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W.I. Fushchych, Scientific Works 2001, Vol. 3, 332–336.

On vector and pseudovector Lagrangians

for electromagnetic field

W.I. FUSHCHYCH, I. KRIVSKY, V. SIMULIK

A Lagrange function in terms of electromagnetic field strengths is constructed which is

?

a 4-vector with respect to the total Poincar? group P (1, 3) and whose Euler–Lagrange

e

equations are equivalent to the Maxwell equations. The advantages of the Lagrange

function proposed in comparison with the known pseudovector with respect to the

?

P (1, 3) group Lagrange function are shown. The conserved quantitites on the basis of

corresponding generalization of Noether theorem are found.

A development of Lagrange approach (L-approach) in electro-dynamics in terms

of field-strength tensor F = (F µ? ) = (E, H) of electromagnetic field, without using

the potentials Aµ , was discussed in [1–4]. It is easy to show, that in terms of (E, H)

there is no scalar, with respect to the Poincar? group P (1, 3), Lagrange function, for

e

which the Euler-Lagrange (EL) equations coincide with the Maxwell equations.

The purpose of this work is to construct a vector (with respect to the total

?

Poincar? group P (1, 3) i.e., P (1, 3) group including the space and time reflections)

e

Lagrange function in terms of (E, H), with the help of which the system of equations

equivalent to the Maxwell equations can be received from the EL equations. The

conserved quantities are constructed on the basis of corresponding generalization of

Noether theorem. Further we will call such Lagrange vector-function a Lagrange

vector.

Let us represent the Maxwell equations

?0 E = rot H ? j, ?0 H = ?rot E,

div E = ?, div H = 0 (1)

in a manifestly covariant form

Qµ = j µ , Rµ = 0, (2)

µ = 0, 1, 2, 3,

where

1 µ???

Qµ ? F,? , Rµ ? ?F,? , ?F µ? ?

µ? µ?

(3)

? F?? ,

2

F = (F µ? ) is a tensor of electromagnetic field:

F = (F µ? ) = (E, H) : F 0i = E i , F ij = ?ijk H k , F µ? = ?F ?µ , (4)

j is a 4-vector of current:

j ? (j µ ) = (?, j), j 0 = ?, j = (j i ), (5)

i = 1, 2, 3,

and ?µ??? is a completely antisymmetric unit tensor, ?0123 = 1:

x = (xµ ) ? R(1, 3), ?µ ? ?/?xµ . (6)

Preprint № 466, Institute for Mathematics and its Applications, University of Minnesota, 1988, 6 p.

On vector and pseudovector Lagrangians for electromagnetic field 333

The explicit form of the components Qµ , Rµ is the following

Qi = (??0 E + rot H)i ? ??0 E i + ?ijk ?j H k ,

Q0 = div E, (7)

Ri = (??0 H + rot E)i ? ??0 H i + ?ijk ?j E k .

R0 = div H, (8)

Now consider the tensor Tµ?? and pseudotensor Tµ?? of 3-rd rank (with respect

?

to P (1, 3) group), which are constructed from 4-vectors Qµ , Rµ (3):

Tµ?? ? a[gµ? (Q? ? j? ) ? gµ? (Q? ? j? )] + b?µ??? R? , (9)

Tµ?? ? a (gµ? R? ? gµ? R? ) + b ?µ??? (Q? ? j ? ), (10)

a, b, a , b are constant coefficients.

Theorem 1. For any ab = 0 = a b each of the set of equations

(11)

Tµ?? = 0,

(12)

Tµ?? = 0,

is equivalent to the initial Maxwell equations (2).

One can easily varify the validity of this statement by rewriting the components

of tensors T , T (11), (12) in the evident form.

? ?

Only the P -tensor set of equations (11) and P -pseudotensor set of equations

?

(12) will be used in this work for the construction of P -vector L-approach for the

electromagnetic field F = (E, H).

Let us introduce in addition to the Lagrange variables for tensor electromagnetic

??

field new Lagrange variables F , F,µ which are dually conjugated to F , F,µ (on the

?

manifold ?0 of the solutions of Maxwell’s equations F = ?F see (3)). The general

?

form of P -vector Lagrange function

??

Lµ = Lµ (F, F,? , F , F,? ), Lµ : R60 > R1 (13)

up to a total 4-divergence terms is the following:

? ? ??

Lµ = a1 Fµ? Q? + a2 Fµ? R? + a3 ?Fµ? R? + a4 ?Fµ? Q? + a5 Fµ? Q? +

(14)

? ?? ? ?

+ a6 Fµ? R? + a7 ?Fµ? R? + a8 ?Fµ? Q? + (q1 Fµ? + q2 ?Fµ? )j ? .

Here we are using also notations

1 µ??? ?

? ? µ? ? ? µ? ?

Qµ ? F,? , Rµ ? F,? , ?F µ? ? (15)

? F?? .

2

?

Theorem 2. The EL equations for P -vector L = (Lµ ) are equivalent to the Maxwell

equations if and only if the following conditions on the coefficients in (14) are

fulfilled

a8 ? a2 = a = ?b = ?q1 ? ?q = 0, a6 ? a4 = a = ?b = 0,

(16)

a1 ? a3 ? a6 ? a8 = a2 + a4 + a5 ? a7 = 0.

?

Proof. The calculation of Lagrange derivatives ?Lµ /?F?? and ?Lµ /? F?? from Lµ (14)

leads to the result that the EL equations for the Lagrange vector (14) may coincide

only with the equations (11), (12), and only in the following form

? (17)

?Lµ /?F?? = Tµ?? = 0, ?Lµ /? F?? = Tµ?? = 0,

and it is possible only if the conditions (16) are fulfilled.

334 W.I. Fushchych, I. Krivsky, V. Simulik

The four component of the Lagrange vector (14) generate four actions

? ? ?

W µ (F, F ) = d3 xLµ F (x), F (x), ?vF (x), ?v F (x) , (18)

where F , F belong to the set ? of twice differentiable functions, and ?µ defines the

?

0

set of extremals of the action (18) with a fixed µ.

Theorem 3. The intersection ?0 = ?µ ?µ of the sets ?µ of extremals of four actions

0 0

(18) given by the Lagrange function Lµ (14) whose coefficients obey the equations

(16), coincides with the set of solutions of Maxwell’s equations (1).

Proof. The validity of this theorem follows from the derivation of the evident form of

EL equations for (14), i.e. from (17) and the theorem I about the equivalence of the

sytems of equations (11), (12) and the Maxwell equations (2), i.e. (1).

?

The P -vector Lagrangian (14), proposed here, has several advantages in compa-

?

rison with the P -pseudovector Lagrangian from [3], which in our notation has the

form

Lµ = Lµ (F, F,? ) = F µ? R? ? ?F µ? (Q? ? j? ). (19)

Firstly, Lagrangian (19) leads only to the pseudotensor system of equations (12),

i.e. it unreasonably separates the pseudo-tensor system of equations (12) in compari-

son with the tensor system of equations (11). That is a direct consequence of a

pseudovector character of Lagrangian (19). Let us note, that without appealing to

? ?

the additional Lagrange variable F it is impossible to construct a P -vector Lagrange

?

function: the demand of function Lµ (F, F,? ) being a P -vector leads to the expression

Lµ = Lµ (F, F,? ) = F µ? Q? + ?F µ? R? , (20)

for which the EL equations are the identities.

Secondly, as it is seen from the terms with the current in (19) the interaction

?

Lagrangian in [3] also is a P -pseudovector one:

LI = ?F µ? j? , LI0 = j · H, LIi = (j ? E ? ?H)i . (21)

A physically unsatisfactority of such an interaction is evident already from the fact,

that density of electric charge in (21) is connected not with the electric field strengths

but with the magnetic field strength H.

Finally, thirdly, during the derivation of conserved quantities the Lagrange function

?

(19) put into correspondence for P -tensor generator of the Poincar? group a pseudo-

e

tensor conserved currents. This shortcoming together with the above mentioned ones,

?

is overcome using the P -vector Lagrange function (14).

Derivation of conservation quantities in the framework of L-approach formulating

here inquires a generalization of Noether theorem for the case of vector Lagrangians.

Theorem 4. Let

q : F (x) > F (x) = g F (x) (22)

? ?

On vector and pseudovector Lagrangians for electromagnetic field 335

be the arbitrary transformation of invariance of equations (2) with j = 0. Then the

tensor of current ?? , constructed on the basis of Lµ (14) (of course with j = 0)

µ

according to the formula

1 ?L? ?L? ? ??

df

q > ?? =

µ ??

? ?? F + ? ?? F ,

2 ?F,µ ? F,µ (23)

?

F ? q F, F ? q F = ??F.

? ? q

is symmetric and its divergence vanishes for any solution of the equation (2) with

µ

(24)

?µ ?? = 0.

Proof. Derivation of currents (23) for Lµ (14) with j = 0 leads to the result

1µ

q > ?? = A F µ? F?? + F

µ µ?

F?? + ?? F ?? F?? ,

?

2 (25)

A = a1 ? a2 + a7 ? a8 = a3 + a4 + a5 + a6 .

Symmetry of the tensor (25) is evident and the equation (24) is a consequency of

the Maxwell equations (2) with j = 0.

Note that in the vector L-approach the four conservation quantities correspond

(according to the Noether theorem) to one generator of invariance transformation.

Let us give the analysis of conserved quantities which are the consequences of

(25). We receive, taking A = 1, that generators of 4-translations ?? according to the

formula (25) give the trivial current

?? > ?µ? (? = ?? ) = (?? )µ? ? ?? T µ? , (26)

q

where T µ? is standard energy-moment tensor for the field

1µ

Tµ = Pµ ,

T? = F µ? F?? + ?? F ?? F?? ,

µ 0

(27)

4

12

P0 ? P ? (E ? H)j .

(E + H 2 ), (28)

2

For the analysis of integral conserved quantities

? ?0µ (x) = ?0µ (?) ? (?0µ )

?µ = d3 x?0µ (x) = const, (29)

q q

it is sufficient to represent the densities ?0µ , ommiting the terms with spacelike

?

derviatives, which do not contribute to the integral ?µ (29). We obtain from the

0µ

formula (25) for the densities ? , corresponding to the rest of the generators of

conformal algebra C(1, 3) (the definition of algebra C(1, 3) see, for example in [5])

the following expressions:

??? > J 0µ = ?? P? ? ?? P? , d > D0µ = P µ ,

µ µ

(30)

j ??

?

K? > K? = 2(?? D + J?? G ?µ ),

0µ µ

(31)

where

D ? xµ Pµ , J?? ? x? P? ? x? P? . (32)

336 W.I. Fushchych, I. Krivsky, V. Simulik

? j, ? ?

As one can see, C(1, 3)-generators q = (?, ? d, k) lead here to the conserved

?

quantities, which are expressed in terms of well-known series of main conservation

стр. 78 |