стр. 102 |

One can demand that the above vectors have to satisfy conditions (22).

Detailed symmetry analysis and construction of exact solutions of the above sug-

gested equations will be carried out in future paper.

Note 6. The classical wave equation

?2u

? c2 ?u = 0

2

?t

in our approach is generalised in the following way

?2u ? ?u

? akl = ?1 (v 2 )vk vl + ?2 (v 2 )?ik ,

akl (?)

v = 0,

?t2 ?xk ?xl

or ??2 vi = 0.

??vi = 0

In one dimensional space the wave equation has the form

?2u ?2v

? ?u

? v2 = 0, ??v + ?3 = 0,

?t2 ?x2

?xk ?x

?1 , ?2 , ?3 are smooth functions of v 2 .

1. Fushchych W.I., Tsifra I.M., On the symmetry of nonlinear electrodynamics equations, Teoret.

Matem. Fizika, 1985, 64, № l, 41–50.

2. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, Reidel, 1987, 214 p.

3. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear

equations of mathematical physics, Kiev, Naukova Dumka, 1989, 336 p.

W.I. Fushchych, Scientific Works 2002, Vol. 4, 436–438.

The complete sets of conservation laws

for the electromagnetic field

W.I. FUSHCHYCH, A.G. NIKITIN

We present a compact and simple formulation of zero- and first-order conserved currents

for the electromagnetic field and give the number of independent n-order currents.

New conservation laws for the electromagnetic field, discovered by Lipkin [1], had

obtained an adequate mathematical and physical interpretation long ago, see e.g. [2–

6]. It happens that these conservation laws are nothing but a small part of the

infinite series of conserved quantities which exist for any self-adjoint linear system

of differential equations; among their number are Maxwell’s equations [7]. As to the

physical interpretation of Lipkin’s zilch tensor it can be connected with conservation

of polarization of the electromagnetic field [5, 6].

The aim of the present letter is to establish certain rules in the bewildering

complexity of the conservation laws and to describe complete sets of them for the

electromagnetic field.

(m) (m)

We say that an arbitrary bilinear function jµ = fµ (Dn F, Dk F ) is a conserved

current if it satisfies the continuity equation

? µ jµ = 0,

(m)

(1)

µ = 0, 1, 2, 3.

Here F = Fµ? is the tensor of the electromagnetic field,

n

n

? µ? ,

D= µ? = 0, 1, 2, 3, m = max(n + k).

?=0

It follows from (1) according to the Ostrgradskii–Gauss theorem that the following

quantity is conserved in time:

(m) (m)

d3 xj0

j0 = .

(m) (m)

We say conserved currents jµ and jµ are equivalent if

(m) (m)

j0 = j0 .

Proposition 1. There exist exactly 15 non-equivalent conserved currents of zero

order for Maxwell’s equation. All these currents can be represented in the form

jµ = Tµ? K ? ,

(0)

(2)

where Tµ? is the traceless energy-momentum tensor of the electromagnetic field and

K ? is a Killing vector satisfying the equations

1

? ? K µ + ? µ K ? ? g µ? ?? K ? = 0. (3)

2

J. Phys. A: Math. Gen., 1992, 25, L231–L233.

The complete sets of conservation laws for the electromagnetic field 437

Proof. This reduces to finding the general solution of the equation

(0) (0)

? 0 j0 = ?0 d3 xj0 (F, F ) = 0, (4)

(0)

where j0 (F, F ) is a bilinear combination of components of the tensor of the electro-

(0)

magnetic field. It is not difficult to find such a solution, decomposing j0 by the

complete set of symmetric matrices of the dimension 6 ? 6

(0)

j0 = ?T Q?, ? = column(F01 , F02 , F03 , F23 , F31 , F12 ),

Q = (?0 Aab + ?1 Aab + ?3 Aab )Zab + ?2 Sa K a ,

0 1 3

Zab = 2?ab + Sa Sb + Sb Sa , a, b = 1, 2, 3,

? I? ?I

S 0 0 0

Sa = ?a , ?0 = ? , ?1 = ,

I?

0 Sa 0I 0

? ?I I?

0 0

?2 = , ?3 = ? ,

0 ?I

?

I0

? ? ? ? ? ?

0 ?1

00 1 0 01 0

S1 = ?0 0 ?1 ? , S2 = ? 0 0 0 ? , S3 = ?1 0 0 ?,

?1 0 0

01 0 00 0

where ? and I are the zero and unit matrices of dimension 3?3, Aab , K a are unknown

0 ?

functions of xµ . In fact substituting (5) into (4) and using the Maxwell equations

?µ F µ? = 0, ? µ ?µ??? F ?? = 0

we come to the relations Aab = Aab = 0, Aab = ?? ab K 0 and to the equations (3) for

1 3 2

K 0 and K a .

(0)

Thus we have found all non-equivalent j0 satisfying (4). The corresponding

(0)

expressions for jµ with µ = 0 can be obtained by Lorentz transformations.

Formula (2) gives an elegant formulation of the classical conservation laws of

Bessel–Hagen [8]. We present a direct (and simple) proof that there are not another

conserved bilinear combination of the electromagnetic field strengths.

In an analogous way it is possible to prove the following assertion.

Proposition 2. There exist exactly 84 conserved currents of first order for the elec-

tromagnetic field. All these currents can be represented in the form

jµ = K ?? Z??,µ + 2?µ??? (? ? K ?? )T ?? ,

(1)

(5)

where T ?? is the energy-momentum tensor, Z??,µ is Lipkin’s zilch tensor, ?µ???

is the completely antisymmetric unit tensor, K ?? is a conformal Killing tensor of

valence 2, satisfying the equations

1

? (µ K ??) = ?? K ?(µ g ??) , K ?? = K ?? , Kµ = 0,

µ

(6)

3

where symmetrization is imposed over the indices in brackets.

Using the relations

? µ Z??,µ = 0, Zµ?, ? = 0, ?? T ?µ = 0, T ? ? = 0,

? ? (????? T ?µ + ??µ?? T ?? ) = Z??, µ + Zµ?, ?

438 W.I. Fushchych, A.G. Nikitin

and the equations (7) we can ensure that the currents (6) really satisfy the continuity

equation (1).

Thus all non-equivalent conserved currents of first order are given by formula (6).

The general solution of the equation (7) is a fourth-order polynomial of xµ depending

on 84 parameters; for the explicit expression of K ?? see e.g. [9]. Formula (6) descri-

bes well known and also ‘new’ conserved currents; the latter depend on the fourth

degree of xµ .

In conclusion we note that in an analogous way it is possible to describe conserved

currents for the electromagnetic field of an arbitrary order m. For m > 1 such currents

are defined by two fundamental quantities i.e. by the conformal Killing tensor of

valence m + 1 and the Floyd–Penrose tensor of valence R1 + 2R2 where R1 = m ? 1,

R2 = 2. The higher order conserved currents will be considered in a separate paper;

here we present only the number of linearly independent currents of order m:

1

(2m + 5) 2m(m + 1)(m + 4)(m + 5) + (m + 2)2 (m + 3)2 ,

Nm = m > 1.

2

For the details about generalized Killing and Floyd–Penrose tensors in applica-

tion to higher symmetries of Poincar?- and Galilei-invariant wave equations see the

e

extended version of our book [10]. Non-Lie symmetries and conservation laws for

Maxwell’s equations are discussed in [11].

1. Lipkin D.M., J. Math. Phys., 1964, 5, 696.

2. Klibble T.W., J. Math. Phys., 1964, 5, 1022.

3. Morgan T.A., J. Math. Phys., 1964, 5, 1659.

4. Michelson J., Niederle J., Lett. Math. Phys., 1984, 8, 195.

5. O’Connel R.F., Tompkins D.R., Nuovo Cimento, 1965, 39, 391.

6. Candlin P.I., Nuovo Cimento, 1965, 37 1390.

7. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986.

8. Bessel-Hagen E., Math. Ann., 1921, 84, 258.

9. Nikitin A.G., Ukr. Math. J., 1991, 43, 786.

10. Fushchych W.I., Nikitin A.G., Symmetries of the equations of quantum mechanics, Moscow, Nauka,

1990 (in Russian); New York, Allerton Press, 1994.

11. Fuschych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, Reidel, 1987.

W.I. Fushchych, Scientific Works 2002, Vol. 4, 439–447.

Symmetry reduction of the Navier–Stokes

equations to linear two-dimensional systems

of equations

W.I. FUSHCHYCH, R.O. POPOVYCH

Побудовано повний набiр нееквiалентних двовимiрних пiдалгебр максимальної в

ceнci Лi (нескiнченовимiрної) алгебри iнварiантноотi рiвнянь Навье–Стокса для в’яз-

кої нестискної рiдини. Отримано анзаци, що редукують рiвняння Навье–Стокса до

лiнiйних систем ДРЧП вiд двох незалежних змiнних. Проведено дослiдження симе-

трiйних властивостей редукованих систем та побудованi деякi їx точнi розв’язки.

In this article, being continuation of our works [1, 2], we construct ans?tze for the

a

Navier–Stokcs (NS) field which reduce the NS equations (NSEs) for an incompressi-

ble viscous fluid to linear systems of partial differential equations (PDEs) in two

independent variables. To solve this problem we use the method described in [3] and

the infinite-dimensional symmetry algebra of the NSEs.

It is known that NSEs

?u

+ (u · ?)u ? ?u + ?p = 0, (1)

div u = 0,

?t

where u = u(x) = {u1 , u2 , u3 } is the velocity field of a fluid, p = p(x) is the pressure,

x = {t, x} ? R4 , ? = {?/?xa }, a = 1, 2, 3, ? = ? · ?, are invariant under the

infinite dimensional algebra A? with basis elements

?

, D = 2t?t + xa ?a ? ua ?ua ? 2p?p ,

?t =

?t

(2)

Jab = xa ?b ? xb ?a + ua ?ub ? ub ?ua ,

R(m) = ma ?a + ma ?ua ? xa ma ?p , Z(?) = ?(t)?p ,

? ?

where m = {ma (t)} and ?(t) are arbitrary differentiable function of t; dot means

differentiation with respect to t. The set of operators (2) determine the maximal in

the sense of Lie invariance algebra of the NSEs [4, 6, 7].

Constructing a complete set of inequivalent two-dimensional subalgebras of A? ,

we choose from it those subalgebras which lie in a linear span of operators Jab , R(m)

and Z(?). It is these subalgebras that allow us to construct ans?tze which reduce the

a

nonlinear NSEs to linear systems of PDEs in two independent variables.

Theorem 1. A complete set of A? -inequivalent two-dimensional subalgebras of A?

is exhausted by such algebras:

1. A1 (m, n) = R(m), R(n) , m · n ? n · m = 0, and ? c1 , c2 ? R c1 m +

? ?

c2 n = 0, where algebras A1 (m , n ) and A (m2 , n2 ) are equivalent if ? {akl }k,l=1,2 ,

1 1 1

det{akl } = 0, B ? O(3), ? ?, ? ? R:

(m2 , n2 )(t) = (B(a11 m1 + a12 n1 ), B(a21 m1 + a22 n1 ))(te2? + ?); (3)

Доповiдi АН України, 1992, № 8, С. 29–36.

440 W.I. Fushchych, R.O. Popovych

2. A2 (?, ?) = J12 + Z(?(t)), R(0, 0, ?(t)) , ? = 0, where algebras A2 (?1 , ? 1 ) and

A2 (?2 , ? 2 ) are equivalent if

? c = 0 ??, ? ? R : (?2 , ? 2 )(t) = (e2? ?1 , c? 1 )(te2? + ?); (4)

dt

3. A3 (?, ?) = J12 + R(0, 0, ?(t) (?(t))2 + Z(?(t)), R(0, 0, ?(t)) , ? = 0, where

algebras A3 (?1 , ? 1 ) and A3 (?2 , ? 2 ) are equivalent if

? ?, ? ? R : (?2 , ? 2 )(t) = (e2? ?1 , e?? ? 1 )(te2? + ?); (5)

4. A4 = D + 2?J12 , R(µ|t|? cos(? ln |t|), µ|t|? sin(? ln |t|), ?|t|? ) + Z(?|t|??3/2 ) ,

? > 0, µ ? 0, ? ? 0, µ2 + ? 2 = 1, ? = 0 if ? = 1/2 and ? ? 0 if ? = 1/2;

5. A5 = D, R(0, 0, |t|? ) + z(?|t|??3/2 ) , ? = 0 if ? = 1/2 and ? ? 0 if ? = 1/2;

6. A6 = ?t +J12 , R(µe?t cos t, µe?t sin t, ?e?t +Z(?e?t ) , µ ? 0, ? ? 0, µ2 +? 2 = 1,

? = 0 if ? = 0 and ? ? 0 if ? = 0;

7. A7 = ?t , R(0, 0, e?t ) + Z(?e?t ) , ? ? {?1; 0; 1}, ? = 0 if ? = 0 and ? ? {0; 1}

if ? = 0;

8. A8 = ?t , J12 + ??3 + ??p , ? ? {0; 1}, ? ? 0 if ? = 1 and ? ? {0; 1} if ? = 0;

9. A9 = ?t , D + ?J12 , ? ? 0;

10. A10 = D, J12 + R(0, 0, ?|t|1/2 ) + Z(?t?1 ) , ? ? 0, ? ? 0;

11. A11 = D + ?J12 , Z(|t|? ) , ? ? 0, ? ? R;

12. A12 = ?t , Z(e?t ) , ? ? {?1; 0; 1};

13. A13 = ?t + J12 , Z(e?t ) , ? ? R;

14. A14 (?, ?) = J12 + R(0, 0, ?(t)), Z(?(t)) , ? = 0, where algebras A14 (?1 , ? 1 )

and A14 (?2 , ? 2 ) are equivalent if ? ?, ? ? R, ? c = 0: (?2 , ? 2 )(t) = (c?1 , e?? ? 1 )(te2? +

?);

15. A15 (?, ?) = J12 + Z(?(t)), Z(?(t)) , ? = 0, where algebras A15 (?1 , ? 1 ) and

A15 (?2 , ? 2 ) are equivalent if ? c1 = 0, ? ?, ?, c2 ? R: (?2 , ? 2 )(t) = (c1 ?1 , e?? ? 1 +

c2 ?1 )(te2? + ?);

16. A16 (m, ?) = R(m(t)), Z(?(t)) , m = 0, ? = 0, where algebras A16 (m1 , ?1 )

and A16 (m2 , ?2 ) are equivalent if ? c1 = 0, ? c2 = 0, ? ?, ? ? R, ? B ? O(3):

(m2 , ?2 )(t) = (c1 Bm1 , c2 ?1 )(te2? + ?);

dt

стр. 102 |