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u1 (1.29a)

r 1 k(2/??1)?1

Rk (?a , ?ab ) (r = 2, . . . , m);

(2) when ? = 0:

ur (r = 1, . . . , m), (u1 u1 )?2k Sjk (wab , wab ),

1 r

dd

(1.29b)

(u1 u1 )1?2k Rk (ur , wab ) (r = 2, . . . , m)

1

a

dd

(for the set of invariants (u1 u1 )?2k Sk (wab ), k does not take the value n); the tensors

dd

r r

?ab , wab are constructed similarly to (1.24) and

ur u1

? a.

a

r

?a =

ur u1

Theorem 4 is proved similarly to Theorem 3.

Second-order differential invariants of the rotation group O(n) 513

The functional independence of the sets of invariants follows from Lemma 2 and 3

taking into account the fact that transformations ur > ?ab , ur > wab (r = 1, . . . , m)

r r

ab ab

and ur > ?a (r = 2, . . . , m) are nondegenerate.

r

a

1.4. Differential invariants of the rotation algebra. The rotation algebra is

defined by the basis operators Jab (1.1).

The second-order invariants of this algebra for m scalar functions ur are construc-

ted with xa , ur , ur , wab similarly to invariants of the Euclid algebra.

r

a

Theorem 5. There is a functional basis of the second-order differential invariants

for the algebra AO(n) that has the form

ur , Sjk (u1 , ur ), Rk (ur , u1 ), Rk (xa , u1 ), r = 1, . . . , m;

ab ab a ab ab

the corresponding basis of invariants for the algebra Jab , D , where D is defined

by (1.28), consists of the expressions

ur Sjk (u1 , ur )

, Rk (ur , u1 )(u1 )2k/??1?k ,

ab ab

(r = 2, . . . , m), a ab

1 1 )k(1?2/?)

u (u

Rk (xa , u1 )(u1 )2/?(k?2)?k+1 , when ? = 0;

ab

u , Rk (ur , u1 )(u1 )?k , Sjk (u1 , ur )(u1 )?k (k = 1 when r = 1),

r

a aa aa

ab ab ab

1 1 2?k

Rk (xa , uab )(uaa ) when ? = 0.

A basis of invariants for the algebra Jab , D, Ka when ? = 0, consists of the

expressions (1.29a) and

1

Rk (xa , ?ab )

, k = 2, . . . , n + 1;

x2 (u1 )(k?1)(1?2/?)

when ? = 0 it consists of the expressions (1.29b) and

1

Rk (xa , wab )

(x2 = xa xa ).

x2 (waa )k?1

1

The proof of this theorem is similar to the proofs of Theorems 2 and 3; notice that

(xa ) is a co variant tensor with respect to the conformal operators.

2. Differential invariants of the Poincar? and conformal algebra

e

In this section, we consider differential invariants of the second order for a set

of m scalar functions

n ? 3.

ur = ur (x0 , x1 , . . . , xn ), (2.1)

The Poincar? algebra AP (1, n) is defined by the basis operators

e

?

Jµ? = xµ p? ? x? pµ , (2.2)

pµ = igµ? ,

?xµ

where µ, ? take the values 0, 1, . . . , n; the summation is implied over the repeated

indices (if they are small Greek letters) in the following way:

x? x? ? x? x? ? x? x? = x2 ? x2 ? · · · ? x2 , gµ? = diag (1, ?1, . . . , ?1). (2.3)

0 1 n

514 W.I. Fushchych, I.A. Yegorchenko

We consider x? and x? equal with respect to summation not to mix signs of

derivatives and numbers of functions.

The quasilinear second-order invariants of the Poincar? algebra were described

e

in [12].

Theorem 6. There is a functional basis of the second-order differential invariants

of the Poincar? algebra AP (l, n) for a set of m scalar functions ur consisting of

e

n(n + 1)

m(2n + 3) + (m ? 1)

2

invariants

ur , Rk (ur , u1 ), Sjk (ur , u1 ).

µ µ? µ? µ?

In this section, everywhere k = 1, . . . , n + 1; j = 0, . . . , k; r = 1, . . . , m.

?

For the extended Poincar? algebra AP (l, n) = pµ , Jµ? , D , where

e

D = xµ pµ + ?ur pur (2.4)

(pur = i(?/?ur ), the summation over r from 1 to m is implied) the corresponding

basis has the following form:

(1) when ? = 0:

Sjk (ur , u1 )(u1 )?k , Rk (ur , u1 )(u1 )?k ;

ur , µ? µ? ?? µ µ? ??

(2) when ? = 0:

ur

, Sjk (ur , u1 )(u1 )k(2/??1) , Rk (ur , u1 )(u1 )2k/??k?1 ,

µ? µ? µ µ?

u1

where Sjk , Rk are defined similarly to (0.3) and the summation over small Greek

indices is of the type (2.2).

For the conformal algebra AC(1, n) = pµ , Jµ? , D, Kµ , where

Kµ = 2xµ D ? x? x? pµ

(D being the dilation operator (2.3)), the corresponding basis consists of the expres-

sions

ur

r 1 1 k(2/??1)

, Rk (?µ , ?µ? )(u1 )k(2/??1)?1 ;

r 1

Sjk (?µ? , ?µ? )(u ) ,

u1

when ? = 0; r = 2, . . . , m, there is no summation over r; the conformally covariant

tensors have the form

ur ur

ur u1 ur ur

µ µ µ? ??

= r ? 1 , ?µ? = ?uµ? + (1 ? ?) r ? gµ?

r r r

?µ .

2ur

u u u

When ? = 0, the corresponding basis of invariants for the conformal algebra has

the form

Sjk (wµ? , wµ? )(u1 u1 )?2k ,

ur , 1

Rk (ur , wµ? )(u1 u1 )1?2k ,

1

r = 2, . . . , m;

?? µ ??

r

the tensors (wµ? ),

gµ? r

wµ? = ur ur ur ? ? ur (ur ur + ur ur )

r

u

1 ? n ??

?? µ? ? µ ?? ? ?µ

are conformally invariant (there is no summation over r).

Second-order differential invariants of the rotation group O(n) 515

The proof of Theorem 6 follows from those of Theorems 2, 3 for x = (x1 , . . . , xn+1 )

if we substitute ix0 instead of xn+1 .

Similarly to the results of Paragraph 1.4, it is possible to construct the invariants

of the algebras Jµ? , Jµ? , D , Jµ? , D, Kµ .

The obtained results allow us to construct new nonlinear many-dimensional equa-

tions, e.g. the equation

u? u ?

u?? ? uµ u? uµ? = (u? u? )2 F (u),

1?n

where F is an arbitrary function, is invariant under the algebra AC(1, n), ? = 0. The

left member of the above equation is equal to wµµ .

There is another quasi-linear relativistic equation with rich symmetry properties

(1 ? u? u? )uµµ ? u? uµ u?µ = 0,

that is, the Born–Infeld equation. The symmetry and solutions of this equation were

investigated in [10, 13]. This equation is invariant under the algebra AP (1, n+1) with

the basis operators

JAB = xA pB ? xB pA ,

A, B = 1, . . . , n + 1, xn+1 ? u.

Let us consider the class of equations

uµ? uµ? = F (uµµ , uµ u? uµ? , uµ uµ , u).

It is evident that they are invariant with respect to the Poincar? algebra AP (1, n)

e

out the straightforward search the conformally invariant equations from this class

with the standard Lie technique requires a lot of cumbersome calculations. The use of

differential invariants turns this problem into one of elementary algebra, e.g. if ? = 0

1

F ? uµ? uµ? = ? S2 (?µ? ) + u2(1?2/?) ?(S1 (?µ? )u2/??1 ),

?

where ?µ? is of the form (1.24) and ? is an arbitrary function. Whence

? + n u? u?

F = u2(1?2/?) ? u2/??1 uµµ ? ?

? u

2(1 ? ?)

1 2uµµ u? u?

? (?2 + n2 )(u? u? )2 ? uµ u? uµ? + .

?2 u2 ?u ?u

It is useful to note that besides the traces of matrix powers (0.3), one can utilize

r r

all possible invariants of covariant tensors ?µ? , wµ? to construct conformally invariant

equations.

3. Differential invariants of an infinite-dimensional algebra

It is well-known that the simplest first-order relativistic equation — the eikonal or

Hamilton equation

u? u? ? u2 ? u 2 ? · · · ? u2 = 0 (3.1)

0 1 n

is invariant under the infinite-dimensional algebra AP ? (1, n) generated by the opera-

tors [10, 14]

X = (bµ? x? + aµ )?µ + ?(u)?u , (3.2)

516 W.I. Fushchych, I.A. Yegorchenko

?bµ? = b?µ , aµ , ? being arbitrary differentiate functions on u. Equation (3.1) is widely

used in geometrical optics.

In this section, we describe a class of second-order equations invariant under the

algebra (3.2).

It is easy to show that the tensor of the rank 2

?µ? = uµ u?? u? + u? u?µ u? ? 2uµ u? u?? (3.3)

is covariant under the algebra AP ? (1, n) (3.2).

Theorem 7. The equations of the form

(3.4)

Sk (?µ? ) = 0, k = 1, 2, . . . ,

Sk being defined as (0.3), are invariant with respect to the algebra AP ? (1, n) (3.2).

The problem of the description of all such equations is more difficult and we do

not consider it here.

Let us investigate in more detail the quasi-linear second-order equation of the form

uµ uµ? u? ? uµ uµ u?? = 0. (3.5)

Theorem 8. When n ? 2, equation (3.5) is invariant with respect to the algebra

AP ? (1, n) with generators of the form

?

X + d(u)xµ ?µ ,

X is of the form (3.2), d(u) is an arbitrary function on u.

The proofs of Theorems 7 and 8 can be easily obtained with the Lie technique

using the criterion of invariance

2

X Sk (?µ? ) = 0,

Sk (?µ? )=0

2

where X is the second prolongation of the operator X [8–10].

4. Differential invariants of the Galilei algebra

4.1. It is well-known that the heat equation

2µut + ?u = 0, ?u ? uaa ,

(4.1)

n?3

u = u(t, x), x = (x1 , . . . , xn ),

is invariant under the generalized Galilei algebra AGI (1, n) with the basis operators

2

? ?

Jab = xa ?b ? xb ?a ,

?t = , ?a = ,

?t ?xa

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