. © Allerton Press, Inc., 2012. Original Russian Text © V.M. Mekhitarian, 2012, published in Izvestiya NAN Armenii, Fizika, 2012, Vol. 47, No. 6, pp. 379–390.
The Invariant Representation of Generalized Momentum V. M. Mekhitarian Institute for Physical Research, NAS of Armenia, Ashtarak, Armenia Received January 26, 2012
Abstract⎯The relativistic-invariance representation of generalized momentum of a particle in the external field is proposed. In this representation the dependence of potentials of the particle–field interaction on the particle velocity is taken into account. An exact correspondence of expressions for the energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions of problems of mechanics with relativistic and classical approaches. The invariance of the generalized momentum representation allows one to describe equivalently a physical system in geometrically conjugated spaces of kinematical and dynamical variables. DOI: 10.3103/S1068337212060011 Keywords: particle, generalized momentum, invariant representation, external field
1. INTRODUCTION Problems of the theory of motion of a charge in a variable magnetic field is of great practical, in addition to theoretical, importance for the topics of particle acceleration (betatron, linear induction accelerator, and so on) [1, 2], plasma physics [3], induction discharge and material processing [4]. Last two decades extensive research is performed in the field of physics and technology of induction discharge, acceleration of plasma, and excitation of media by the magnetic field induction; this research is aimed at creation of high-power plasmatrons [5], plasma-jet engines [6], sources of radiation [7], and lasers [8, 9]. Although the induction discharge is known as long as 125 years [10], and the induction accelerator of electrons, betatron, 90 years [10], the existing theory of motion and acceleration of charges in a variable magnetic field does not satisfactory describe the processes of acceleration and heating of charges by the induced electric field. In particular, the Wideroe condition of circular motion of a charge in the betatron at 2:1 ratio of fields has not hitherto been confirmed and the theory of charge acceleration by the electric field induced in betatron was not correspondingly developed: only problems of stability of motion were solved [1, 10]. In case of the induction discharge, its existence in the central region of solenoid, where the induced electric field and the current are zero, is not explained [7]. Consistent construction of the theory of motion of charges in electromagnetic fields rests upon the equations of electromagnetic fields and equations of motion of charges. If one considers motion of a single charged particle in a given field, it may be assumed that the currents and fields of this particle do not affect, within certain limits, the particle motion. In such formulation the solution of the problem may be performed in two stages: i) determination of electric and magnetic fields, E and B, for given boundary conditions and sources and ii) solution of equations of motion of the particle with the mass m and the charge q in the fields E and B. Electromagnetic fields are described by the Maxwell equations which have in the Hertz–Heaviside representation the appearance 1 ∂E 4π rot B = + j, div E = 4πρ, c ∂t c (1) 1 ∂B rot E = − , div B = 0. c ∂t The fields E and B are solutions to these equations for the given boundary conditions and sources ρ and j. These fields exist independently of the presence of the considered particle and of the state of its motion. Directions and magnitudes of fields are determined from the Maxwell equations by only the symmetry of the problem and the boundary conditions. 249
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The next stage is solution of the equation of motion of the charged particle in the given fields dp = F ( q, r, v, E, B ) , (2) dt where it is still to be defined how the force F acting upon the moving particle depends on the velocity of the particle, its charge, and the fields E, B. This dependence and the equation of motion are no longer associated with the Maxwell equations (fields E and B are already present and determined), hence for determination of the force in Eq. (2) another laws should be used. Conventionally the Faraday law is used which is formulated as follows: the work of the induction force FI over a charge q along a closed curve L is determined by the rate of variation of the flux Ф of magnetic field B through the area S of the closed curve: q dΦ q d (3) v∫L FI ⋅ dL = − c dt = − c dt ∫∫S B ⋅ d S, independently of causes of the flux variation. This formulation joins two different phenomena: arising of an electric field in some point of space at a time variation of the magnetic field in this point and arising of a force acting upon a charged particle when it moves in a variable or constant magnetic field. Relationships 1 ∂B 1 ∂E (4) v∫L E ⋅ dL = − c ∫∫S ∂t d S or v∫L B ⋅ dL = c ∫∫S ∂t d S obtained from Eqs. (1) in integral form with partial derivatives in integrands express only the properties of the fields and there is no need in an extraneous particle with a charge q and a mass m and in its motion. This is the property of only the fields, rather than of the interaction of fields with the charged particle or effect of particle motion. The question of the properties of interaction between the fields and particle cannot be discussed within the Maxwell equations, since the considered particle is not available in the equations of fields. R. Feynman describing in his Physics Lectures the attempts to unite the Faraday law (3) and the field properties (4) declares that such a beautiful generalization is found to stem from an angle deep underlying principle. Neverthless, in this case there does not appear to be any such profound implication. We have to understand the “rule” as the combined effects of two quite separate phenomena. Induction accelerator of electrons, betatron, proposed by R. Wideroe as early as in 1922 had to confirm directly the formula for the Lorentz force dp q = qE + [ v × B ] (5) dt c in case of variable magnetic fields. It follows from the formula that for ensuring circular orbital motion of electron in a variable magnetic field the mean value of the deflecting field should be in ratio 2:1 (Wideroe ratio) with the value of field on the cyclic orbit (accelerating field). But Wideroe “saw no one accelerated electron” on his accurately mounted arrangement and abandoned the further attempts to realize his intention [11]. In the next 20 years scientists did not succeed to obtain even a single circuit of an electron accelerated by the magnetic field induction, although the Wideroe condition of circular motion is a direct consequence of the Lorentz formula (5). As late as in 1941 Kerst and Serber reported creation of a working betatron with the acceleration energy of 2.3 MeV [12, 13]. They solved the problem in a straightforward way – they produced inhomogeneous distribution of magnetic field on the orbit (a well) and revealed that for keeping the electron on the orbit the fields are needed which decrease with radius r near the orbit as r − n with 1 > n > 0. Stability of the orbit in non-uniform magnetic field was no longer determined by the Wideroe condition and contemporary betatrons operate at almost any ratio of fields – it is always possible to obtain a stable motion by an appropriate choice of the magnetic field inhomogeneity on the orbit. Calculations by Wideroe and Yasinskii concerning acceleration of electrons by the induced electric field (transformer approach) did not provide corresponding results and have been refused. Kerst, Serber, Terletskii, and others developed the theory of induction acceleration considering only conditions of stable motion of a single electron at its small deviations from the equilibrium orbit in inhomogeneous magnetic field (barrier) [1]. The fact that the ratio 2:1 of the fields does not provide any multiple acceleration (even a single stable circuit) suggests an idea on the principal limitation of the basis of derivation of Wideroe condition, i.e., formula (5) for the Lorentz force, in case of variable magnetic fields. JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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Principal difficulties arise also at description of magnetic resonance phenomena where the motion of a magnetic moment is considered in a variable magnetic field. It is known that the equation of motion of a magnetic moment in the form dL (6) = − [ω L × L ] , dt where ω L = ( q 2mc ) B is the Larmor angular frequency of rotation, relates to the case of a constant magnetic field (Larmor theorem) [14]. Scalar multiplication of Eq. (6) by L provides L2 = const which makes apparent that this equation does not anyway describe excitation of the moment in a variable magnetic field. In spite of this, Bloch added in 1946 relaxation terms to Eq. (6) for description of nuclear magnetic resonance and used it for the case of variable fields [15]: dL L − L 0 (7) + = − [ω L × L ] , dt τˆ where τˆ are typical times of longitudinal and transverse relaxations and L 0 is the equilibrium value of the moment. The Bloch equation (7) was then frequently used for description of phenomena of magnetic resonance in variable magnetic fields. Such a consideration is incorrect, since the equation for the particle moment L has the form above only in case of constant magnetic field; in case of variable field Eqs. (6) and (7) must contain an obvious term proportional to S ⋅ dB dt responsible for excitation of any current loop (magnetic moment) with the area S by the induction B of variable magnetic field. Based on traditionally used expression (5) for the Lorentz force, it is possible to derive the exact equation of motion for the moment in time-variable homogeneous magnetic field. In this case electric field E is given in the form E = (1 2c ) [r × dB dt ] and it is obtained from expression (5): dv q ⎡ dB ⎤ q r× = + [ v × B ]. dt 2c ⎢⎣ dt ⎥⎦ c After vector multiplying by r, transformations, and rearrangement of rhs expressions, we obtain m
(8)
d⎛ q q ⎞ (9) ⎜ m [r × v ] + ⎣⎡[r × B ] × r ⎦⎤ ⎟ = − ⎣⎡ B × [r × v ]⎦⎤ . dt ⎝ 2c 2c ⎠ For a particle with the mass and the charge distributed in the volume identically, it can be written for an element of mass dm and charge dq = ( q m ) dm d⎛ q q ⎞ ⎡⎣r × [ B × r ]⎤⎦ dm ⎟ = − ⎡ B × [r × v ]⎤⎦ dm. (10) ⎜ [r × v ] dm + dt ⎝ 2mc 2mc ⎣ ⎠ After integration over volume V it is obtained d L + Iω L = − [ω L × L ] , (11) dt where L is the angular momentum, ω L = ( q 2mc ) B the Larmor angular velocity of rotation, and Iˆ ( Iˆ = I i ,k = ∫ ⎡⎣ ri 2 δi ,k − ri rk ⎤⎦ dm) the tensor of the particle’s moment of inertia. In case of a spherical top the
(
)
V
moment of inertia Iˆ = I is a scalar, hence the equation for generalized angular momentum J = L + Iω L becomes dJ = − [ω L × J ]. (12) dt With allowance for Bloch relaxation terms one has dω L dL L − L 0 + +I = − [ ω L × L ]. (13) τˆ dt dt Scalar multiplying by L yields dω L d ⎛ L2 ⎞ dB , = −M ⎜ ⎟ = −L dt ⎝ 2 I ⎠ dt dt JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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where M = ( q 2mc ) L is the magnetic moment of the particle. This equation makes apparent why and how does the kinetic energy of rotation, L2 2 I , of a particle with magnetic moment M change in a variable magnetic field B. It is seen that in case of variable magnetic field absent in the Bloch equation (7) is just the term responsible for excitation of a magnetic moment by the induction of variable magnetic field. At rotation in the plane perpendicular to the magnetic field direction the angular velocity in the constant field is obtained from expression (5) to be ωC = ( q mc ) B (cyclotron frequency), while from Eq. (13) it equals ω L = ( q 2mc ) B (Larmor precession frequency). So, at establishing of a new value of the constant magnetic field, during the process of change, according to Eq. (13), the Larmor frequency of rotation ( q 2mc ) B is observed just which should be established. On the other hand, in a constant magnetic field, formula (5) for the Lorentz force yields ( q mc ) B. This means that only the case of precession is described, i.e., the case of the transverse field where the moment directed along the field is absent. Electron paramagnetic resonance at excitation by a longitudinal variable magnetic field cannot be described by not only Bloch equation (7), but also Eq. (13). Limitation of the Lorentz force formula (5) for description of charge motion comes from derivation of this formula. In derivation of the equation of motion of a charged particle in the external field [14] with the given field potentials φ and A from the Lagrangian in the form L = −mc 2 1 − v 2 c 2 − qϕ + ( q c ) A ⋅ v
(15)
it is assumed that interaction of a particle with the field does not depend on the velocity v of the particle motion. As a consequence, the generalized momentum is obtained from formula (15) as P = ∂L ∂v = p + qA c and the equations of motion read dp q ∂A q q = − grad ϕ − + [ v × rot A ] = qE + [ v × B] , (16) dt c ∂t c c i.e., formula for the Lorentz force is obtained by substitution into the particle Lagrangian of the potentials φ and A of the field (acting on a charge in rest in the given point) instead of potentials qϕ ' and qA ' of interaction of the field with the moving particle. Therefore, use of formula (5) produces contradictions in a number of problems. In particular, in the problem of a hydrogen-like atom with the nucleus charge Ze [16] a limitation, not physically substantiated, arises for the nucleus charge, Z < 137, indicating the absence of stable atoms with atomic number Z > 137. By the same reasons, there are no sufficiently sound equations describing the motion of elementary particles in strong interactions, and in nuclear physics they are forced to use the Schrödinger equation, although absurdity of such approach is in this case apparent. In the present work the relativistic-invariant representation of the generalized momentum of a particle in the external field is proposed, where the dependence of the field-particle interaction potentials on the velocity of particle motion is taken into account. 2. INVARIANCE OF REPRESENTATION OF GENERALIZED MOMENTUM. CLASSICAL CORRESPONDENCE If electromagnetic field is given by its potentials (ϕ, A), then the electric E and magnetic B fields are determined by the formulas ∂ϕ ∂A B = rot A, E = − − . (17) ∂r ∂τ If the charge does not move, it feels just these fields E and B. But if the charge in this point has a nonzero velocity v, it feels the field in different way and the interaction energy is different. In order to determine the force and the interaction energy for a charge moving at the velocity v, the principle of relativity of motion can be employed. The effective values of force or energy of interaction with the field of a charge moving at velocity v should be the same as in case where the charge does not move, but the field moves at velocity −v. So, for a moving charge, the effective values E' and B' of interaction fields (ϕ ', A ') are represented as [14]
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A|| + ϕβ ⎞ ⎫ ⎟ ,⎪ 2 1 − β2 ⎟⎠ ⎪ ⎝ 1− β ⎪ ϕ ' 2 − A ' 2 = ϕ2 − A 2 , E + [β × B ] ⎪ 2 2 2 2 , E ' = E|| + ⊥ ⎬ E' − B' = E − B , 2 1− β ⎪ E '⋅ B ' = E ⋅ B. ⎪ B − [β × E ] ⎪ , B ' = B|| + ⊥ ⎪ 1 − β2 ⎭ In this case the generalized momentum of the particle in the external field acquires the form ⎛ ϕ+β⋅A
( ϕ ', A ') = ⎜⎜
, A⊥ +
2 2 ⎞ 1 ⎛ mc + qϕ + q ( β ⋅ A ) ( mc + qϕ ) β + qA|| P= ⎜ , + qA ⊥ ⎟ , ⎟ c⎜ 1 − β2 1 − β2 ⎝ ⎠ and for the modulus I of the 4-vector of generalized momentum P it is obtained
(18)
(19)
2 2 I 2 = P 2 = ε 2 − p 2 = ⎡( mc 2 + qϕ ) − ( qA ) ⎤ c 2 . (20) ⎢⎣ ⎥⎦ Thus the generalized momentum of a particle in an external field is not only invariant with respect to transformations from a reference system to another one, but it also has invariant representation (19) in terms of particle velocity and the value of the invariant I is in every point given by the expression (20). In other words, this property is inherent to not only the representation of the proper momentum of the particle (mechanical part), but also the generalized momentum. Invariant representation of the generalized momentum in terms of particle velocity means that the nature of physical processes is such that the change in the state of the system is described mathematically by a rotation of the vector of generalized momentum in four-dimensional space. This result will here be generalized for the case of representation of generalized momentum of any systems by the statement that independently of the system’s state of motion the generalized 4-momentum has always invariant representation
P = ( ε, p ) → P 2 = ε 2 − p 2 = inv,
(21)
where ε and p are the energy and the momentum of the system. By writing the expression for the invariant ε 2 − p 2 in the form
( mc2 + qϕ) − ( qA ) = p 2 + m2c 2 + 2mϕ + q 2 ϕ2 − A 2 , E2 ε = 2 = p2 + ( ) c c2 c2 dividing by 2m and grouping, the formula is obtained, 2
2
2
(22)
ε 2 − m 2c 2 E 2 − m2 c 4 p 2 q2 = = + ϕ + q (23) ( ϕ2 − A 2 ) , 2m 2mc 2 2m 2mc 2 of correspondence between the system energy E and its classical notion H. The classical correspondence in the form H = p 2 2m + U ( τ, r ) [17] will be complete and exact if the potential energy of interaction U and the system energy H will be defined as H=
q2 E 2 − m2 c 4 2H 2 2 ϕ − = , H , → E = ± mc 2 1 + 2 . A (24) ( ) 2 2 2mc 2mc mc For example, the potential energy U of an electron in the field of Coulombian potential ϕ = Ze r and in homogeneous magnetic field B with the vector potential A = [r × B ] 2 is equal to U = qϕ +
e2 Ze2 1 Z 2e4 e2 B 2 2 2 2 2 r sin θ. ϕ − A = − + − ( ) 2mc 2 r 2mc 2 r 2 8mc 2 Expression (23) may in non-relativistic case be written as U = −eϕ +
(25)
2
H=
e ⎞ e ⎛ e ⎞ e2 mv 2 e 1 ⎛ 2 p − A + p − A ⋅ A + e ϕ + ϕ ≈ + ( v ⋅ A ) + eϕ, ⎜ ⎟ ⎜ ⎟ 2m ⎝ 2mc 2 2 c ⎠ mc ⎝ c ⎠ c
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where p − eA c ≈ mv is conventionally called proper (mechanical) momentum of the particle. The expression obtained for the classical energy H differs essentially from the traditional expression by the presence of (absent in classical electrodynamics) energy of interaction of the particle with the magnetic field (term linear in vector-potential A). In case of homogeneous magnetic field B when the vectorpotential may be represented as [ B × r ] 2, one has H = mv 2 2 + B ⋅ M + eϕ.
(27)
Appearance of these additional terms in representation of Hamiltonian in operator form in quantum mechanics [18] results in a conclusion that magnetic interaction is a purely quantum phenomenon. However, in the framework of the approach above these distinctions are absent and the magnetic interaction enters similarly in the form H M = q 2 A2 2mc 2 . 3. REPRESENTATION OF GENERALIZED PARTICLES If one represents the generalized momentum in the form ⎛ ⎞ ⎛ 1 p ε η ⎞ 2 ⎟=I⎜ 1 , ⎟, (28) , P = ε 1 − (p ε) ⎜ 2 2 ⎟ 2 ⎟ ⎜ ⎜ 1 − η2 1 − η 1 1 p p − ε − ε ( ) ( ) ⎝ ⎠ ⎝ ⎠ it is possible to introduce the concept of generalized velocity η = p ε and the invariant I in the form
I = ε 1 − ( p ε ) . For a particle in the external electromagnetic field expression (19) yields qA qA β+ 2 & + 2 ⊥ 1 − β2 β − β 0 & − β 0 ⊥ 1 − β2 mc + qϕ mc + qϕ η= = , qA β β 1 − ⋅ 0 1+ ⋅β mc 2 + qϕ 2
ε=
p=
( mc
mc 2 + qϕ + q ( β ⋅ A ) c 1− β 2
2
+ qϕ ) β + q A & c 1 − β2
+
=
I 1 − η2
(29)
(30)
,
q I A⊥ = η. c 1 − η2
(31)
This means that the system is represented (at I > 0 ) as a generalized particle with the velocity η and the rest energy ε 0 : ε 0 = ( mc 2 + qϕ ) 1 −
( qA )
2
( mc2 + qϕ )
2
(32)
.
If particle velocity in the considered point equals β = −β 0 = −
qA , mc 2 + qϕ
(33)
then it corresponds to a resting generalized particle ( η = 0 ). But if the particle velocity is β = 0, this corresponds to motion of generalized particle at the velocity qA (34) η = β0 = . mc 2 + qϕ Invariant representation of the 4-momentum of a generalized particle is given in the form ⎛ π + β ⋅ π π + β ⋅ π β2 ⎞ ⎡β × [ π × β ]⎦⎤ ⎞ ⎛ π0 + β ⋅ π π0β + π|| ⎟ ⎜ ⎟, (35) , 0 , P=⎜ 0 β+ ⎣ = + π ⊥ ⎜ 1 − β2 ⎟ ⎜ 1 − β2 ⎟ β2 1 − β2 1 − β2 ⎝ ⎠ ⎝ ⎠ where P0 = ( π 0 , π ) is the four-momentum of a resting particle with P 2 = π02 − π 2 . Expression (35) may be represented in the matrix form JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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⎛ ⎞ β ⎛ β⋅π ⎞ β2 + π0 , ⎛ π0 ⎞ ⎜ π0 + ⎟ ⎜ ⎟ 2 2 2 ⎜ ⎟ 1− β ⎝ β ⎠ 1− β 1+ 1− β ⎜ ⎟ π1 l ⎜ ⎟ P= = T ⎜ ⎟, 2 ⎜ π2 ⎟ ⎜ ⎛β⋅π⎞β ⎟ β ⎛β⎞ β ⎜⎜ ⎟⎟ π0 + ⎜π+ ⎟ ⎜ ⎟ ⎜ ⎟ 1 − β2 ⎝ β ⎠ ⎝ π3 ⎠ 1 − β2 1 + 1 − β 2 ⎝ β ⎠ β ⎟ ⎜ ⎝ ⎠ l where T is the representation (transformation) matrix whose explicit form is given as [19]
)
(
(36)
)
(
β3 ⎞ 0 0 0 ⎞ ⎛1 β⎟ ⎜ ⎟ ⎟ ⎜ 0 β1β1 β1β 2 β1β3 ⎟ ⎛1 0 ⎟ 0 0 0⎟ ⎜ β2 β2 β2 ⎟ ⎜ 2 0 1 β ⎜ ⎟ ⎟+ Tl = ⎜ β2β1 β2β 2 β2β3 ⎟ . (37) ⎜0 0 ⎟ 1 + 1 − β2 1 − β 2 ⎜ 0 0 0 0⎟ ⎜ β2 β2 β2 ⎟ ⎜ 0 0 ⎜ ⎟ ⎝ ⎟ β3β1 β3β2 β3β3 ⎟ ⎜ ⎟ ⎜ 0 β2 0 0 0⎟ β2 β2 ⎟⎠ ⎝ ⎠ Matrices of invariant representation of a four-vector which preserve the vector magnitude in the fourspace form a Poincare group (nonhomogeneous Lorentz group). In addition to displacements and rotations the group contains representations of space-time reflections P, T and inversion PT = I . So, from the mathematical point of view, the group of transformation of a four-vector and the group of representation of this 4-vector are identical as a consequence of the principle of relativity of motion and conservation of four-dimensional properties of generalized momentum. This, in its turn, means that an arbitrary 4-vector is representable in terms of some basic four-vectors «at rest». Below are given representations of generalized momentum of the particles above at rest, β1 β
⎛ ⎜ 0 ⎜ 0 0⎞ ⎜ β1 ⎟ ⎜ 0 0⎟ β ⎜β + 1 0⎟ 1 − β2 ⎜ β2 ⎟ ⎜ 0 1⎠ ⎜β ⎜ β3 ⎜ ⎝β
β2 β
)
(
P+ = ε 0 (1,0 ) ,
P+2 = +ε 02 ,
P− = ε 0 ( 0, n 0 ) , P−2 = −ε 20 ,
(38)
P0 = ε 0 (1, n 0 ) , P0 2 = 0,
and in motion,
(
)
1 − β2 (1, β ) ,
P+ = ε 0
(
P− = ε0
(
P0 = ε0
1 − β2
)(
)(β ⋅ n , n 0
0
P+2 = +ε02 ,
)
+ n 0 ⊥ 1 − β2 ,
P−2 = −ε02 ,
)
1 − β2 1 + β ⋅ n 0 , β + n 0|| + n0 ⊥ 1 − β2 ,
(39)
P02 = 0.
It is suitable to use in last formulas the direction n of motion (momentum) in the observer's reference system. Then
(
P+ == ε 0 P− ==
P0 ==
)
1 − β2 (1, β ) , ε0
1 − (β ⋅ n)
2
( β ⋅ n, n ) ,
ε 0 1 − β2 (1, n ) , 1− β ⋅n
n=
n 0 ⊥ + n 0||
(
1 + β ⋅ n0 n=
1 − β2 1− β
2
β + n 0|| + n 0 ⊥ 1 − β2 1 + β ⋅ n0
)
2
(40)
,
,
where the last formula describes the Doppler effect. If the generalized momentum of a particle in an external field is considered, it may also be represented via expressions (40) depending on the sign of the invariant (20). JOURNAL OF CONTEMPORARY PHYSICS (ARMENIAN Ac. Sci.)
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4. CONCLUSION So, allowance for the dependence of interaction on the velocity of particle motion leads to invariance of representation of the generalized momentum. Relativistic-invariant representation of generalized momentum of a particle in an external field allows establishing exact correspondence of expressions for energy and potential energy for the classical Hamiltonian which makes identical the solutions to the problems of mechanics with relativistic and classical approaches. Invariance of representation of generalized momentum allows describing equivalently a physical system in geometrically conjugate coordinate and momentum (kinematical and dynamical) spaces by kinematical and dynamical (canonical) variables. These spaces must be mutually representable and equivalent in description of properties of physical systems. Respectively, representation of a physical system in kinematical and dynamical spaces should be expressed via the metrical and differential correspondence of kinematical and dynamical spaces. Physical and geometrical properties of these conjugate spaces may completely be described in the framework of variational approaches which will be given in the subsequent work. ACKNOWLEDGMENT Author is grateful to Prof. V. Chaltykyan (IPR, NAS of Armenia), for helpful discussions and valuable remarks during preparation of the manuscript. REFERENCES 1. Anan’ev, L.M., Vorob’ev, A.A., and Gorbunov, V.I., Induktsionnyi uskoritel’ elektronov – betatron (Induction Accelerator of Electrons – Betatron), Moscow: Gosatomizdat, 1961. 2. Vakhrushin, Yu.P. and Anatskii, A.I., Lineynye induktsionnye uskoriteli (Linear Induction Accelerators), Moscow: Atomizdat, 1978. 3. Hopwood, J., Plasma Sources Sci. Technol., 1992, vol. 1, p. 109. 4. Lieberman, M.A. and Lichtenberg, A.J., Principles of Plasma Discharges and Materials Processing, New York: Wiley, 1994. 5. Bottin, B. et al., Predicted and Measured Capability of the VKI 1.2 MW Plasmatron Regarding Re-Entry Simulation, ESA SP-426, 1998. 6. LaPointe, M.R. and Mikellides, P.G., High-Power Magnetoplasmadynamic and Pulsed Inductive Thrusters, NASA Glenn Research Center, OAT, OSS, Project ASTP, 2002. 7. Lister, G.G., Lawler, J.E., et al., Rev. Modern Phys., 2004, vol. 76, p. 542. 8. Razhev, A.M., Mekhitarian, V.M., Churkin, D.S., JETP Lett., 2005, vol. 82, p. 259. 9. Razhev, A.M. and Churkin, D.S., Opt. Precis. Eng., 2011, vol. 19, p. 237. 10. Hittorf, W., Ann. Phys. Chem., 1884, vol. 21, p. 90. 11. Valoshek, P., The Infancy of Particle Accelerators: Life and Work of Rolf Wideröe, Wiesbaden: Vieweg, 1994. 12. Kerst, D.W., Phys. Rev., 1941, vol. 60, p. 47. 13. Kerst, D.W. and Serber, R., Phys. Rev., 1941, vol. 60, p. 53. 14. Landau, L.D. and Lifshitz, E.M., Teoriya polya (Theory of Fields), Moscow: Fizmatlit, 2003. 15. Bloch, F., Phys. Rev., 1946, vol. 70, p. 460. 16. Berestetskii, V.B., Lifshitz, E.M., and Pitaevskii, L.P., Kvantovaya elektrodinamika (Quantum Electrodynamics), Moscow: Fizmatlit, 2002. 17. Landau, L.D. and Lifshitz, E.M., Mekhanika (Mechanics), Moscow: Fizmatlit, 2004. 18. Landau, L.D. and Lifshitz, E.M., Kvantovaya Mekhanika (Quantum Mechanics), Moscow: Fizmatlit, 2004. 19. Tonnelat, M.-A., Osnovy elektromagnetizma i teorii otnositel’nosti (The Principles of Electromagnetic Theory and Relativity), Moscow: IIL, 1962.
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