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2
x2 ?2cx?1 x2 +1
5 ln ? arctg
ax
(dx)2
2bx
dx dx
(ax)2 ?(bx)2
x2 +1 x2 ?2cx?1
6 (dx)2
dx ax?bx
(x2 ?2cx?1)dx (ax)2 ?(bx)2
x2 +1
7 (ax?bx)2 (dx)2
dx
(ax)2 +(dx)2
x2 ?1 x2 +1
8 arctg ? 2 arctg ax
(cx)2
2bx
cx dx
(ax)2 +(dx)2
x2 +1 2cx
9 + (? + 2) arctg ax
(bx?1)2
bx?1 bx?1 dx
(x +1)2 ?4(dx)2
2
x2 +2bx?1 2
2cx +1
10 + ? arcth x2dx (ax)2
ax ax
(x2 ?1)2 ?4(cx)2
x2 ?2dx+1
? arcth x2cx
11 2 arctg bx
2 ?1 (ax)2 ?(bx)2
ax?bx ax
(x2 +1)dx?2axbx (x2 ?1)2 ?4(cx)2
x2 ?2cx?1
12 ln ? arctg ax
(ax)2 +(dx)2 x2 +2cx?1 (ax)2 +(dx)2
dx
(x2 ?1)2 ?4(cx)2
x2 ?2dx+1
arcth ax ? arcth x2cx
13 bx
2 ?1 (ax)2 ?(bx)2
ax?bx
(x2 ?1)2 ?4(cx)2
x2 +1 x2 +1
+ arcth x2cx
14 arctg arctg ? 2 arctg
ax
2 ?1 (ax)2 ?(dx)2
2bx 2bx
dx
2 2
(x2 +1)2 ?4(dx)2
(x ?2cx?1)(x ?2dx+1)
arcth ax ? 2arcth x2dx
15 (ax?bx)2 2 +1 (ax)2 ?(bx)2
bx



Пiдставивши анзац (4) в систему рiвнянь (3), одержимо

va v A ? 2?A v A ?1 + ?A ? A (?1 )2 = 0,
? ?
wA wA ? 2?A wA ?1 + ?A ? A (?2 )2 = 0,
? ? (7)
vA wA ? ?A wA ?1 ? ?A v A ?2 + ?A ? A ?1 ?2 = 0.
? ? ??

Розглянувши систему (8) разом з табл. 1, де вказанi вiдповiднi значення iнварi-
антних змiнних ?, v та w, одержимо редукованi системи рiвнянь для визначення
функцiй ?1 (?) i ?2 (?). Наведемо декiлька таких систем:

(c )2 ? 2a c ?1 + (a )2 (?1 )2 = 0,
1) ? ?
(d )2 ? 2a d ?2 + (a )2 (?2 )2 = 0,
? ?
c d ? a d ?1 ? a c ?2 + (a )2 ?1 ?2 = 0;
? ? ??
?1 + (?1 )2 ? 2??1 ?1 + (? 2 + 1)(?1 )2 = 0,
3) ? ?
?4 + (?2 )2 ? 2??2 ?2 + (? 2 + 1)(?2 )2 = 0,
? ?
?1 ?2 ? ??2 ?1 ? ??1 ?2 + (? 2 + 1)?1 ?2 = 0;
? ? ??
(? + 2)2
(? ) + 4 1 ? ? 2(? + 2)?1 ?1 + (? + 2)2 (?1 )2 = 0,
12
9) ? ?
2
?
(? + 2)2 2 2
?2 (1 ? ?2 ) + (? + 2)?2 ?2 ?
? (? ) = 0,
?
4
2?1 ?2 ? 2(? + 2)?2 ?1 ? (? + 2)?1 ?2 + (? + 2)2 ?1 ?2 = 0;
? ? ??
294 В.I. Фущич, М.I. Сєров, Ю.Г. Подошвєлев

4
? ? ?1 = 0,
11) 1+ ?
2
?
4 + ?2 ? ? ?2 = 0,
?
2?2 + ?1 + ?2 = 0.
? ?

Номер системи вiдповiдає номеру iнварiантiв в табл. 1.
Якщо розв’язати редукованi рiвняння i використати вiдповiднi їм iнварiанти
i анзац (4), то одержимо розв’язки системи (3). Наведемо деякi з них:
u1 = aµ xµ , u2 = bµ xµ ;
(aµ xµ )2 ? xµ xµ ;
u1 = aµ xµ , u2 =
u1 = xµ xµ + (bµ xµ )2 , u2 = bµ xµ ;
u1 ? ax = m1 (u2 ? bx), xA xA = m2 (u2 ? bx),
де aµ , bµ — сталi вектори; aµ aµ = ?bµ bµ = 1, aµ bµ = 0, µ = 0, 1; m1 , m2 — довiльнi
сталi.
Одержанi iнварiанти алгебри AC(2, 2) та розв’язки системи (3) можна розмно-
жити за допомогою перетворень iнварiантностi. Цi перетворення мають вигляд
cAB (xB ? ?B xA xA )
xA > ,
1 ? 2?A xA + ?A ?A xA xA
де x2 ? u1 , x3 ? u2 , cAB , ?A — довiльнi сталi параметри.

1. Fushchych W.I., Shtelen W.M., The symmetry and some exact solutions of the relativistic
eikonal equation, Lett. Nuovo Cim., 1982, 34, № 16, 498.
2. Fushchych W.I., Serov N.I., The symmetry and some exact solutions of the nonlinear many-
dimensional Liouville, d’Alembert and eikonal equations, J. Phys. A, 1983, 16, 3645–3658.
3. Fushchych W.I., Serov N.I., Shtelen W.M., Some exact solutions of many-dimensional nonli-
near d’Alembert, Liouville, eikonal, and Dirac equations, in Group-Theoretical Methods in
Physics, London, Harwood Acad. Publ., 1984, 489–496.
4. Штелень В.М., Касательные преобразования релятивистского уравнения Гамильтона–
Якоби, в сб. Теоретико-алгебраические методы в задачах математической физики, Киев,
Ин-т математики, 1983, 62–65.
5. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations
of nonlinear mathematical physics, Dordrecht, Kluwer Acad. Publ., 1993, 436 p.
6. Баранник А.Ф., Баранник Л.Ф., Фущич В.И., Редукция и точные решения уравнения
эйконала, Укр. мат. журн., 1991, 43, № 4, 461–474.
7. Фущич В.И., Баранник Л.Ф., Баранник А.Ф., Подгрупповой анализ групп Галилея, Пуан-
каре и редукция нелинейных уравнений, Киев, Наук, думка, 1991, 300 с.
8. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978,
400 с.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 295–315.

On covariant realizations of the Euclid group
R.Z. ZHDANOV, V.I. LAHNO, W.I. FUSHCHYCH
We classify realizations of the Lie algebras of the rotation O(3) and Euclid E(3) groups
within the class of ?rst-order di?erential operators in arbitrary ?nite dimensions. It
is established that there are only two distinct realizations of the Lie algebra of the
group O(3) which are inequivalent within the action of a di?eomorphism group. Using
this result we describe a special subclass of realizations of the Euclid algebra which
are called covariant ones by analogy to similar objects considered in the classical
representation theory. Furthermore, we give an exhaustive description of realizations
of the Lie algebra of the group O(4) and construct covariant realizations of the Lie
algebra of the generalized Euclid group E(4).


1 Introduction
The standard approach to constructing linear relativistic motion equations contains
as a subproblem the one of describing inequivalent matrix representations of the
Poincar? group P (1, 3). So that if one succeeds in obtaining an exhaustive (in some
e
sense) description of all inequivalent representations of the latter, then it is possible to
construct all possible Poincar?-invariant linear wave equations (for more details see,
e
e.g. [1–3]). It would be only natural to apply the same approach to describing nonli-
near relativistically-invariant models with the help of the Lie’s in?nitesimal technique.
However, in the overwhelming majority of the papers devoted to symmetry classi?-
cation of nonlinear di?erential equations admitting some Lie transformation group G
the realization of the group was ?xed a priory. As a result, only particular classes
of partial di?erential equations invariant with respect to a prescribed group G were
obtained. One of the possible reasons for this is that the problem of describing inequi-
valent realizations of a given Lie transformation group reduces to constructing general
solution of some over-determined system of nonlinear partial di?erential equations (in
contrast to the case of the classical matrix representation theory where one has to
solve nonlinear matrix equations).
We recall that given a ?xed realization of a Lie transformation group G, the
problem of describing partial di?erential equations invariant under the group G is
reduced with the help of the in?nitesimal Lie method to integrating some over-
determined linear system of partial di?erential equations (called determining equa-
tions) [4–7]. However, to solve the problem of constructing all di?erential equations
admitting the transformation group G whose realization is not ?xed a priori one has
• to construct all inequivalent (in some sense) realizations of the Lie transforma-
tion group G,
• to solve the determining equations for each realization obtained.
And what is more, the ?rst problem, in contrast to the second one, reduces to solving
nonlinear systems of partial di?erential equations. In this respect one should men-
tion the Lie’s classi?cation of integrable ordinary di?erential equations based on his
Commun. Math. Phys., 2000, 212, P. 535–556.
296 R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych

classi?cation of complex Lie algebras of ?rst-order di?erential operators in one and
two variables [8]. However, it seems impossible to give an exhaustive description of
all Lie algebras of ?rst-order di?erential operators. Till now there is no complete
classi?cation of them even for the case of ?rst-order di?erential operators in three
variables, though a partial classi?cation was obtained by Lie a century ago [8].
The classi?cation problem is substantially simpli?ed if we are looking for inequi-
valent realizations of a speci?c Lie algebra. It has been completely solved by Ri-
deau and Winternitz [9], Zhdanov and Fushchych [10] for the generalized Galilei
(Schr?dinger) group G2 (1, 1) acting in the space of two dependent and two inde-
o
pendent variables.
Yehorchenko [11] and Fushchych, Tsyfra and Boyko [12] have constructed new
(nonlinear) realizations of the Poincar? groups P (1, 2) and P (1, 3), correspondingly
e
(see also [13, 14]). Some new realizations of the Galilei group G(1, 3) were suggested in
[15]. A complete description of covariant realizations of the conformal group C(n, m)
in the space of n + m independent and one dependent variables was obtained by
Fushchych, Zhdanov and Lahno [16, 17] (see, also [18]). It has been established,
in particular, that any covariant realization of the Poincar? group P (n, m) with
e
max{n, m} ? 3 in the case of one dependent variable is equivalent to the standard
realization. But given the condition max{n, m} < 3, there exist essentially new reali-
zations of the corresponding Poincar? groups.
e
The present paper is devoted mainly to classi?cation of inequivalent realizations of
the Euclid group E(3), which is a semi-direct product of the three-parameter rotation
group O(3) and of the three-parameter Abelian translation group T (3), acting in the
space of three independent (x1 , x2 , x3 ) and n ? N dependent (u1 , . . . , un ) variables.
Being a subgroup of such fundamental groups as the Poincar? and Galilei groups, the
e
Euclid group plays an exceptional role in modern mathematical and theoretical physi-
cs, since it is admitted both by equations of relativistic and non-relativistic theories.
In particular, group E(3) is an invariance group of the Klein–Gordon–Fock, Maxwell,
heat, Schr?dinger, Dirac, Weyl, Navier–Stokes, Lam? and Yang–Mills equations.
o e
The paper is organized as follows. The second section contains the necessary notati-
ons, conventions and de?nitions used throughout the paper. In the third section we
give an exhaustive classi?cation of inequivalent realizations of the Lie algebra of the
rotation group O(3) within the class of ?rst-order di?erential operators. The fourth
section is devoted to description of covariant realizations of the Euclid algebra AE(3).
We give a complete classi?cation of them and, furthermore, demonstrate how to reduce
the realizations of AE(3) realized on the sets of solutions of the Navier–Stokes, Lam`, e
Weyl, Maxwell and Dirac equations to one of the two canonical forms. In the forth
section the results obtained are applied to describe covariant realizations of the Lie
algebra of the generalized Euclid group AE(4).


2 Basic notations and de?nitions
It is a common knowledge that investigation of realizations of a Lie transformation
group G is reduced to study of realizations of its Lie algebra AG whose basis elements
are the ?rst-order di?erential operators (Lie vector ?elds) of the form

(1)
Q = ?? (x, u)?x? + ?i (x, u)?ui ,
On covariant realizations of the Euclid group 297

where ?? , ?i are some real-valued smooth functions of x = (x1 , x2 , . . . , xm ) ? Rm
and u = (u1 , u2 , . . . , un ) ? Rn , ?x? = ?x? , ?ui = ?ui , ? = 1, 2, . . . , m, i = 1, 2, . . . , n.
? ?

Hereafter, a summation over the repeated indices is understood.
In the above formulae we have two “sorts” of variables. The variables x1 , x2 , . . . , xm
and u1 , u2 , . . . , un will be referred to as independent and dependent variables, respecti-
vely. The di?erence between these becomes essential when we consider AG as an
invariance algebra of some system of partial di?erential equations for u1 (x), . . . , un (x).
Due to properties of the corresponding Lie transformation group G basis operators
Qa , a = 1, . . . , N of a Lie algebra AG satisfy commutation relations
c
(2)
[Qa , Qb ] = Cab Qc , a, b = 1, . . . , N,

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