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(14)
The designations Ci and F i mean constants C i and functions F i , U # means
?
U ? ? 1 . Taking into account formula (12) we see that the 1st order LB symmetries
appear to be point ones. Considering the coefficients at C i and using (12) it is easy
to obtain the symmetry operators
1
XU ?U ? T ?X ,
e1 = ?X , e2 = e3 = X?X + 2T ?T ,
2
1 1
T + X 2 ?U ? XT ?X ? T 2 ?T ,
e4 = U e5 = ?T , e6 = U ?U .
2 4
As is the case for every linear partial differential equation, there is also infinite-
dimensional subalgebra e? = F 1 (X, T )?U , where F 1 is an arbitrary solution of
equation (14). This example takes 47 seconds of CPU time and 260 Kbytes memory
Computer algebra application for determining symmetries 501

on the ES 1045 computer (under the OS/VS) with the internal performance of about
700 000 op/sec.
The LBF program consists of two main parts; the computation of determining
system, and the solution of that system. The 1st part is the improved and generalised
version of the program described in [2].
We use the sequencing of u? ···il and derivatives of functions F i , included in
i1
symmetry generators, in the following way: sets of lower indices are ordered lexico-
graphically; after that the items are ordered in accordance with their upper indices.
The position of the item in such a row we shall call “ordinal”. This ordering permits to
perform the considerable part of calculations using the fast and non-wasting memory
of PL/I integer arithmetic only.
The program executes sequentially the following steps.
(1) Reading the input data and transforming them into internal form introducing
the ordering mentioned above.
(2) Computation of the differential consequences up to the required order and elimi-
nation of dependences thereof. The program tries to solve the system together
with differential consequences with respect to derivatives of the highest ordi-
nals which are included linearly in relations. To do this, the program uses the
Gauss excluding method. Note that differential consequences depend on the hi-
ghest order derivatives only linearly. Some equations may not contain linearly
included derivatives. The program marks such equations.
If the problem of classification is considered, then the system of equations
contains arbitrary parameters or functions. Some combinations of these items
may be used as denominators during the excluding process. Everywhere in the
program in similar situations such different denominators are memorised to be
printed at the end of program execution. It is necessary to consider separately
the cases when such combinations are zeros.
(3) The determination of symmetry variables, i.e. variables which the generators
depend on. (The generators do not depend on derivatives of an order higher than
the symmetry order and on derivatives excluded in the previous step.)
(4) Computation of canonical LB operator (9) and the result of its action on system
(1), i.e. computation of invariance conditions.
(5) Transition to manifold. The program eliminates the derivatives, excluded in step
(2), in the invariance conditions. If there are equations unresolved in step (2) the
program adds them to invariance conditions (11) having preliminarily multiplied
them by indefinite factors.
(6) Separation of the determining system with respect to free variables, i.e. deriva-
tives of an order higher than the symmetry order. The program separates the
equations not only with respect to different powers of the free variables but also
with respect to arbitrary different independent functions of such variables. It is
possible that the functions independent in general may be dependent in some
particular cases. For example, if uxx is a free variable, then from A sin(uxx ) +
B cos(uxx ) = 0 it follows that A = 0 and B = 0, but from Au2 + Buk = 0
xx xx
it follows that A = 0 and B = 0 if k = 0, but A + B = 0 otherwise. Another
obvious example Af (uxx )+Bf (uxx ) = 0 requires to consider the particular case
502 W.I. Fushchych, V.V. Kornyak

f = euxx . For subsequent consideration of similar particular cases the program
memorises all different separation factors containing arbitrary functions and
parameters. For the sake of economy of memory, during the separation process,
zeros are deleted immediately, i.e. when a one-term determining equation arises,
it and its differential consequences are substituted at once into all remaining
expressions.
(7) Exclusion of indefinite factors. Using the Gauss method the program excludes
indefinite factors if they were introduced at step (5).
At the end of the first part of the program the state of the determining
system (i.e. expressions for generators, dependences of functions, equations)
considered is for the example:
?
u = F 1; F 1 = F 1 (x, t, u, ux , ut );
Ft1 ? Fxx ? 2ut ux Fuux ? 2ut Fxux ? 2ux Fxu ? u2 Fux ux ? u2 Fuu = 0,
1 1 1 1 1 1
t x
1 1 1 1
Fxut + ut Fux ut + ux Fuut = 0, Fut ut = 0.

(8) Simplification of the determining system. The fragment of the program respon-
sible for this step is
BC = ‘1’B;
DO WHILE(BC);
BB = ‘1’B;
DO WHILE(BB);
BA = ‘1’B;
DO WHILE (BA);
CALL REDSYS(1); CALL ORTSYS(?); BA = SMONINT;
END;
BB = INVOL;
END;
BC = RESTINT;
END;
Here, BA, BB, BC are the control variables. The very inner loop contains
the calls of the most efficient procedures. The REDSYS procedure is auxiliary.
It reduces the determining system to some canonical form. Argument “1” means
that the reduction begins from the first equation. (There are calls of REDSYS
from other procedures with different values of argument.) The call of the
ORTSYS(K) procedure reduces the determining system to the orthonomic form
[4], i.e. the derivatives of the highest ordinals (leading derivatives) are singled
out and substituted (with theirs differential consequences of the order up to K),
equations are ordered in accordance with the increase of ordinals of leading deri-
vatives. ORTSYS deletes the equations not containing derivatives after having
performed the corresponding substitutions. The SMONINT procedure integrates
the systems of monomial equations, i.e. systems of the form {Fij1 ···ik = 0}. Here,
Computer algebra application for determining symmetries 503

the lower index il means the differentiation with respect to the il th symmetry
variable. The procedure yields the expression for F j simultaneously consideri-
ng all monomials for the given j. It allows the number of iterations to be
reduced. SMONINT substitutes also F j and its derivatives in other expressi-
ons and separates the determining system after that. If there are no monomial
equations in the determining system, then BA takes the value ‘?’B and the loop
is completed.
After this loop the state of the determining system for the heat equation is
?
u = F 3 + F 4 u + F 2 ux + F 1 ut ;
F 1 = F 1 (t), F 2 = F 2 (x, t), F 3 = F 3 (x, t), F 4 = F 4 (x, t);
2Fx ? Ft1 = 0, 2Fx ? Ft2 + Fxx = 0, Ft3 ? Fxx = 0, Ft4 ? Fxx = 0.
2 4 2 3 4

The INVOL procedure performs the further integrations or reduces the
remaining part of the determining system to the system in involution by Riquier–
Janet method [4]. The method consists in calculating differential consequences
of equations and in excluding dependences out of them. INVOL tries to separate
and integrate the relations arising during this process. If it fails in integration,
then the system is reduced to the one in involution, and BB takes the value
‘?’B; if otherwise, BB = ‘1’B, and the whole process is repeated. The INVOL
procedure integrates the monomials and often arising equations of the form
Fij1 ···ik = P , where P is the polynomial of the variables of differentiations.
As a rule, the combination of the above-mentioned operations is sufficient to
reveal the major part of dependences of the symmetry generators, because these
dependences are often polynomial. For instance, our example is completed by
the INVOL procedure. In some cases it is possible to go further in integration of
the remaining part of the determining system. This is effected by the RESTINT
procedure. In gaining experience, it is possible to add in this procedure some
particular and rare methods of integration. Now RESTINT contains the pro-
cedure for solving the ordinary differential equations or systems with constant
(with respect to differentiation variable) coefficients up to the 4th order, and the
procedure for solving differential equations of the 1st order with variable coeffi-
cients. The last procedure yields in non-polynomial cases the formal expressions
for indefinite integrals. BC takes the value ‘1’B if further simplification after
RESTINT is possible, and ‘?’B, if otherwise.
(9) The last step of the LBF program is output. The internal designations are
replaced by more expressive ones. The expressions for generators are reduced to
some form simplifying the extraction of different one-dimensional subalgebras,
as in the example above. The program removes also superfluous functions or
constants if they arise during integration. For example, if F 1 (X, Y ) and F 2 (X)
are arbitrary functions, then F 1 (X, Y ) + F 2 (X) is equivalent to F 1 (X, Y ). In
general, the output may contain two more items, in addition to those presented
in the example: the list of different denominators containing arbitrary functions
or parameters, and the analogous list of separation factors.
504 W.I. Fushchych, V.V. Kornyak

Let us demonstrate the result of the program application in obtaining the 1st order
symmetries of a more complicated non-linear equation [5]
?u ?u ?u
? 2u ? ?u µ (15)
= 0,
?? ?x ?xµ
where
?2 ?2 ?2 ?2 ?u ?u
2= ? ? 2 ? 2, = u2 t ? u2 ? u2 ? u2 .
? x y z
??2 ?x2 µ ?x
?y ?z ?x µ
t

The LBF program gives the general solution having in terms of operators the form:
e1 = ?x , e2 = ?y , e3 = ?z , e4 = ??t , e5 = ?x + x??t ,
e6 = ?t ?y + y??t , e7 = ?t ?z + z??t , e8 = y?x ? x?y , e9 = x?z ? z?x ,
e10 = z?y ? y?z , e11 = ?(u)?u , e12 = ?? ,
e13 = x?x + y?y + z?z + t?t + 2? ?? ,
x2 + y 2 + z 2 ? t2
? 2? ?(u)?u ,
2
e14 = x? ?x + y? ?y + z? ?z + t? ??t + ? ?? +
4
x y z
e15 = ? ?x + ?(u)?u , e16 = ? ?y + ?(u)?u , e17 = ? ?z + ?(u)?u ,
2 2 2
2
t ?u
e18 = ? ??t ? ?(u)?u , e? = ?(x, y, z, t, ? ) exp ? ?u ,
2 2
where
?u2 ?u2
?(u) = exp ? exp du,
2 2
?(x, y, z, t, ? ) is an arbitrary solution of equation
??
? 2u = 0. (16)
??
This example takes 5 min 18 s and 320 Kbytes on ES 1045. Generators e1 –e10 create
the Poincar? algebra. As it follows from the above operators, equation (15) turned
e
out to be automorphic, i.e. all its solutions lie on one group orbit. It allows to reduce
non-linear equation (15) to linear one (16) using standard techniques of symmetry
analysis.
4. Conclusion
There are some problems in connection with the considered one where the com-
puter algebra may be successfully applied. For example, to complete the symmetry
analysis of the system of differential equations it is important to learn the subgroup
structure of the symmetry group, i.e. to classify the subalgebras of Lie-algebra of
symmetries into conjugacy classes. This problem is also important in many other
fields. In [9] the algorithm for classification of subalgebras of finite-dimensional Lie-
algebras and its computer implementation were described.
Modern development of symmetry analysis includes several approaches considering
non-local symmetries, i.e. symmetries depending on integrals or even more general
operators acting in the functional space of the dependent functions [7]. Some class of
such symmetries and corresponding algorithms and programs were considered in [6].
Computer algebra application for determining symmetries 505

We are grateful to Professors B. Buchberger, B. F. Caviness and J. A. van Hulzen
for their interest in this work.

1. Eliseev V.P., Fedorova R.N., Kornyak V.V., A REDUCE program for determining point and contact
Lie symmetries of differential equations, Comput. Phys. Commun., 1985, 36, 383.
2. Fedorova R.N., Kornyak V.V., Determination of Lie–B?cklund symmetries of differential equations
a
using FORMAC, Comput. Phys. Commun., 1986, 39, 93.
3. Fedorova R.N., Kornyak V.V., A REDUCE program for computing determining equations of Lie–
B?cklund symmetries of differential equations, Dubna, JINR, 1987, R11-87-19 (in Russian).
a
4. Finikov S.P., Carton’s exterior forms method, Moscow–Leningrad, Gostechizdat, 1948, 432 p. (in
Russian).
5. Fushchych W.I., Symmetry in the problems of mathematical physics, in Algebraic-Theoretical Studi-
es in Mathematical Physics, Kyiv, Inst. of Math., 1981, 6–28 (in Russian).
6. Fushchych W.I., Kornyak V.V., Computation on a computer of non-local symmetries of linear
systems in mathematical physics, In International Conference on Computer Algebra and its Appli-
cations in Theoretical Physics, Dubna, JINR, 1987, D11-85-791, 345–350 (in Russian).
7. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, D. Reidel, 1987,
217 p.
8. Ibragimov N.H., Transformation groups in mathematical physics, Moscow, Nauka, 1983, 286 p. (in
Russian).
9. Kornyak V.V., Classification of subalgebras for finite-dimensional Lie-algebra using computer, In
international Conference on Computer Algebra and its Applications in Theoretical Physics, Dubna,
JINR, D11-85-791, 339–344 (in Russian).
10. Ovsiannikov L.V., Group analysis of differential equations, Moscow, Nauka, 400 p. (in Russian).
11. Schwarz F.A., A REDUCE package for determining Lie symmetries of ordinary and partial di-
fferential equations, Comput. Phys. Commun., 1982, 27, 179.
12. Schwarz F.A., Automatically determining symmetries of partial differential equations, Computing,
1985, 34, 91.
13. Wolf T., Analytic solutions of differential equations with computer algebra systems, Preprint N 87/5,
Friedrich Schiller Universitat, Jena, 1987.
14. Zhdanov R.Z., On application of Lie–B?cklund method to study of symmetries of the Dirac equati-
a
ons, in Group-Theoretical Studies of Mathematical Physics Equations, Kyiv, Inst. of Math., 1985,
70–73 (in Russian).
W.I. Fushchych, Scientific Works 2001, Vol. 3, 506–510.

On vector and pseudovector Lagrangians
for electromagnetic field
W.I. FUSHCHYCH, I.Yu. KRIVSKY, V.M. 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
equivalent to the Maxwell equations. The advantages of the known pseudovector with
?
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