PHYSICS
УДК 539
Corresponding Member of NAS RA A. G. Bagdoev, A. V.
Shekoyan
Generalized Nonlinear Equations of Magnetohydrodynamical
Media
(Submitted 17/III 2006)
Keywords: Maxwell equations, magnetohydrodynamics, Galilean
transformations
From elder classes of school we were
fascinating by the theory of relativity, which we studied by popular
literature.
At the ripe age the sceptical
attitude to the theory of relativity arised in ourselves. In the middle of 60 th
of XX th century A.G. Bagdoev informed the president of AS of Armenia V. A.
Ambartsumyan, and some years ago also the leading Armenian physicists about his
judgments on this question.
It is difficult
to say that they completely agreed with him, but there weren't also any remarks
very much to the point. Recently we acquainted with the works of Academician of
RAS S.S Grigoryan [1], where he showed insolvency of the general theory of
relativity in explanation of various Cosmogonic problems, in particular, of
generation of the black holes, and on the other hand he showed the applicability
of the Newton's theory, developed by Mitchell and Laplace to this problems. This
encouraged us to write this article, concerning to the critical analysis of the
principal sources of the special theory of relativity. There are experimental
data of American physicists from LosAlamos about variability of the speed of
light and facts of constancy in big volumes of temperature in cosmic objects,
which leads to the necessity to take into account the fact of the presence of
the super light speed. These facts, which are in contradiction with special
theory of relativity, are brought in the article of Pyotr Obraztov, published
"In
science world" N2, 2005.
The basic argument
for creation the special theory of relativity was the thesis due to which the
linear Maxwell equations are invariant with respect to more general Lorents
transformations, in contrast to nonlinear equations of mechanics, which are
invariant with respect to Galilean transformations and therefore one must fit
the equations of mechanics to these more general
transformations.
Following to this logics one
must prefer to general nonlinear equations of hydromechanics, which are
invariant with respect to Galilean transformation, the obtained from them after
linearisation for small disturbances, wave equation, which is invariant with
respect to more general Lorents transformation. Of course nobody will insist on
it. Then the question arises whether it is not better to fit Maxwell equations,
derived in earth conditions for relatively small electromagnetic fields with
respect to the enorme light speed, and write down nonlinear Maxwell equations,
which by analogy with mentioned example, must satisfy to more particular but
more natural Galilean transformations.
By the
way the equations of this type are equations of magnetohydrodynamics, obtained
from matching solution of Maxwell equations for moving media [2, 3]

(1) 
where current density is connected
with
by Ohm law

(2) 
with equations of motion of media

(3) 
For
not too small variated in time processes or (which is the same) for not too
large frequencies, the item with displacement current in (1) can be neglected,
and as a result the induction equation can be obtained from (1), (2)

(4) 
and instead of (3)

(5) 
and equations (4), (5) are invariant
with respect to Galilean transformation, where to the same accurancy
can be taken as the invariant.
These equations were successfully applied to study nonlinear waves in
pulsars [4, 5], where the great frequencies and large magnetic fields take
place. In these conditions the Alfen velocity has the same order as light
velocity. In such problems we must take into account nonlinear effects and
solve nonlinear Maxwell equations coupled with hydrodynamical equations Usually
in magnetohydrodynamics frequencies of process w are
much smaller than electroconductivity s, which allows
to neglect the displacement current.
In the
case of w ~ s for example for magnetic stars [4, 5] w ~ 10^{14}[1/sec], we have
also to take into account the displacement currents in magnetohydrodynamic
equations and, as it was mentioned, to take into account nonlinearity for
considered great magnetic fields, i.e. to write down nonlinear Maxwell
equations.
As an example of taking into
account the displacement current in nonlinear equations (1)(3) which satisfy
the condition of invariancy with respect to the Galilean transformations, as
well as in (4), (5) we can replace the displacement current in (1) on
where derivative in time is taken for moving particle. Then system (1)(3) again
will be invariant respectively to Galilean transformations, where
is a simply parameter, which
can be excluded as in derivation of (4).
Taking into consideration the improved value of displacement current we
have equations of moving electroconducting media in the form (1)(3), where
Note that taking into account the displacement current demands to revise
orders of parameters. To do it let us simplify the system of equations, making
the iteration in the first equation

(6) 
Placing (6) in (2) we obtain the induction equation with the accounting
of displacement current

(7) 
Here
m = 1. Taking into
consideration that in propagating waves [d /dt] ~ w, С ~
[(w)/(c_{n})], where c_{n} is
wave speed, from (7) we obtain orders w ~ s,
v ~
c_{n}, c_{n} ~ c, E ~
H  
(8) 
where [3, 4] c_{n} ~ a_{1}, a_{1 }= is Alfen speed. From the equation of motion (3) we obtain

(9) 
It
is evident that orders (8), (9) take place for general nonsimplified system of
equations as well.

(10) 

(11) 

(12) 
System (10)(12) due to (8), (9) describes the motion of
electroconducting fluids in great magnetic fields and with great velocities and
pressures, and is invariant with respect to Galilean
transformation.
Of course it must be
considered only as a possible example of modification of Maxwell equations for
moving media, and accounting of nonlinearity, the mentioned equations are
invariant with respect to Galilean transformations.
Note that system (10)(12) is closed together with the equation of
continuity of mass and the state equation of fluid. At the same time it takes
place the [2] equation of continuity of charge density r_{e} in form

(13) 
If
we place a usual Maxwell equation (1) due to which
= [1/(4p)][(¶)/(¶t)] in (13), we obtain from
(13)

(14) 
which is the[2] known Culon law,
derived initially for static problem, and postulated by Maxwell for dynamic
problems, by the way, for inmoved media. In considered statement of problem for
moving media equations (10)(12) yield

(15) 
From (13) and (15) we obtain

(16) 
After resolving the equations (10)(12), from (16) we obtain r_{e}.
For
inmoving media
= 0 and (16) coincides with (14), and
in general case of moving media the correctness of (16) is not less based than
of equation (14). Also note that as it is mentioned in [2] it is not necessary
to derive (2) from Lorents transformations, in which the velocity of inertial
system is supposed to be constant, but in (2)
is a
variable velocity of particle, and as it is done in [2] to assume (2) as an
experimental Ohm law for moving media. Thus, on our opinion, the special theory
of relativity can be considered as an asymptotic theory taking place for large
velocities and bounded fields, which allows it to describe the motion of
elementary particles with corresponding shortening of length and time, but does
not make a fetish of its results in application to nonlinear equations of
mechanics, as well as to electrodynamic equations, particularly for large
electromagnetic fields.
In applications to
these nonlinear equations Galilean transformations are more natural than more
general by form, but more particular by applications, Lorents
transformations.
Also note that equations of
motion of twocomponent plasma and equations of motion of mixtures of gas and
fluid in clouds are not invariant with respect to Lorents transformation, and
for the former ones the dispersion law for waves is obtained, which gives
electromagnetic waves for large frequencies and magnetohydrodynamic waves for
small ones [6].
And it is not posed the
condition of invariancy of equations with respect to any transformations in
inertial systems, which is the basic argument of the special theory of
relativity.
Institute of Mechanics NAS RA
Литература
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2005. С. 164169.
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Н.С. Основы радиоэлектр. М. Энергия. 1969. 879 с.
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4. Багдоев А.Г., Седракян Д.М.  Астрофизика. 2003. Т.46.
Вып. 1. С. 8794.
5. Тамм И.Е.  Основы теории электричества. М. Наука. 1966. 624
с.
6. ФранкКаменецкий Д.А. Лекции по физике плазмы. М. Атомиздат. 1964. 283с.