ñòð. 117 |

on the basis of hte corresponding generalization of Noether theorem are found.

A development of Lagrange approach (L-approach) to electrodynamics in terms

of field strength tensor F = (F µ? ) = (E, H) of the elcectromagnetik 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

e

function, for which the Euler–Lagrange (EL) equations coincide with the Maxwell

equations.

?

The aim of this paper is a construction of a P (1, 3) vector Lagrangian in terms

of (E, H), i.e. a Lagrange function Lµ which is the vector with respect to the total

?

Poincar? group P (1, 3) (including both P (1, 3) and the spase-time reflections) and for

e

which the EL equations are exactly equivalent to the original Maxwell equations. In

what follows such a Lagrangian Lµ will be called a Lagrange vector.

Let us represent the Maxwell equations

?0 E = curl H ? j, ?0 H = ?curl E, (1)

div E = ?, div H = 0

in a manifestly covariant form

µ = 0, 3 ? 0, 1, 2, 3.

Qµ = j µ , Rµ = 0, (2)

Here

1 µ???

Qµ ? F µ? = ?? F µ? (x), Rµ ? ?F µ? , ?F µ? ? (3)

? F?? ,

,? ,?

2

F = (F µ? ) is the field strength tensor:

F µ? = ?F ?µ ,

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

j is a current density:

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

i = 1, 2, 3,

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

x = (xµ ) ? (x0 , x1 , x2 , x3 ), ?µ ? ?/?xµ . (6)

Nuovo Cimento B, 1989, 103, ¹ 4, 423–429.

On vector and pseudovector Lagrangians for electromagnetic field 507

The componets Qµ , Rµ of the vectors Q ? (Qµ ) and R ? (Rµ ) are connected with

the field strengths E ? (E i ) and H ? (H i ) as

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

Q0 = div E, (7)

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

R0 = div H, (8)

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

?

the P (1, 3) group), which are constructed with the help of the 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)

where a, b, a , b are arbitrary constants.

Theorem 1. For any a, b, a , b = 0 each of the sets of equations

(11)

Tµ?? = 0,

(12)

Tµ?? = 0

is equivalent to the original Maxwell equations (2).

One can easily verify the validity of this assertion by rewriting the components of

tensors T , T (11), (12) in the explicit form.

? ?

Just the P -tensor set of equations (11) and P -pseudotensor set of equation (12)

?

will be used in this work for the construction of P -vector L-approach to the electro-

magnetic field F = (E, H).

Let us introduce in addition to the Lagrange variable for tensor eletromagnetic

??

field the 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 Lagrangian

??

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 have used also notations

1 µ??? ?

? ? ? ? ?

Qµ ? F µ? , Rµ ? F µ? , ?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 coefficients in (14) obey the conditions

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

(16)

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

Proof. The straightforward calculations of Lagrange derivatives for the Lagrangian

Lµ (14) lead to the result

? (17)

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

if and only if conditions (16) are fulfilled.

508 W.I. Fushchych, I.Yu. Krivsky, V.M. Simulik

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

? ? ? ?

d3 x Lµ (F (x), F (x), ?? F (x), ?? F (x)), F, F ? ?,

W µ (F, F ) = (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 of the sets ?µ of extremals of four actions

0 0

µ

(18), given by the Lagrangian L (14) whose coefficients obey conditions (16) coin-

µ

cides with the set of solutions of Maxwell equations (1).

Proof. The validity of this assertion follows from the derivation of the explicit form

of EL equations for (14), i.e. from (17) and the Theorem 1 about the equivalence of

the set of eqs. (11) or (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 notations 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 eqs. (12), i.e. it

gives rise to the pseudotensor system of eqs. (12) in favour of the tensor system of eqs.

(11). That is a direct consequence of the pseudovector character of Lagrangian (19).

? ?

Let us note that without appealing to the additional Lagrange variable F ? (F µ? ) it

?

is impossible to construct a P -vector Lagrangian: 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 is seen from the terms with the current in (19), the interaction

?

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

Lµ = ?F µ? j? , L0 = j · H, Li = (j ? E ? ?H)i . (21)

I I

I

A physical unsatisfactoriness of such an infraction is evident already from the fact that

the density of electric charge in (21) is connected not with the electric-field stengths

E but with the magnetic-field strength H.

Finally, the calculation of conserved quantities on the basis of Lagrangian (19)

?

gives the result that a P -tensor generator of the Poincar? group is corresponded by

e

? -pseudotensor conserved currents. This defect, together with the above-mentioned

P

?

ones, is eliminated by using the P -vector Lagrangian (14).

Derivation or conserved quantities in the framework of the L-approach formulated

here demands a generalization of Noether theorem for the case of vector Lagrangian.

Theorem 4. Let

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

? ?

be an arbitrary invariance transformation of eqs. (2) with j = 0. Then the conserved

?

current ?? , constructed on the basis of the P -vector Lagrangian Lµ (14) (of course

µ

with j = 0) with the help of the formula

?L? ?L? ?

df ? ??

q > ?? = F ? q F, F ? q F = ??F,

µ ?? ??

(23)

? ?? F + ?? F , ? q

?F ,µ ?F ,µ

On vector and pseudovector Lagrangians for electromagnetic field 509

is symmetric and its divergence vanishes for any solutions of eqs. (2) whith j = 0:

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 .

Summetry of tensor (25) is evident and eq. (24) is a consequence of the Maxwell

equations (2) with j = 0.

Note that in the vector L-approach the correspond (according to the Noether

theorem) to one generator of invariance transformation.

Let us give a short discussions of conserved quantities which are the concequences

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

the formula (25) give the trivial current

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

q

where T µ? is standard energv-momentum tensor for the field F = (E, H):

1µ

T? = F µ? T?? + ?? F ?? F?? , Tµ = Lµ ,

µ 0

(27)

4

1

L0 ? (E 2 + H 2 ), Lj ? (E ? H)j . (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 write down the densities ?0µ , omitting the terms with spacelike

?

derivatives, which are not contributed in integral ?µ . We obtain from formula (25) for

the densities ?0µ , corresponding to the rest of generators of conformal algebra C(1, 3)

(for the definition of algebra C(1, 3) see, for example, [5]), the following expressions:

?

?

J?? > J?? = ?? L? ? ?? L? , d > D0µ = Lµ ,

0µ µ µ

(30)

?

K? > K? = 2(?? D + J?? g ?µ ),

0µ ?

(31)

where

D ? xµ Lµ , J?? ? x? L? ? x? L? . (32)

? j, ? ?

As one can see from (30)–(32), the C(1, 3)-generators q = (?, ? d, k) lead here

?

to the conserved quantities, which are expressed in terms of well-known series of

main conserved quantities for the electromagnetic field F = (E, H), found by Bessel-

Hagen [6] on the basis of the L-approach for vector field A = (Aµ ) of potentials,

namely

d3 x L? (x), d3 x (x? L? (x) ? x? L? (x)),

P? = J?? =

(33)

d x (2x? D(x) ? x L? (x)).

3 3 2

D= d xD(x), K? =

510 W.I. Fushchych, I.Yu. Krivsky, V.M. Simulik

It is interesting to note that formula (25) gives the identical zero for the generator

q = ? of duality transformations. In order to obtain nontrivial conservation laws with

?

the help of ? let us remind ref. [1], where new invariance algebra A32 ? C(1, 3) of

free Maxwell’s equations was found. A subset of the generators of the algebra A32

? j, ? ?

has the form of composition q = ?? of C(1, 3) generators q = (?, ? d, k) and the

? q ?

generator ?. Formula (25) gives nontrivial conservation laws just for the generators

? j, ? ?

q = (??, ?? ?d, ?k). The corresponding integral conservation laws expressed in terms

?

of series

d3 x Z? (x), d3 x (x? Z? ? x? Z? ),

µ µ µ µ µ

Z? = Z?? =

(34)

ñòð. 117 |