Lectures on the Theory of Group Properties of Differential Equations

lectures on the Theory of Group Properties of Differential Equations 微分方程群性质理论讲义 Edited by N. H. Ibragimov 也 1lt 奇'J:P...

Author: L. V. Ovsyannikov | Nail H. Ibragimov

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lectures on the Theory of Group Properties of Differential Equations 微分方程群性质理论讲义 Edited by N. H. Ibragimov

也 1lt 奇'J:P.ι ~l

HIGHEREOυCAτlON PRESS

•

NONLINEAR PHYSICAL SCIENCE

5ÔnÆ

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NONLINEAR PHYSICAL SCIENCE Nonlinear Physical Science focuses on recent advances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physical science and nonlinear mathematics with engineering applications. Topics of interest in Nonlinear Physical Science include but are not limited to: - New findings and discoveries in nonlinear physics and mathematics - Nonlinearity, complexity and mathematical structures in nonlinear physics - Nonlinear phenomena and observations in nature and engineering - Computational methods and theories in complex systems - Lie group analysis, new theories and principles in mathematical modeling - Stability, bifurcation, chaos and fractals in physical science and engineering - Nonlinear chemical and biological physics - Discontinuity, synchronization and natural complexity in the physical sciences

SERIES EDITORS Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805, USA Email: [email protected]

Nail H. Ibragimov Department of Mathematics and Science Blekinge Institute of Technology S-371 79 Karlskrona, Sweden Email: [email protected]

INTERNATIONAL ADVISORY BOARD Ping Ao, University of Washington, USA; Email: [email protected] Jan Awrejcewicz, The Technical University of Lodz, Poland; Email: [email protected] Eugene Benilov, University of Limerick, Ireland; Email; [email protected] Eshel Ben-Jacob, Tel Aviv University, Israel; Email: [email protected] Maurice Courbage, Universit´e Paris 7, France; Email: [email protected] Marian Gidea, Northeastern Illinois University, USA; Email: [email protected] James A. Glazier, Indiana University, USA; Email: [email protected] Shijun Liao, Shanghai Jiaotong University, China; Email: [email protected] Jose Antonio Tenreiro Machado, ISEP-Institute of Engineering of Porto, Portugal; Email: [email protected] Nikolai A. Magnitskii, Russian Academy of Sciences, Russia; Email: [email protected] Josep J. Masdemont, Universitat Politecnica de Catalunya (UPC), Spain; Email: [email protected] Dmitry E. Pelinovsky, McMaster University, Canada; Email: [email protected] Sergey Prants, V.I.Il’ichev Pacific Oceanological Institute of the Russian Academy of Sciences. Russia; Email: [email protected] Victor I. Shrira, Keele University, UK; Email: [email protected] Jian Qiao Sun, University of California, USA; Email: [email protected] Abdul-Majid Wazwaz, Saint Xavier University, USA; Email: [email protected] Pei Yu, The University of Western Ontario, Canada; Email: [email protected]

L.y. Ovsyannikov

Lectures on the Theory of Group Properties of Differential Equations 微分方程群'性质理论讲义 Weifen Fangcheng Qunxingzhi Li lun Jiangyi

Edited by N.H. Ibragimov Translated by E. D. Avdonina, N.H. Ibragimov

·北京剧盹

Aωh(Jr

E"it(J~

LV

\l ail

Ov 0 as one can see below. If L(S) = Lr ; the system (S) is said to admit the space of operators L r :

1.5.2 First-order ordinary differential equations Let us find the space L admitted by the ordinary differential equation y0 = f (x; y): Here the system (S) consists of one equation p = f (x; y);

S:

(1.5.3)

where

dy dx We seek the operator admitted by Eq. (1.5.3) in the form p = y0 =

X =ξ

∂ ∂ +η ; ∂x ∂y

where the coordinates ξ and η depend on x and y: The prolonged operator has the form

∂ ∂ Xe = ξ +η + [ηx + p(ηy ∂x ∂y

ξx )

p2 ξ y ]

∂ ∂p

The determining equations are obtained by acting by the operator Xe on Eq. (1.5.3) and then by replacing the variable p by its value f (x; y) (transition to the manifold S). This provides the determining equation

ηx + f (ηy

ξx )

f 2 ξy = ξ f x + η f y :

(1.5.4)

Since equation (1.5.4) contains two unknown functions, one of them can be chosen to be arbitrary. Hence, L(S) = L∞ : In order to construct the general solution of Eq. (1.5.4), we set that

η = ξ f +θ;

(1.5.5)

where θ is a new unknown function of x; y: Substituting this expression into the determining equation one can see that it takes the form

∂θ ∂θ +f = fy θ : ∂x ∂y

(1.5.6)

1.5 Groups admitted by differential equations

31

This equation is obviously satisfied when θ = 0: Hence, equation (1.5.3) admits the operator ∂ ∂ X0 = + f (x; y) ; ∂x ∂y as well as the operators of the form X = ξ X0 = ξ 0

∂ ∂ +f ∂x ∂y

with an arbitrary function ξ = ξ (x; y): One can also see from here that the admitted space L is infinite-dimensional. Let us demonstrate that if θ = θ (x; y) is a non-vanishing solution of Eq. (1.5.6), then the function 1 M= θ is the integrating factor for Eq. (1.5.3). Indeed, writing the initial equation in the form f dx = 0

dy

and multiplying it by M; one obtains the conditions of the integrating factor in the form ∂ M ∂ (M f ) + = 0: ∂x ∂y Substituting here M = 1=θ one can see that the result coincides with Eq. (1.5.6). Thus, we conclude that if equation (1.5.3) admits the operator X =ξ for which η

∂ ∂ +η ∂x ∂y

f ξ 6 0; then the function M=

1

η

fξ

is an integrating factor. Generally speaking, the problem of finding θ from Eq. (1.5.6) is not easier than the problem of integrating the initial equation (1.5.3). However, the indicated property provides an effective tool for integrating Eq. (1.5.3) if the admitted operator X is known incidentally. One can easily verify that this property is the basis of the whole “elementary” theory of integration of ordinary differential equations by quadrature. For instance, consider the equation y0 =

y x(y + ln x)

It is readily verified that it admits the operator

32

1 One-parameter continuous transformation groups admitted by differential equations

X = xy and that

η

fξ =

∂ ∂x

y2 y + lnx

Thus, the given equation is equivalent to the equation y + lnx dy y2

1 dx = 0; xy

with a total differential in the left-hand side.

1.5.3 Second-order ordinary differential equations Let us find the space L admitted by the equation y00 = f (x; y; y0 ): Denoting p = y0 and r = y00 ; one has the system (S) of one equation r = f (x; y; p):

S:

(1.5.7)

We are looking for the admitted operator in the form X = ξ (x; y)

∂ ∂ + η (x; y) ∂x ∂y

(1.5.8)

In order to write the determining equation, one has to make the second prolongation e Xe of the operator X : It has been shown in x1.4 that the first prolongation Xe of X is given by the formulae (1.4.11) and (1.4.17). According to Eqs. (1.5.2), the determining equation is written e e X(r

f (x; y; p))jr= f = 0:

It takes the following form in detail:

ηxx + p(2ηxy

ξxx ) + p2 (ηyy

= ξ fx + η fy + [ηx + p(ηy

ξx )

2ξxy )

p3ξyy + f (ηy

p2 ξ y ] f p :

2ξx

3pξy) (1.5.9)

An important peculiarity of Eq. (1.5.9) is that it holds identically with respect to the variables x; y; p; while the unknown functions ξ ; η depend on the variables x; y only. Therefore, equation (1.5.9) splits into a system of equations that already do not contain p: The equation readily splits if the function f (x; y; p) is holomorphic with respect to the variable p and can be expanded into the Taylor series:


33 ∞

f (x; y; p) =

∑ fn (x; y)pn :

n=0

Substituting this expansion into Eq. (1.5.9), we have to equate the terms with the same powers of p: This is the operation of splitting which, generally speaking, leads to an infinite system of equations for ξ ; η : The latter system is also called a system of determining equations. Carrying out the indicated operation of Eq. (1.5.9), one obtains the following system of determining equations p0 : ηxx = f0x ξ + f0yη

f0ηy + 2 f0ξx ;

p1 : 2ηxy

ξxx = f1x ξ + f1yη + 2 f2ηx + f1ξx + 3 f0ξy ;

p2 : ηyy

2ξxy = f2x ξ + f2yη + 3 f3ηx + f2 ηy + 2 f1ξy ;

p3 :

ξyy = f3x ξ + f3yη + 4 fvηx + 2 f3ηy

f3 ξx + f2 ξy ;

::::::::::::: Of course, there is no chance here that the solution of the system provides some Lr with r > 0: On the other hand, the question on the maximum possible value of r is of interest. The answer is given by the following theorem by S. Lie. Theorem 1.9. Equation (1.5.7) cannot admit the space Lr of operators of the form (1.5.8) with r > 8: Proof. Let us rewrite the first four determining equations in a compact form

ηxx = ϕ1 ;

2ηxy

ξxx = ϕ2 ;

ηyy

2ξxy = ϕ3 ;

ξyy = ϕ4 ;

(1.5.10)

where ϕσ (σ = 1; : : : ; 4) are some linear functions of ξ ; η and the first derivatives of ξ ; η with respect to x; y: Equations (1.5.10) provide definite expressions for the third derivatives:

ξxxx = φ1 ;

ξxxy = φ2 ;

ηxxx = ω1 ;

ηxxy = ω2 ;

ξxyy = φ3 ; ηxyy = ω3 ;

ξyyy = φ4 ; ηyyy = ω4 ;

(1.5.11)

where every function φσ ; ωσ depends only on ξ ; η as well as on their first and second derivatives. The general solution of the system (1.5.11) depends on 6 + 6 = 12 arbitrary constants and due to linearity of Eqs. (1.5.11) is a linear form of these constants itself. Moreover, equations (1.5.10) impose four independent relations on these constants. Therefore, no more than 12 4 = 8 arbitrary constants remain, which means that the space of solutions has the dimension r 8: The actual number of arbitrary constants remaining in the solution can be less than 8 ; since the complete system of the determining equations contains other equations as well. Let us demonstrate that the dimension 8 is reached by the equation

34


y00 = 0: In this case the complete system of the determining equations is reduced to Eqs. (1.5.10) and (1.5.11), where ϕσ = φσ = ωσ = 0: The general solution of Eqs. (1.5.11) in this case has the form

ξ = A1 + A2 x + A3y + A4xy + A5 x2 + A6 y2 ; η = B1 + B2x + B3 y + B4xy + B5x2 + B6 y2 with the arbitrary constants Ai ; Bi : Equations (1.5.10) impose on these constants the relations B5 = 0; B4 A5 = 0; B6 A4 = 0; A6 = 0 by virtue of which A5 ; A6 ; B5 ; B6 are expressed in terms of other constants. Finally, one obtains the general solution of the determining equations in the form

ξ = A1 + A2 x + A3y + A4xy + B4x2 ; η = B1 + B2 x + B3y + A4y2 + B4 xy; where A j ; B j are already independent. The family of operators X in the form (1.5.8) with the above expressions for ξ and η generates a space L8 :

1.5.4 Heat equation Turning to partial differential equations, let us find operators admitted by the heat equation uy = uxx : The following notation of derivatives is introduced: ux = p;

uy = q;

uxx = r;

uyx = s;

uyy = t:

The equation of the manifold S in the extended space has the form S:

q = r:

(1.5.12)

Let us look for the admitted operator in the form X = ξ (x; y; u)

∂ ∂ ∂ + η (x; y; u) + ζ (x; y; u) ∂x ∂y ∂u

Prolonging this operator with respect to u(x; y); one obtains

∂ ∂ Xe = X + α +β ; ∂p ∂q

(1.5.13)


35

where according to Eqs. (1.4.8)

α = Dx (ζ )

pDx (ξ )

qDx(η );

β = Dy (ζ )

pDy (ξ )

qDy(η );

and operators of total differentiation have the form Dx =

∂ ∂ +p ; ∂x ∂u

Dy =

∂ ∂ +q : ∂y ∂u

The expanded form of the expressions α ; β is

α = ζx + pζu

p(ξx + pξu)

q(ηx + pηu );

β = ζy + qζu

p(ξy + qξu)

q(ηy + qηu ):

In order to write the determining equation, one has to find the second prolongation

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ e Xe = Xe + ρ +σ +τ = X +α +β +ρ +σ +τ ; ∂r ∂s ∂t ∂p ∂q ∂r ∂s ∂t which provides e e X(q

r) = β

ρ:

Therefore, the determining equation (1.5.2) is written (β

ρ )jr=q = 0:

(1.5.14)

Equation (1.5.14) shows that one has to calculate only the expression for the cooree dinate ρ in the prolonged operator X: Using the operator of total differentiation ex = ∂ + p ∂ + r ∂ + s ∂ D ∂x ∂u ∂p ∂q and the formula (1.4.15) one obtains ex α ρ =D

e xξ rD

exη : sD

All preliminary formulae are ready now and one can write out the determining equation (1.5.14) in detail:

36


ζy + qζu

p(ξy + qξu )

= ζxx + pζxu

p(ξxx + pξxu )

+p[ζxu + pζuu +q(ζu

q(ηy + qηu )

ξx

p(ξxu + pξuu )

2pξu

q(ξx + pξu)

q(ηxx + pηxu )

qηu)

(1.5.14 0)

q(ηxu + pηuu )]

s(ηx + pηu )

s(ηx + pηu ):

Since we have already turned to the manifold S of Eq. (1.5.12) by setting r = q; the determining equations (1.5.14 0) should hold identically in the variables x;

y;

u;

p;

q;

s;

t:

However, according to the assumption (1.5.13), the coordinates ξ ; η ; ζ of the unknown operator X depend only on x; y; u: Therefore, we split the determining equation with respect to four independent variables p; q; s and t: Note that the variable t is not involved in Eq. (1.5.14 0). It is manifest from Eq. (1.5.14 0) that splitting with respect to s yields the equation

ηx + pηu = 0; which in its turn splits with respect to p and gives two equations:

ηx = 0;

ηu = 0:

Using these equations in (1.5.14 0) and splitting with respect to q one obtains

ζu

pξu

ηy = ζu

2ξx

3pξu;

whence, splitting with respect to p one obtains

ξu = 0;

ηy = 2ξx :

In view of this information, equation (1.5.14 0) takes the form

ζy

pξy = ζxx + pζxu

pξxx + p(ζxu + pζuu ):

Finally, splitting the latter equation with respect to p one obtains

ζuu = 0;

ζy = ζxx ;

ξy = ξxx

2ζxu:

Thus, equation (1.5.14 0) generates the following system of determining equations

ηx = 0;

ηu = 0;

ξy = ξxx

2ζxu ;

ηy = 2ξx ; ζuu = 0;

ξu = 0;

ζy = ζxx :

(1.5.15)


37

In order to find the space L(S); one has to construct the general solution of the system (1.5.15). The equation ζuu = 0 shows that ζ is linear with respect to u at most, i.e. ζ = a(x; y)u + b(x; y): Furthermore, the equations

ηx = 0;

ηu = 0

show that η = η (y) and therefore the equation ηy = 2ξx yields 1 ξ = η 0 (y)x + c(y): 2 Upon substituting the expression for ζ ; equation ζy = ζxx takes the form ay u + byu = axx u + bxx; whence, splitting with respect to the variable u; one obtains two equations: ay = axx ;

by = bxx :

Similar substitution brings the equation ξy = 2ζxu to the form 1 2ax = η 00 (y)x + c0 (y): 2 Now the equation ay = axx becomes ay =

1 00 η : 4

Hence, all the determining equations (1.5.15) are satisfied if we find the functions a(x; y); b(x; y); η (y); c(y) as solutions of the equations ax =

1 00 η (y)x 4

1 0 c (y); 2

1 00 η (y); 4

ay =

by = bxx :

The compatibility condition for the first two equations (1.5.16) is written

η 000 (y)x + c00 (y) = 0; whence

η 000 (y) = 0;

c00 (y) = 0:

The latter equations have the general solution

η = A1 + 2A2y + 4A3y2 ;

c = B1 + 2B2 y:

Substituting these functions into ax ; ay one obtains

(1.5.16)

38


a(x; y) = B3

B2 x

A3 (x2 + 2y):

Summing up the above results one obtains the following general solution of the determining equations (1.5.15)

ξ = B1 + 2B2y + A2x + 4A3xy; η = A1 + 2A2y + 4A3y2 ; ζ = [B3

B2 x

(1.5.17)

A3 (x2 + 2y)]u + b(x; y);

where A1 ; A2 ; A3 ; B1 ; B2 ; B3 are arbitrary constants and b(x; y) is any solution of Eq. (1.5.16). It follows that the space L of operators admitted by the heat equations is infinite dimensional. Indeed, one can find an infinite set of linearly independent functions satisfying Eq. (1.5.16). Let us try to investigate the structure of this space L∞ in detail. Note that the operators with the coefficients (1.5.17) contain the operators of the form ∂ X 0 = b(x; y) (1.5.18) ∂u The set of the operators (1.5.18) with all b(x; y) satisfying Eq. (1.5.16) is a linear space L0 : This property is guaranteed by linearity and homogeneity of Eq. (1.5.16). On the other hand, the set of all operators obtained when b 0 is a six-dimensional space L6 : Since the operators from L0 and L6 are linearly independent, one obtains the decomposition of L∞ into the direct sum of subspaces: L∞ = L6 L0 :

(1.5.19)

Let us dwell on operators from L0 and find out to which groups G1 they correspond. According to Theorem 1.2, in order to construct G1 with X 0 of the form (1.5.18) one has to solve the system of equations

∂ x0 = 0; ∂a

∂ y0 = 0; ∂a

∂ u0 = b(x0 ; y0 ) ∂a

with the initial conditions x0 ja=0 = x;

y0 ja=0 = y;

u0 ja=0 = u:

One can readily solve the system and verify that the corresponding transformations Ta 2 G1 have the form Ta :

x0 = x;

y0 = y;

u0 = u + ab(x; y):

Since the function ab(x; y); together with b(x; y); is a solution of Eq. (1.5.16), i.e. of the original heat equation, the above transformations Ta consist merely in adding some solution of the equation uy = uxx to the solutions u of the same equation. It is clear that the presence of such transformations Ta admitted by the equation uy = uxx


39

is a trivial consequence of linearity of this equation. This property is obviously valid for all linear equations (and systems of equations). Therefore, the valuable part of the derived L(s) is encapsulated in the subspace L6 : As any finite dimensional linear vector space, L6 has a basis. In the given case, the latter can be obtained by fixing the nonzero value of one of the arbitrary constants A; B and assuming that the rest are vanishing in Eqs. (1.5.17). Thus, one obtains the following basis of the space L6 of operators admitted by the equation uy = uxx : ∂ ∂ ∂ ∂ X1 = ; X2 = ; X3 = x + 2y ; ∂x ∂y ∂x ∂y

∂ ; ∂u 1 2 ∂ (x + 2y)u ; 4 ∂u

X4 = 2y X5 = xy

∂ ∂ + y2 ∂x ∂y

∂ ∂x

xu

(1.5.20) X6 = u

∂ ∂u

Every operator (1.5.20), or any linear combination of these operators (with constant coefficients) generates a one-parameter group G1 admitted by the equation uy = uxx : According to Theorem 1.7, one can map any known solutions into new solutions by applying the transformations Ta 2 G1 : Let us find the general formula for the transformation of solutions. Let a transformation Ta 2 G1 be given by Eqs. (1.1.2), and let

Φ : uk = ϕ k (x) (k = 1; : : : ; m) be a solution of the system (S) admitting the group G1 : Then Ta maps the manifold Φ into the manifold Φ 0 : u0k = ϕ 0k (x0 ); which is also a solution of (S). Equations (S) are written in the variables (x; u) in the form gk (x; u; a) = ϕ 0k ( f (x; u; a)): Since ϕ 0k (x) is a solution of (S), then omitting the prime one can formulate the following rule of transformation of solutions. If uk = ϕ k (x)

(k = 1; : : : ; m)

is a solution of the system (S), then the functions uk = ϕ 0k (x; a) determined implicitly from the system of equations gk (x; u; a) = ϕ k ( f (x; u; a))

(k = 1; : : : ; m);

(1.5.21)

define a new solution of the system (S) for any value of the parameter a: Let us apply this procedure to some operators (1.5.20).

40


Example 1.18. The operator X1 generates the group G1 : x0 = x + a;

y0 = y;

u0 = u:

The formula (1.5.21) yields the transformation of the solution u = ϕ (x; y) into the solution u = ϕ (x + a; y): Example 1.19. The operator X3 generates the group G1 : x0 = ax;

y0 = a2 y;

u0 = u;

and equations (1.5.21) provides u = ϕ (ax; a2 y): Example 1.20. The operator X4 generates the group G1 ax a 2 y

x0 = x + 2ay; y0 = y; u0 = e

u:

Here, equation (1.5.21) takes the form e

ax a 2 y

u = ϕ (x + 2ay; y)

and provides the following formula for transformation of solutions: 2

u = eax+a y ϕ (x + 2ay; y): Example 1.21. Consider the operator X5 : In order to construct transformations Ta 2 G1 ; one has to integrate the system of ordinary differential equations (Theorem 1.2): dx0 = x0 y 0 ; da

dy0 = y02 ; da

du0 = da

1 02 (x + 2y0)u0 4

with the following initial conditions a = 0 : x0 = x;

y0 = y;

u0 = u:

The solution is given by x0 =

x 1

ay

;

y0 =

y 1

ay

;

u0 =

p

1

ay e

ax2 4(1 ay)

u:

Equation (1.5.21) has the form p

1

ay e

ax2 4(1 ay)

u=ϕ

x 1

;

ay 1

y ay

and provides the following formula for transformation of the solution ϕ (x; y) into the solution: ax2 1 x y u= p e 4(1 ay) ϕ ; : 1 ay 1 ay 1 ay


41

The formula is far from being p as evident as the previous ones. In particular, assuming that ϕ = 1; multiplying by a and passing over to the limit as a ! ∞; one obtains the well-known fundamental solution of the heat equation: 1 u= p e y

x2 4y

There exists another method for constructing new solutions from known ones. It is not connected with constructing finite transformations Ta ; but applicable only to linear homogeneous equations. Consider a solution depending on the parameter a: Since the parameter a is not involved in (S), one can obtain a new solution by differentiating the considered solution with respect to the parameter a: Let us apply this observation to solutions provided by the formula (1.5.21). If we denote the solution derived from the formula (1.5.21) by uk = ϕ¯ k (x; a); then we have the identities gk (x; ϕ¯ (x; a); a) ϕ k ( f (x; ϕ¯ (x; a); a)): Differentiating them with respect to a; letting a = 0; invoking definition (1.1.10), and taking into account that ϕ¯ k (x; 0) = ϕ k (x); we obtain

∂ ϕ¯ k ∂ϕk + η k (x; ϕ ) = ξ i (x; ϕ ) i ∂ a a=0 ∂x

Since (S) is linear, the functions

∂ ϕ¯ k ∂ a a=0

should also provide a solution for the system (S). Hence, the equations uk = ξ i (x; ϕ )

∂ϕk ∂ xi

η k (x; ϕ ) (k = 1; : : : ; m)

provide a solution together with uk = ϕ k (x): It is easy to remember these equations since the right-hand side is the result of application of the operator X to the difference ϕ k (x) uk taken on the initial solution. Finally, one makes the following conclusion for linear equations. If uk = ϕ k (x) (k = 1; : : : ; m) is a solution of the linear (homogeneous) system (S) admitting the operator X; then the functions uk = X (ϕ k (x) uk )juk =ϕ k (x) (k = 1; : : : ; m) (1.5.22)

42


generate a solution of the system (S) as well. For instance, applying this method to X4 from Eqs. (1.5.20), one obtains the following solution: u = 2yϕx + xϕ : Likewise, applying the procedure to the operator X5 and any solution ϕ (x; y); one obtains the following solution: 1 u = xyϕx + y2ϕy + (x2 + 2y)ϕ : 4

1.5.5 Gasdynamic equations Let us consider the space of operators admitted by equations of gas dynamics. The system of equations describing the one-dimensional polytropic gas flow (with plane waves) has the form ut + uux +

1 px = 0; ρ

ρt + uρx + ρ ux = 0;

(1.5.23)

pt + upx + γ pux = 0: The spatial coordinate x and time t are independent variables, the velocity u; pressure p and density ρ are functions of x and t: The isentropic exponent γ 6= 0 is a constant. We look for the infinitesimal operator admitted by the system (1.5.23) in the form X =ξ

∂ ∂ ∂ ∂ ∂ +η +ω +τ +σ ∂t ∂x ∂u ∂ρ ∂p

For the sake of simplicity, we will use the already known result and assume that the coordinates of the operator X depend on the variables t; x; u; ρ ; p as follows:

ξ = ξ (t; x); η = η (t; x); ω = ω (t; x; u); τ = τ (t; x; ρ ); σ = σ (t; x; p): We will write the first prolongation of the operator X in the form

(1.5.24)


43

∂ ∂ ∂ ∂ Xe = X + ζut + ζux + ζρ t + ζρ x ∂ ut ∂ ux ∂ ρt ∂ ρx +ζ pt

∂ ∂ + ζ px ∂ pt ∂ px

The operators of total differentiation have the form Dt =

∂ ∂ ∂ ∂ + ut + ρt + pt ; ∂t ∂u ∂ρ ∂p

Dx =

∂ ∂ ∂ ∂ + ux + ρx + px ∂x ∂u ∂ρ ∂p

The additional coordinates of the prolonged operator are derived by the formulae (1.4.10):

ζut = Dt (ω )

ut Dt (ξ )

ux Dt (η ) = ωt + (ωu

ζux = Dx (ω )

ut Dx (ξ )

uxDx (η ) = ωx

ζρt = Dt (τ )

ρt Dt (ξ )

ρx Dt (η ) = τt + (τρ

ζρx = Dx (τ )

ρt Dx (ξ )

ρx Dx (η ) = τx

ζ pt = Dt (σ )

pt Dt (ξ )

pxDt (η ) = σt + (σ p

ζ px = Dx (σ )

pt Dx (ξ )

px Dx (η ) = σx

ξt )ut

ξx ut + (ωu ξt )ρt

ξxρt + (τρ ξt )pt

ξx pt + (σ p

ηt ux ; ηx )ux ; ηt ρx ; ηx )ρx ; ηt px ; ηx )px :

Acting by the operator Xe; on every equation of the system (1.5.23) we obtain the equations 1 τ ζut + uζux + ζ px + ω ux px = 0; ρ ρ2

ζρt + uζρx + ρζux + ωρx + τ ux = 0; ζ pt + uζ px + γ pζux + ω px + γσ ux = 0: In order to write these equations on the manifold S given by Eqs. (1.5.23), it suffices to eliminate the variables ut ; ρt ; pt by means of Eqs. (1.5.23). This yields three determining equations: 1 ωt (ωu ξt ) uux + px ηt ux + u ωx ρ 1 1 + ξx uux + px + (ωu ηx )ux + [σx + ξx (upx ρ ρ + γ pux ) + (σ p

ηx )px ] + ω ux

τ px = 0; ρ2

(I)

44


τt

(τρ

ξt )(uρx + ρ ux )

ηt ρx + u[τx + ξx (uρx 1 ηx )ρx ] + ρ ωx + ξx uux px ρ

+ ρ ux ) + (τρ + (ωu

σt

(σ p

ηx )ux + ωρx + τ ux = 0;

(II)

ξt )(upx + γ pux )

ηt px + u[σx + ξx (upx 1 ηx )px ] + γ p ωx + ξx uux + px ρ

+ γ pux ) + (σ p

+ (ωu + ηx )ux + ω px + γσ ux = 0:

(III)

Equations (I)—(III) should hold identically with respect to the variables t; x; u; ρ ; p; ux ; ρx ; px : We split Eqs. (I)—(III) with respect to the variables ux ; ρx ; px ; i.e. equate the coefficients of these variables to zero. Note that equation (II) contains only one term with px ; namely, ξx px : Therefore, ξx = 0: Taking this equation into account we make further splitting and obtain the following equations ux : px :

u(ωu 1 (ωu ρ

u(τρ

px : Equation (I) (ux ) yields

ξt ) p(σ p

ux :

u(σ p

ξt )

ω = (ηx

ηt + u(ωu

1 ξt ) + (σ p ρ

ρ (τρ

ux :

ρx :

ξt )

ηx ) + ω = 0; ηx )

τ = 0; ρ2

ξt ) + ρ (ωu

ηx ) + τ = 0;

ηt + u(τρ

ηx ) + ω = 0;

ξt ) + p(ωu

ηx ) + σ = 0;

ηt + u(σ p + ηx ) + ω = 0:

(I)

(II)

(III)

ξt )u + ηt :

Due to this expression for ω equations (II) (ρx ) and (III) (px ) are satisfied identically. Furthermore, substituting the expression for ω into Eq. (II) (ux ); one obtains

τ

ρτρ = 0


45

and substituting it into Eq. (III) (ux ); one obtains

σ

pσ p = 0:

Integration of these equations provides

σ = a(t; x)p;

τ = b(t; x)ρ ;

where a(t; x) and b(t; x) are arbitrary functions so far. Substituting these expressions into Eq. (I) (px ); one obtains 2ηx + 2ξt :

b=a Thus,

τ = (a

2ηx + 2ζt )ρ :

This completes the procedure of splitting of the determining equations with respect to the variables ux ; ρx ; px and provides the equations

ξx = 0; ω = (ηx

ξt )u + ηt ;

(1.5.25)

σ = a(t; x)p; τ = (a

2ηx + 2ξt )ρ ;

and the remaining equations (I)—(III) 1 ωt + uωx + σx = 0; ρ τt + uτx + ρωx = 0; σt + uσx + γ pωx = 0:

(I ) (II ) (III)

Equations (I )—(III ) admit further splitting with respect to the variables u; ρ ; p: Equation (I ) yields that ax = 0; whence σx = 0: Upon substitution of the expression for ω ; from Eqs. (1.5.25) the equation (I ) takes the form u(ηxt

ξtt ) + ηtt + u(u(ηxx

ξtx ) + ηtx ) = 0:

Then the split with respect to u provides u2 : ηxx = 0;

u1 : 2ηtx = ξtt ;

u0 : ηtt = 0:

In view of the above results, equation (III ) takes the form at + γηtx = 0: Likewise, equation (II ) yields

(1.5.26)

46


2ηxt + 2ξtt ) + ηtx = 0;

(at which is equivalent to

3 at + ξtt = 0 2 due to the previous equations. Expressing at via ξtt ; one obtains two formulae: at = whence

3 ξtt ; 2 (γ

at =

γ ξtt ; 2

3)ξtt = 0:

(1.5.27)

Let γ 6= 3; then ξtt = 0 and we have the following system of determining equations: ηxx = 0; ηxt = 0; ηtt = 0; ξtt = 0; ax = 0; at = 0: Its general solution is a = a1 ;

ξ = a2 t + a3 ;

η = a4 t + a5 x + a6 ;

where ai (i = 1; : : : ; 6) are arbitrary constants. Therefore, if γ 6= 3; the system (1.5.23) admits a six-dimensional space of operators L6 : 3 If γ = 3; then at = ξtt : Equations (1.5.26) demonstrate that ξttt = 0: Hence, 2 in this case the general solution has the form

ξ = a0 t 2 + a2 t + a3 ; a=

3a0t + a1;

η = a0tx + a4t + a5 x + a6: As compared to the above, the dimension of the space is increased by one, i.e. one obtains L7 : Since L6 results from L7 when a0 = 0; the case γ = 3 is considered further. Equations (1.5.25) provide

ω = ( a0 t + a5

a2 )u + a0x + a4 ;

τ = ( a0 t + a1

2a5 + 2a2)ρ ;

σ = ( 3a0t + a1)p: Let us introduce the constant a05 = a5 a2 instead of a5 for the sake of convenience. Since a05 is an independent arbitrary constant, the prime will be omitted. Finally, the general solution of the determining equations has the form

1.6 Lie algebra of operators

47

ξ = a 0 t 2 + a2 t + a3 ; η = a0tx + a4t + a2x + a5x + a6; ω = a0 (x

tu) + a5u + a4;

τ = ( a0 t + a1

(1.5.28)

2a5)ρ ;

σ = ( 3a0t + a1)p: The following basis of the space of operators L6 (γ 6= 3) corresponds to this solution: X1 =

∂ ; ∂t

(time translation)

X2 =

∂ ; ∂x

(space translation)

∂ ∂ +t ; ∂x ∂t

(dilation)

∂ ∂ + ; ∂x ∂u

(the Galilean translation)

X3 = x X4 = t

X5 = x

∂ ∂ +u ∂x ∂u

X6 = ρ

∂ ∂ +p : ∂ρ ∂p

2ρ

∂ ; (dilation) ∂ρ (dilation)

In the case L7 (γ = 3) one more basis operator is added X7 = t 2

∂ ∂ + tx + (x ∂t ∂x

tu)

∂ ∂u

tρ

∂ ∂ρ

3t p

∂ ; ∂p

which is an operator of some projective transformation of the space E 5 (t; x; u; ρ ; p):

1.6 Lie algebra of operators 1.6.1 Commutator. Definition of a Lie algebra We have already introduced in x1.5 the operations of addition of operators and their multiplication by constants. We will consider one more operation now. Let ∂ ∂ X = ξ i i and Y = η i i ∂x ∂x be two operators in E N :

48


Definition 1.7. Commutator of the operators X and Y is a new operator [X;Y ] determined by the formula [X;Y ] = (X η i

Y ξ i)

∂ ∂ xi

(1.6.1)

It can be obtained by the following principle. Consider the expression X(Y F(x))

Y (XF(x))

as a result of the action of some operator on the function F(x): Expansion of this expression cancels out the derivatives of the second order and provides the result of the action of the commutator on F: Therefore, the formula [X ;Y ] = X η i

Yξ i

∂ = XY ∂ xi

YX

(1.6.2)

holds. The operation defined by the formula (1.6.1) maps any two operators X and Y into their commutator [X ;Y ]: This operation is referred to as the operation of commutation. Let us formulate some properties of the operation of commutation. 1 The commutator is bilinear with respect to X and Y; i.e. for any constants α and β the following identity holds: [(α X + β Y ); Z] = α [X; Z] + β [Y; Z]: 2 The commutator is skew symmetric: [X;Y ] =

[Y; X]:

3 The Jacobi identity [[X;Y ]; Z] + [[Y; Z]; X] + [[Z; X];Y ] = 0 holds for any three operators X;Y and Z: These properties can be easily proved by means of the representation (1.6.2) of the commutator. Definition 1.8. A linear space L of operators is referred to as a Lie algebra of operators if for any X and Y belonging to L; their commutator belongs to L as well. Some examples of spaces L of the operators admitted by differential equations have been considered in x1.5. One can easily verify that all of them are Lie algebras of operators. To this end, it is sufficient to calculate commutators of the basis operators. In what follows, it will be demonstrated that this circumstance is not incidental. For this purpose, we have to study some additional properties of the operation of commutation.


49

1.6.2 Properties of commutator Lemma 1.4. The commutator is invariant with respect to changes of coordinate systems in E N : Proof. Consider a new system of coordinates given by the equations yi = yi (x)

(i = 1; : : : ; N):

According to the transformation formula (1.2.6) for operators, we have X = X 0 = X(yi )

∂ ; ∂ yi

Y = Y 0 = Y (yi )

∂ ∂ yi

Calculating the commutator in the variables (y); one has [X 0 ;Y 0 ] = [X 0Y (yi ) = [X ;Y ](yi )

Y 0 X(yi )]

∂ = [XY (yi ) ∂ yi

Y X(yi )]

∂ ∂ yi

∂ = [X ;Y ]0 : ∂ yi

Hence, [X 0 ;Y 0 ] = [X ;Y ]0 and Lemma 1.4 is proved. Theorem 1.10. If a manifold M E N is invariant with respect to the operators X and Y; then it is also invariant with respect to their commutator [X;Y ]: Proof. Let us make the same assumption as in the proof of Theorem 1.5, namely that M is given by the equations x1 = 0; : : : ; xs = 0: The necessary and sufficient conditions of invariance of such M with respect to the operators ∂ ∂ X = ξ i i ; Y = ηi i ∂x ∂x are obtained in Theorem 1.5 in the form

ξ σ jM = ξ σ (0; : : : ; 0; xs+1 ; : : : ; xn ) = 0; η σ jM = η σ (0; : : : ; 0; xs+1 ; : : : ; xn ) = 0 (σ = 1; : : : ; s): Note that all coordinates [X;Y ]σ = X η σ

Yξ σ = ξ i

∂ ησ ∂ xi

ηi

∂ξσ ∂ xi

(1 σ s)

of the commutator are identically equal to zero on the manifold M : Indeed, [X;Y ]σ can be written in the form

50


ξτ

σ 0 ∂η ∂ησ +ξτ 0 τ τ ∂x ∂x

ητ

∂ξσ ∂τ

ητ

0

∂ξσ ; ∂τ0

where τ = 1; : : : ; s; τ 0 = s + 1; : : : ; N: The terms with indices τ are equal to zero, because ξ τ and η τ are equal to zero according to the conditions of invariance. The terms with τ 0 are also equal to zero, because the operations of differentiation with respect to τ 0 and of transition to the manifold M are permutable. Theorem 1.11. The operation of prolongation is permutable with the operation of commutation: e Ye ] = [X;Y g ]: [X; Proof. We use the invariance of the commutator and the operation of prolongation with respect to the system of coordinates. Let us introduce a system of coordinates in E N so that the operator X becomes an operator of translation along one of the coordinates. It is always possible according to Theorem 1.3. As it has been demonstrated above, the operator of translation is “does not prolong”, so the prolongation of the operator X has the form Xe = X: We assume that the operator of translation X is ∂ X= 1 ∂x The alternative assumption ∂ X= 1 ∂u is considered likewise. Further, one has

∂ ∂ ∂ ∂ Ye = Y + ζik k = ξ i i + η k k + ζik k ; ∂ x ∂ u ∂ pi ∂ pi

ζik = Di (η k )

e Ye ] : Let us compute the commutator [X; e Ye ] = [X; Ye ] = [X;Y ] + X; ζik ∂ [X; ∂ pki k ∂ ζi ∂ = [X;Y ] + : ∂ x1 ∂ pki On the other hand, since [X ;Y ] =

∂ξi ∂ ∂ ηk ∂ + i ; 1 i ∂x ∂x ∂ x ∂ uk

one has g ] = [X;Y ] + ζi0k [X;Y where

∂ ; ∂ pki

pkj Di (ξ j ):


51

ζi0k = Di

∂ηk ∂ x1

pkj Di

Hence, [Xg ;Y ] = [X ;Y ] +

∂ξ j ∂ x1

=

∂ k ζ : ∂ x1 i

∂ ζik ∂ e Ye ]; = [X; ∂ x1 ∂ pki

and the theorem is proved.

1.6.3 Lie algebra of admitted operators Theorem 1.12. Given any system of differential equations (S), the linear space L(S) of operators admitted by the system (S) is a Lie algebra. Proof. If (S) admits X and Y; i.e. X;Y 2 L(S); then the manifold S is invariant with respect to Xe and Ye in the prolonged space. According to Theorem 1.10, S is also invariant with respect to [Xe ; Ye ]; and according to Theorem 1.11, e Ye ] = [X;Y g ]: [X; Thus, S is invariant with respect to [Xg ;Y ]; which means that (S) admits the commutator [X;Y ] by definition, so that [X;Y ] 2 L(S) and Theorem 1.12 is proved. Theorems 1.8 and 1.12 demonstrate that a commutator of any two operators from L(S) is an operator in L(S): In particular, in case of a finite-dimensional Lr ; the commutator of any two basis operators is a linear combination of basis operators. It is convenient to write this circumstance in the table of commutators, where the intersection of the k-th row and the l-th column gives the commutator [Xk ; Xl ]: As an example, we provide the following table of commutators for basis operators of the Lie algebra L6 admitted by the heat equation uy = uxx :

X1

X2

X3

X4

X5

X6

1 X4 2

0

X1

0

0

X1

X6

X2

0

0

2X2

2X1

X3

X1

2X2

0

X4

2X5

0

2X1

X4

0

0

0

1 X3 + X6 2

2X5

0

0

0

0

0

0

0

X4

X6

X5

1 X4 2

X6

0

0

1 X6 2

X3

0

Chapter 2

Lie algebras and local Lie groups

2.1 Lie algebra 2.1.1 Definition and examples We consider a linear vector space L of elements (vectors) u; v; : : : over a field of real (or complex) numbers. Definition 2.1. A linear space L is said to be a Lie algebra if a binary operation of multiplication [u; v] satisfying the following properties is defined: 1 Linearity [α u + β v; ω ] = α [u; ω ] + β [v; ω ]; 2 Antisymmetry [u; v] = [v; u]; 3 Jacobi identity [[u; v]; ω ] + [[v; ω ]; u] + [[ω ; u]; v] = 0: The multiplication is also termed an operation of commutation and the product [u; v] is called the commutator of the vectors u; v: Example 2.1. The set of operators admitted by a system of differential equations. The operation of commutation is defined in x1.6. Example 2.2. The set of vectors in the three-dimensional Euclidian space where the operation of commutation is the vector product: [a; b] = a b: Example 2.3. The set of linear transformations A; B; : : : of a linear space. The operation of commutation is defined by [A; B] = AB BA: Depending on the dimension of the linear space L; one can single out infinite Lie algebras (L is infinite-dimensional) and finite Lie algebras (L is finite-dimensional). In the latter case, the Lie algebra is denoted by Lr if the dimension of L is r: Thus, one has the Lie algebra L3 in Example 2.2. Definition 2.2. A linear mapping ψ of a Lie algebra L to a Lie algebra L0 is called an isomorphism if: (a) it is a one-to-one mapping,

54

2 Lie algebras and local Lie groups

(b) it preserves the commutator of any two vectors. If only the condition (b) is satisfied, the mapping ψ is called a homomorphism. The set J of vectors from L mapped into the zero of the Lie algebra L0 is called the kernel of the homomorphism ψ : Isomorphism of L onto itself is called an automorphism. In this definition, preservation of the commutator means that

ψ [u; v] = [ψ (u); ψ (v)]: Example 2.4. The Lie algebra L3 from Example 2.2 is isomorphic with respect to the Lie algebra of the matrices 0 1 0 a1 a2 0 a3 A A = @ a1 a2 a3 0 if the operation of commutation for A is defined as in Example 2.3. The isomorphism ψ is determined as follows. If a vector a has the coordinates (a1 ; a2 ; a3 ); then ψ (a) = A:

2.1.2 Subalgebra and ideal Definition 2.3. A linear subspace N L is called a subalgebra of the Lie algebra L if [u; v] 2 N for any u; v 2 N: The subalgebra J L is termed an ideal of L if [u; v] 2 J for any u 2 J and v 2 L: One can easily verify that a homomorphism (and an isomorphism in particular) maps a subalgebra N L into a subalgebra N 0 L0 ; and an ideal J into an ideal J 0 : Further, the kernel J of a homomorphism of the Lie algebra L into the Lie algebra L0 is an ideal in L: Indeed, if u 2 J and v 2 L; then ψ (u) = 0 implies that

ψ ([u; v]) = [ψ (u); ψ (v)] = [0; ψ (v)] = 0; so that [u; v] 2 J: The set Z of all vectors u 2 L; such that [u; v] = 0 for any v 2 L is an ideal in L: This ideal is termed the center of the Lie algebra L: Here we will verify only the property of Z to be a subalgebra. Let u1 ; u2 2 Z and v 2 L: Using the Jacobi identity [[u1 ; u2 ]; v] + [[u2; v]; u1 ] + [[v; u1]; u2 ] = 0 and noting that the second and the third terms vanish due to the assumptions that [u2 ; v] = 0 and [u1 ; v] = [v; u1] = 0; we obtain [[u1 ; u2 ]; v]: Hence, [u1 ; u2 ] 2 Z:

2.1 Lie algebra

55

The remaining properties are verified trivially. If J is an ideal in L; one can introduce an equivalence relation. Namely, two vectors u and v from L are equivalent, u v; if u v 2 J : One can easily verify that the introduced equivalence relation is reflexive, symmetric and transitive. The equivalence relation splits L into classes U;V; : : : of equivalent vectors. The set of these classes is a Lie algebra if one introduces the basic operations by the following rule. If u 2 U; v 2 V; then α U + β V and [U;V ] are classes containing the elements α u + β v and [u; v]; respectively. Definition 2.4. The Lie algebra of classes of equivalent vectors introduced above is called the quotient algebra of the Lie algebra L with respect to its ideal J and is denoted by L=J : There exists a “natural” homomorphism ψ of the Lie algebra L to the quotient algebra L=J: It is given by the formula ψ (u) = U if u 2 U: The ideal J is the kernel of the homomorphism ψ :

2.1.3 Structure of finite-dimensional Lie algebras Let us consider the case of a finite-dimensional Lie algebra Lr : Since Lr is a linear space, it has a basis fuα g of r linearly independent vectors uα

(α = 1; : : : ; r):

Any vector of Lr is represented uniquely as a linear combination of the basis vectors. In particular, γ [uα ; uβ ] = Cαβ uγ (α ; β = 1; : : : ; r); (2.1.1) where the summation over γ = 1; : : : ; r is assumed in the right-hand side. The numγ bers Cαβ are called the structure constants of Lr in the basis fuα g: Structure constants change together with the basis. Let fu¯α g be a new basis in Lr defined by β u¯α = pα uβ : Comparing two expressions for the commutator: γ

[u¯α ; u¯β ] = [pσα uσ ; pβτ uτ ] = pσα pβτ [uσ uτ ] = pσα pβτ Cσ τ uγ and

γ ε ε [u¯α ; u¯β ] = C¯αβ uε = C¯αβ p ε uγ ;

one obtains the following rule for the change of structure constants: γ γ ε C¯αβ pε = Cσ τ pσα pβτ :

56

2 Lie algebras and local Lie groups γ

It follows that Cαβ is a tensor of the third order which is twice covariant and once contravariant. The structure constants determine the Lie algebra Lr completely, since they allow to find the commutator of any two elements in the coordinate form. Namely, if u = aα uα ;

v = bα uα ;

then, by virtue of Eqs. (2.1.1) γ

[u; v] = aα bβ [uα ; uβ ] = Cαβ aα bβ uγ :

(2.1.2)

The properties 2 and 3 of the operation of commutation (see Definition 2.1) γ can be expressed in terms of structure constants Lr : One can readily verify that Cαβ satisfies the following Jacobi relations: γ

γ

Cαβ = Cβ α ;

σ σ ε Cαβ Cσε γ +Cβσγ Cσε α +Cγα Cσ β = 0

(2.1.3)

for all α ; β ; γ ; ε = 1; : : : ; r (summation over σ = 1; : : : ; r). The property of isomorphism of the Lie algebras Lr is also expressed in terms of structure constants. Theorem 2.1. Algebras Lr and L0r are isomorphic if and only if they have the same γ structure constants. Namely, if Lr and L0r have the same Cαβ ; they are isomorphic; γ

if Lr and L0r are isomorphic, then one can find such bases in them that Cαβ of both Lie algebras coincide in these bases. Proof. Let Lr and L0r be isomorphic and let fuα g be a basis of Lr : If ψ is an isomorphism of Lr onto L0r ; then fψ (uα )g is a basis of L0r : One has γ

γ

[ψ (uα ); ψ (uβ )] = ψ [uα ; uβ ] = ψ (Cαβ uγ ) = Cαβ ψ (uγ ); γ

where Cαβ are structure constants of Lr : The resulting equation demonstrates that γ

the same Cαβ provide the structure constants of L0r in the basis fψ (uα )g: Conversely, let fuα g and fuα0 g be bases in Lr and L0r ; respectively, defined so that γ

[uα ; uβ ] = Cαβ uγ and

γ

[uα0 ; uβ0 ] = Cαβ uγ0 : Let us define an isomorphism ψ by the relation

ψ (uα ) = uα0 on the basis vectors and extend it linearly on the whole Lr : if u = aα uα ;

2.2 Adjoint algebra

then

57

ψ (u) = aα uα0 :

It is manifest that ψ is a one-to-one mapping. Preservation of the commutator follows from (2.1.2). Theorem 2.1 is proved. It is important to point out that there exists a Lie algebra Lr for every set of γ γ constants Cαβ ; satisfying the Jacobi relations (2.1.3), for which these Cαβ are its structure constants. In order to construct such Lr one has to take any r-dimensional vector space, choose some basis fuα g in it and introduce the operation of commutation by the formula (2.1.2). Then equations (2.1.3) guarantee that the defined commutator satisfies all axioms of Definition 2.1. Finally, let us introduce several other notions connected with a Lie algebra. The Lie algebra L is said to be simple if it does not contain ideals other than zero (consisting of one zero vector) or other than the algebra L itself. One can readily verify that the linear span of all commutators [u; v] of vectors of the Lie algebra L is an ideal in L: It is called the derived algebra of the Lie algebra L and is denoted by L(1) : One can construct the sequence of derived algebras L(k) (k = 1; 2; : : :) by determining L(k) as the derived algebra of the Lie algebra L(k 1) : A Lie algebra L is said to be solvable if L(k) = f0g (null algebra) for a certain k < ∞: The Lie algebra L is said to be semi-simple if it does not contain solvable ideals.

2.2 Adjoint algebra 2.2.1 Inner derivation Let L be a Lie algebra and a 2 L: According to property 1 of Definition 2.1, the formula v = [u; a]; u 2 L defines a linear mapping of L into itself. This mapping is referred to as the inner derivation of the Lie algebra L and is further denoted by Da : Thus Da u = [u; a]:

(2.2.1)

The term “inner derivation” is justified by the fact that the mapping Da acts on the commutator [u; v] according to the formula similar to the derivation of a product of functions, namely Da [u; v] = [Da u; v] + [u; Dav]: (2.2.2) Equation (2.2.2) is easily proved by using the Jacobi relations. When the vector a runs through the whole Lie algebra L; one obtains a set of inner derivation fDa g; where one can determine linear operations of summation and multiplication by numbers that turn it into a linear vector space LD :

58


α Da + β Db = Dα a+β b:

(2.2.3)

The null of the space LD is D0 such that D0 u = [u; 0]: It maps any vector u 2 L into 0: We introduce in LD the operation of commutation defined by the formula [Da ; Db ] = D[a;b] : (2.2.4) Theorem 2.2. The linear space LD with the operation of commutation (2.2.4) is a Lie algebra. Proof. Since the validity of axioms 1 and 2 of Definition 2.1 is evident, one has only to verify the axiom 3 : According to the definition (2.2.4), one has [[Da ; Db ]; Dc ] = [D[a;b] ; Dc ] = D[[a;b]c] : Therefore, for any u 2 L f[[Da ; Db ]; Dc ] + [[Db; Dc ]; Da ] + [[Dc ; Da ]; Db ]gu = [u; f[[a; b]; c] + [[b; c]; a] + [[c; a]; b]g] = [u; 0] = D0 u; which was to be proved.

2.2.2 Adjoint algebra Definition 2.5. The Lie algebra LD constructed according to Eqs. (2.2.1), (2.2.3), (2.2.4) is called the algebra of inner derivations or the adjoint algebra of the Lie algebra L: Theorem 2.3. The adjoint algebra LD is isomorphic to the quotient algebra L=Z of the Lie algebra L with respect to its center Z: Proof. There exists a natural homomorphism ψ of L on LD ; namely

ψ (a) = Da : Equation (2.2.3) shows that ψ is linear, whereas equation (2.2.4) entails that it preserves the commutator. To complete the proof one has to verify that the kernel J of the homomorphism ψ coincides with Z: By definition, a 2 J if and only if ψ (a) = D0 : If a 2 Z; then according to Eq. (2.2.1), Da u = 0; so that ψ (a) = D0 ; i.e. a 2 J : Conversely, if a 2 J; then Da = D0 and [a; b] =

[b; a] =

Da b =

D0 b = 0

2.2 Adjoint algebra

59

for any vector b 2 L; so that a 2 Z: Thus, J = Z: Let us consider the case of a finite-dimensional Lr with a basis fuα g: Any vector u 2 Lr is written in the form u = xα uα ; where xα (α = 1; : : : ; r) are coordinates of the vector u in the basis fuα g: Let us introduce “basis” inner derivations by the following formula: Dα = Du α (α = 1; : : : ; r): Then,

γ

Dβ u = [u; uβ ] = xα [uα ; uβ ] = Cαβ xα uγ

and one arrives at the formula γ

Dβ (xα uα ) = Cαβ xα uγ :

(2.2.5)

Let us introduce the operators γ

Eβ = Cαβ xα

∂ ∂ xγ

(β = 1; : : : ; r)

(2.2.6)

acting in the r-dimensional space of the points x (x1 ; : : : ; xr ) and consider their linear combinations E = eβ Eβ with constant (i.e. independent of x) coefficients e1 ; : : : ; er : Let fEg be the set of all such operators E: Theorem 2.4. The set fEg is a Lie algebra of operators, isomorphic to the adjoint algebra LD of the Lie algebra L: Proof. The set fEg is a linear space of operators according to construction. The axioms 1 —3 of Definition 2.1 always hold (see x1.6) for operators with a usual definition of the commutator (see Definition 1.7). Therefore, in order to prove that fEg is a Lie algebra, one has only to verify that [Eβ ; Eθ ] 2 fEg for any β ; θ = 1; : : : ; r: This is provided by straightforward calculations invoking the properties of structure constants (2.1.3). One has ∂ ∂ ∂ γ γ [Eβ ; Eθ ] = Cαβ xα γ (Cσε θ xσ ) Cαθ xα γ (Cσε β xσ ) ∂x ∂x ∂ xε

∂ ∂ γ γ ε ε = (Cαβ Cγθ +Cθ α Cγβ )xα ε ∂ xε ∂x γ γ γ ε α ∂ ε α ∂ Cβ θ Cγα x = C C x = Cβ θ Eγ : αγ βθ ∂ xε ∂ xε γ

ε = (Cαβ Cγθ

=

γ

ε Cαθ Cγβ )xα

Further, using the fact that equation (2.2.4) entails the equality γ

[Dβ ; Dθ ] = Cβ θ Dγ

60


and comparing Eqs. (2.2.5) and (2.2.6) one concludes that the mapping ψ (Eα ) = Dα is an isomorphism. The Lie algebra of the operators fEg is said to be a representation of the adjoint algebra LD :

2.2.3 Inner automorphisms of a Lie algebra Every operator from fEg generates a one-parameter group G1 of transformations of the r-dimensional space E r of points (x): If γ

E = eβ Eβ = eβ Cαβ xα

∂ ; ∂ xγ

then, according to Theorem 1.2, the transformations composing the corresponding G1 can be obtained by integrating the following system of ordinary differential equations with the initial conditions: dx0γ γ = eβ Cαβ x0α ; dt

x0γ (0) = xγ (γ = 1; : : : ; r):

(2.2.7)

Since (2.2.7) is a system of linear homogeneous equations with constant coefficients, the solution of (2.2.7) is a linear form of the initial data x1 ; : : : ; xr and can be written in the form γ x0γ = fσ (t)xσ (γ = 1; : : : ; r): (2.2.8) Equations (2.2.8) determine the desired transformations in E r composing the group G1 : These transformations are linear. We will interpret them as transformations of coordinates of the vector u 2 Lr ; i.e. as transformations of the vectors u 2 Lr given in the basis fuα g: These transformations are denoted by the symbol At so that A0 is an identical transformation. Theorem 2.5. The transformations At are automorphisms of the Lie algebra Lr : γ

γ

Proof. It is manifest that the mapping u0 = At u is one-to-one (since fσ (0) = δσ ) and linear. It remains to verify that this mapping preserves the commutator. It is sufficient to prove this property for the basis vectors only, i.e. to show that At [uα ; uβ ] = [At uα ; At uβ ]: One has

γ

γ

At [uα ; uβ ] = Cαβ At uγ = Cαβ fγσ (t)uσ ; γ

γ

σ [At uα ; At uβ ] = [ fα (t)uγ ; fβθ (t)uθ ] = Cγθ fα (t) fβθ (t)uδσ :

Setting one obtains

γ

σ qαβ = Cαβ fγσ (t)

γ

σ Cγθ fα (t) fβθ (t);

2.3 Local Lie group

61 σ dqαβ

σ τ σ = eε Cτε qαβ ; qαβ (0) = 0 dt upon simple but tedious calculations based on Eqs. (2.2.7) and (2.1.3). The uniqueness of the solution of the above system of equations provides σ qαβ (t) 0;

which proves the theorem. The above automorphisms (2.2.8) are also referred to as inner automorphisms of the Lie algebra Lr :

2.3 Local Lie group 2.3.1 Coordinates in a group They say that it is possible to introduce coordinates in the group G if elements g 2 G can be put into one-to-one correspondence with the points a of a set Ω E r where E r is an r-dimensional Euclidian space. If an element of G corresponding to the point a 2 Ω is denoted by ga ; then the formula ga gb = gc implies

c = ϕ (a; b);

where ϕ (a; b) is a function determined on Ω Ω : The function ϕ (a; b) can be called a multiplication law of elements of the group G: Sometimes the multiplication law can be determined not for the whole group G; but only for some subset Gr 2 G; which leads to the notion of a local group. Definition 2.6. A subset Gr of the group G containing the unit element g0 is called a local Lie group if the following conditions are satisfied: (i) there is a one-to-one correspondence between elements of Gr with the points a 2 Q of an open sphere Q E r with the center 0; so that g0 $ 0; (ii) there exists ε > 0 such that ga gb 2 Gr and ga 1 2 Gr for any points a; b with jaj < ε ; jbj < ε ; (iii) the multiplication law c = ϕ (a; b) is a thrice continuously differentiable function of coordinates of the points a and b: Remark 2.1. In general Gr is not a group. Therefore the notion of a local Lie group can be defined without the supposition that Gr is included into a group G: Then it is a set with an associative operation of multiplication containing a unit and an inverse transformation of elements. However, these operations are determined not for all elements, but only for those that are “sufficiently close” to the unit element in the meaning of Definition 2.6.

62


The properties of multiplication of elements lying in a sufficiently small neighborhood of the unit are investigated in local Lie groups. Therefore local Lie groups Gr and G0r such that G0r Gr are indistinguishable in this theory if they differ only by the size of the sphere Q and by the value of ε : Let the points a 2 Q have coordinates aα (α = 1; : : : ; r): Then the multiplication law ϕ (a; b) in the group multiplication ga gb = gc :

c = ϕ (a; b)

is written in the coordinates as follows: cα = ϕ α (a; b) = ϕ α (a1 ; : : : ; ar ; b1 ; : : : ; br ) (α = 1; : : : ; r):

(2.3.1)

Investigation of local Lie groups is reduced to investigation of properties of the functions ϕ (a; b) in a neighborhood of the origin of coordinates. Of course, since the system of coordinates in E r can be chosen arbitrarily, we are interested only in the properties of ϕ (a; b) independent of this choice. In what follows, the system of coordinates in E r is called a system of coordinates in Gr and is denoted by the symbol ∑a : The transition from the system of coordinates ∑a to a system ∑a¯ is given by the following equations: aα = f α (a) ¯ = f α (a¯1 ; : : : ; a¯2 ) (α = 1; : : : ; r);

(2.3.2)

where the functions f α (a) ¯ are thrice continuously differentiable and satisfy the condition α ∂f 6= 0: (2.3.2 0 ) ∂ a¯β a=0 ¯ Let us point out some simple properties of the multiplication law. Since the unit element g0 corresponds to the point a = 0 (aα = 0; α = 1; : : : ; r); the equations g0 g0 = g0 ; yield

ϕ (0; 0) = 0;

ga g0 = ga ;

ϕ (a; 0) = a;

g0 gb = gb

ϕ (0; b) = b:

(2.3.3)

By virtue of Eqs. (2.3.3), the Taylor expansion of ϕ (a; b) yields

ϕ α (a; b) = aα + bα + rβαγ aβ bγ + O(jaj2 + jbj2):

(2.3.4)

The free index α appearing in this formula runs through the values 1 ; : : : ; r even though it is not written explicitly. Many formulae follow this rule in what follows.

2.3.2 Subgroups Let us introduce the auxiliary functions

2.3 Local Lie group

63

Aβα (a) =

∂ ϕ α (a; b) : ∂ bβ b=0

(2.3.5)

According to Eqs. (2.3.3), we have Aβα (0) = δβα : Using these functions, one can write the expansion of ϕ (a; b) as follows:

ϕ α (a; b) = aα + Aβα (a)bβ + O(jbj2 ):

(2.3.6)

A family of elements g(t) 2 Gr depending on a real parameter t is called a curve if the coordinates aα (t) of these elements are continuous differentiable functions of t when jtj < ε : The vector e with the coordinates daα (t) α e = (α = 1; : : : ; r) (2.3.7) dt t=0 is called the directing vector of the curve g(t): Definition 2.7. The curve g(t) is said to be a one-parameter subgroup (or, for the sake of brevity, a subgroup G1 ), if g(s)g(t) = g(s + t) for all admissible values of s;t: The property of the curve g(t) to be a subgroup of G1 is written in the coordinates by the equation aα (t + s) = ϕ α (a(t); a(s)): (2.3.8) Theorem 2.6. A subgroup G1 with the directing vector e satisfies the system of equations daα = Aβα (a)eβ ; aα (0) = 0: (2.3.9) dt Conversely, whatever the vector e is, the solution of the system (2.3.9) determines the subgroup G1 with the directing vector e: Proof. Differentiating Eq. (2.3.8) with respect to s; setting s = 0 and invoking Eqs. (2.3.5) and (2.3.7), one obtains Eqs. (2.3.9). In order to prove the converse it suffices to verify that (2.3.8) holds, since equations (2.3.7) follows from Eqs. (2.3.9) and (2.3.3). One has daα (t + s) = Aβα (a(t + s))eβ ; ds

aα (t + s)js=0 = aα (t)

by construction. Further, for solution of the system (2.3.9) one has aα (t) = eα t + O(jtj2)

64


and aα (s + u) = aα (s) + eβ Aβα (a(s))u + O(juj2 ) = aα (s) + Aβα (a(s))aβ (u) + O(juj2) = ϕ α (a(s); a(u)) + O(juj2); where the latter equality follows from Eq. (2.3.6). Therefore, we have the equation

ϕ α (a(t); a(s + u)) = ϕ α (a(t); ϕ (a(s); a(u))) + O(juj2 ): Note that the value ϕ α (a(t); ϕ (a(s); a(u))) is a coordinate of the element g(t)[g(s); g(u)] = [g(t); g(s)]g(u) (since the multiplication is associative) and therefore it equals

ϕ α (ϕ (a(t); a(s)); a(u)): Thus, differentiating the above equation with respect to u; setting u = 0 and invoking Eqs. (2.3.7), (2.3.5) and (2.3.3), one obtains d ϕ α (a(t); a(s)) = Aβα (ϕ (a(t); a(s)))eβ ; ϕ α (a(t); a(s))js=0 = aα (t): ds Hence, the left and the right-hand sides of Eq. (2.3.8) satisfy one and the same differential equation and the same initial condition, and hence they coincide according to the theorem on uniqueness of the solution of the Cauchy problem. Theorem 2.6 is proved. Corollary 2.1. For any vector e there exists one and only one subgroup G1 ; having this vector e as its directing vector.

2.3.3 Canonical coordinates of the first kind Definition 2.8. The system of coordinates ∑a in a local Lie group Gr is referred to as a canonical system of coordinates of the first kind if the curve g(t) :

aα = eα t

is a subgroup of G1 for any vector e: Theorem 2.7. One can introduce a canonical coordinate system ∑a of the first kind in any local Lie group Gr : Proof. Let aα = f α (e;t) be a solution of the system (2.3.9) depending on the vector e: Since multiplication of the vector e by k is equivalent to multiplication of t by k;

2.3 Local Lie group

65

the uniqueness of the solution entails that the function f α has the property f α (ke;t) = f α (e; kt): Setting here t = 1 and replacing k by t; one obtains f α (e;t) f α (te; 1): Then, we introduce a new system of coordinates ∑a¯ in Gr defined by the equation aα = f α (a; ¯ 1): First we verify the condition (2.3.2 0 ): ∂ aα ∂ f α (0; : : : ; a¯β ; : : : ; 0; 1) ∂ f α (0; : : : ;t; : : : ; 0; 1) = = β ∂t ∂ a¯β a=0 ∂ a¯β ¯ a¯ =0 t=0 =

∂ f α (0; : : : ; 1; : : : ; 0;t) α α = Aβ (0) = δβ : ∂t t=0

Finally we demonstrate that ∑a¯ is a canonical system of coordinates of the first kind. Indeed, if a¯ α = eα t; then aα = f α (et; 1) = f α (e;t); so that the curve g(t) with the coordinates aα = aα (t) is a subgroup G1 : Theorem 2.7 is proved. Corollary 2.2. One can draw a one-parameter subgroup through every element of a local Lie group Gr sufficiently close to the unit element g0 : In what follows, the symbol a 1 denotes a point from E r corresponding to the element ga 1 ; so that ga 1 = ga 1 : Let us introduce auxiliary functions ∂ ϕ α (a; b) α Vβ (b) = ; Vβα (0) = δβα ; (2.3.10) ∂ bβ a=b 1 where the second equation holds due to Eqs. (2.3.3). Now let us formulate and prove the so-called Lie’s fundamental theorems.

2.3.4 First fundamental theorem of Lie Theorem 2.8. The functions ϕ α (a; b) satisfy the system of equations γ

Vα (ϕ )

∂ ϕα γ = Vβ (b); ∂ bβ

ϕ α (a; 0) = aα :

(2.3.11)

Conversely, given twice continuously differentiable functions Vβα (b); such that Vβα (0) = δβα ; with which the system (2.3.11) has a single solution ϕ (a; b) with any

66


values aα ; there exists a local Lie group Gr with the multiplication law ϕ (a; b) and with the given auxiliary functions Vβα (b): Proof. Let us replace b by b + ∆ b in the formula gc = ga gb with a unaltered. Then, c is replaced by c + ∆ c: Multiplying the equation gc+∆ c = ga gb+∆ b from the left by gc 1 = gb 1 ga 1 one obtains gc 1 gc+∆ c = gb 1 gb+∆ b: The latter equality is written in terms of the functions ϕ as

ϕ γ (c 1 ; c + ∆ c) = ϕ γ (b 1 ; b + ∆ b): Noting that

∆ cα = and

∂ϕα β ∆ b + O(j∆ bj2 ) ∂ bβ

ϕ α (b 1 ; b + ∆ b) = Vβα (b)∆ bβ + O(j∆ bj2 );

one transforms the above equality into γ

Vα (ϕ )

∂ϕα β γ ∆ b = Vβ (b)∆ bβ + O(j∆ bj2 ); ∂ bβ

whence equations (2.3.11) follow. Let us prove now the converse statement. The assumptions about the functions Vβα (b) guarantee that the solution ϕ α (a; b) of the system (2.3.11) is determined

and is thrice continuously differentiable with respect to the variables aα ; bβ from a certain neighborhood ω of the origin of coordinates in the space E r of the points a: Further considerations refer to this neighborhood ω without special notice. We determine the operation of multiplication a b for points E r by the formula a b = ϕ (a; b) and prove that E r (specifically, some sphere Q E r ) is a local Lie group with this multiplication. Since ϕ (a; b) is smooth, our operation of multiplication is determined in a sufficiently small vicinity of the origin of coordinates. Therefore, it remains to prove the validity of group axioms only. First, let us establish a relation between the functions Vβα given by Eqs. (2.3.11), and the functions Aβα determined by the solution ϕ (a; b) of the system (2.3.11) according to the formulae (2.3.5). This relation is given by γ

γ

Vα (a)Aβα (a) = δβ ;

(2.3.12)

so that the matrix (Vβα ) is the inverse matrix to (Aβα ): Equations (2.3.12) follow directly from Eq. (2.3.11) upon setting b = 0: By virtue of (2.3.12), equations (2.3.11)

2.3 Local Lie group

67

can be rewritten in the equivalent form

∂ϕα γ = Aαγ (ϕ )Vβ (b); ∂ bβ

ϕ α (a; 0) = aα :

(2.3.13)

Let us prove that the introduced multiplication is associative. If we set u = ϕ (a; b);

v = ϕ (b; c);

ω = ϕ (u; c);

ω¯ = ϕ (a; v);

we will have to prove only the equality w = w: ¯ Turning to coordinates and using Eqs. (2.3.13) and (2.3.12), one obtains

∂ wα γ = Aαγ (w)Vβ (c); ∂ cβ

wα jc=0 = uα ;

∂ w¯ α ∂ ϕ α (a; v) ∂ vσ γ γ = = Aατ (w)V ¯ στ (v)Aσγ (v)Vβ (c) = Aαγ (w)V ¯ β (c); β ∂ vσ ∂c ∂ cβ w¯ α jc=0 = ϕ α (a; b) = uα : One can see that wα and w¯ α satisfy one and the same system of differential equations of the form (2.3.13) and the same initial conditions. By virtue of the uniqueness of the solution it follows that wα = w¯ α : Further, it follows from ϕ (a; 0) = a that the point 0 is the right unit. Letting a = 0 in Eqs. (2.3.13), one can see that the solution is ϕ α = bα ; so that ϕ (0; b) = b; i.e. the point 0 is the left unit as well. Finally the system of equations ϕ α (a; b) = 0 (α = 1; : : : ; r) determines the functions bα = bα (a) in a vicinity of the point 0: Setting (a 1 )α = bα (a) one obtains the inverse a 1 to the element a: To complete the proof one has only to verify that the functions α ∂ ϕ (a; b) α V β (b) = ∂ bβ a=b 1 coincide with the given functions Vβα (b): Setting a = b into account the equations

γ

γ

Vβ (b) = δα

∂ ϕ α ∂ bβ a=b

= 1

in Eqs. (2.3.11) and taking

Vβα (0) = δβα ;

ϕ (b 1 ; b) = 0; one obtains

1

∂ ϕ γ ∂ bβ a=b

γ

1

= V β (b):

Theorem 2.8 is proved. In order to proceed we have to tackle the problem of solvability of systems of the form (2.3.13). In general, let us discuss the problem on determination of functions ui = ui (x) = ui (x1 ; : : : ; xr ) (i = 1; : : : ; m) by solving the system of the form

68


∂ ui = fαi (x; u); ∂ xα

ui (x0 ) = ui0

(i = 1; : : : ; m; α = 1; : : : ; r):

(2.3.14)

Definition 2.9. The system (2.3.14) is said to be totally integrable if it has a solution for any initial data x0 ; u0 : Lemma 2.1. In order for the system (2.3.14) to be totally integrable, it is necessary and sufficient that the equations i i ∂ fαi ∂ fαi j ∂ fβ ∂ fβ j + f = + fα ∂ xβ ∂ u j β ∂ xα ∂ u j

(2.3.15)

hold identically with respect to the independent variables x; u: Proof. Necessity. Calculating the derivative

∂ 2 ui ∂ xα ∂ xβ by two ways, one arrives at Eqs. (2.3.15) on the solution and, in particular, at the initial point x0 ; u0 : Since the point x0 ; u0 is arbitrary, equations (2.3.15) are identities with respect to the independent variables x; u: Sufficiency. Let us determine the functions vi = vi (t; e) as the solution of the system of ordinary differential equations ( e is a constant vector)

∂ vi = eα fαi (te; v); ∂t

vi (0) = ui0 :

(2.3.16)

For the sake of simplicity we can prove without loss of generality that there exists a solution of the system (2.3.14) with the initial data at the point x0 = 0: We set ui (x) = vi (1; x) and prove that this is a solution of the problem (2.3.14). To this end we note that the following equation is satisfied: vi (t; e) = vi (1;te): It is derived in the same way as the similar property of the function f α (e;t) in the proof of Theorem 2.7. We will verify Eqs. (2.3.14) demonstrating that Rαi (x) =

∂ ui ∂ xα

fαi (x; u) 0:

To this end let us introduce the functions Sα (t) = tRαi (et): Differentiating with respect to t; using Eqs. (2.3.16), the identities (2.3.15) and the definition of Sαi ; one obtains

2.3 Local Lie group

69

∂ fβ j dSαi = eβ S ; dt ∂uj α i

Sαi (0) = 0:

Thus, Sαi (t) satisfy the system of linear homogeneous ordinary differential equations and have the zero initial values. Therefore, Sαi (t) 0 and Lemma 2.1 is proved. Note that the system (2.3.14) always has a unique solution for sufficiently smooth right-hand parts fαi (x; u); independently of conditions (2.3.15).

2.3.5 Second fundamental theorem of Lie The following theorem has the same relation to the system (2.3.13) as Lemma 2.1 to the system (2.3.14). Theorem 2.9. The system (2.3.13) is completely integrable if and only if the functions Vβα (b) satisfy the system of equations

∂ Vβα ∂ bγ

∂ Vγα ∂ bβ

= Cσατ Vβσ Vγτ ;

Vβα (0) = δβα ;

(2.3.17)

where Cβαγ (α ; β ; γ = 1; : : : ; r) are some constants. Proof. If the system (2.3.13) is completely integrable, then according to Lemma 2.1, the right-hand sides of Eqs. (2.3.13) should satisfy equations of the form (2.3.15). Due to the special form of these right-hand sides, one can reduce the corresponding Eqs. (2.3.15) to the form " α # " α # ∂ Vβ (ϕ ) ∂ Vγα (ϕ ) ∂ Vβ (b) ∂ Vγα (b) β γ β γ Aσ (ϕ )Aτ (ϕ ) = Aσ (b)Aτ (b) ; ∂ ϕγ ∂ bγ ∂ ϕβ ∂ bβ upon simple transformations where only the relations (2.3.12) are used. Since the variables b and ϕ = ϕ (a; b) are independent, the resulting equality can be an identity with respect to b; ϕ only if the common value of both expressions is a constant " α # ∂ Vβ (b) ∂ Vγα (b) β γ Aσ (b)Aτ (b) = Cσατ = const: ∂ bγ ∂ bβ Multiplying both sides by Vβσ Vγτ one obtains Eqs. (2.3.17) which are thereby equivalent to the conditions of complete integrability of the system (2.3.13). Theorem 2.9 is proved. Definition 2.10. Equations (2.3.17) are referred to as the Maurer-Cartan equations. The constants Cβαγ in Eqs. (2.3.17) are called the structure constants of the local Lie group Gr :

70


Note that the Maurer-Cartan equations can be transformed into equivalent equations for the auxiliary functions Aβα (a): Namely, multiplying Eqs. (2.3.17) by β γ

Aαv Aπ Aρ and using the identities (2.3.12) one obtains Aσβ

∂ Aαγ ∂ aσ

Aσγ

∂ Aβα ∂ aσ

= Cβσγ Aασ :

(2.3.18)

2.3.6 Properties of canonical coordinate systems of the first kind In what follows one can see that the local Lie group Gr is completely determined by the set of its structure constants (the third Lie theorem). First, let us deduce two properties of a canonical coordinate system of the first kind. Lemma 2.2. The necessary and sufficient condition for the system of coordinates ∑a to be canonical of the first kind is that the functions Vβα satisfy the relations bβ Vβα (b) = bα

(α = 1; : : : ; r):

(2.3.19)

Proof. Necessity. Let ∑a be canonical of the first kind. Then the curve aα = eα t (α = 1; : : : ; r) is a subgroup G1 for any vector e (e1 ; : : : ; er ) and hence satisfies Eqs. (2.3.9). Substituting aα = eα t there, one obtains eα = Aβα (et)eβ : When t = 1 and e = b the above equation yields bα = Aβα (b)eβ : γ

Whence, multiplying by Vα (b) and applying Eq. (2.3.12) we get Eqs. (2.3.19). Sufficiency. Equations (2.3.19) entail that aα = Aβα (a)aβ : Setting aα = eα t one obtains

aα = Aβα (a)eβ t

or

daα = Aβα (a)eβ : dt By virtue of Theorem 2.6, it follows that the curve aα = eα t is a subgroup G1 : Therefore, according to Definition 2.8, the system of coordinates ∑a is canonical of the first kind. The lemma is proved. Lemma 2.3. In a canonical coordinate system of the first kind, the functions Vβα (b) are determined uniquely by the structure constants.

2.3 Local Lie group

71

Proof. Let

Wβα (t) = tVβα (te):

It is sufficient to prove that Wβα (t) is determined uniquely by Cβαγ ; because Vβα (e) = Wβα (1): Note that equations (2.3.19) imply the relations β

bα

∂ β Vβα ∂ bγ

= δγα

Vγα :

Therefore, by virtue of Eqs. (2.3.17), dWβα dt

= Vβα + teγ = Vβα + δβα

∂ Vβα (te) ∂ bγ

= Vβα + teγ

∂ Vγα ∂ bβ

+ teγ Cσατ Vβσ Vγτ

Vβα +Cσατ teτ Vβσ = δβα +Cσατ eτ Wβσ :

Thus, the functions Wβα satisfy the system of equations dWβα dt

= δβα +Cσατ eτ Wβσ ;

Wβα (0) = 0:

(2.3.20)

Since the solution of the system (2.3.20) is unique, it is completely determined if the set of the structure constants Cβαγ is given. Thus, Lemma 2.3 is proved.

2.3.7 Third fundamental theorem of Lie Theorem 2.10. The structure constants Cβαγ of the local Lie group Gr satisfy the Jacobi relations α Cβαγ = Cγβ ;

σ σ τ Cαβ Cστ γ +Cβσγ Cστ α +Cγα Cσ β = 0:

(2.3.21)

Conversely, given any set of constants Cβαγ satisfying the relations (2.3.21), there exists a local Lie group Gr whose structure constants coincide with the given Cβαγ : Proof. We set b = 0 in Eqs. (2.3.17). Since Vβα (0) = δβα ; one obtains Cβαγ =

∂ Vβα ∂ bγ

! ; ∂ bβ b=0

∂ Vγα

(2.3.22)

whence the first Jacobi relation (2.3.21) follows. Further, applying the operation ∂ =∂ bε to Eqs. (2.3.17) and using these equations once more one obtains

72


∂ 2Vβα

∂ bγ ∂ bε

∂ 2Vγα ∂ bβ ∂ bε

=

Cσατ

∂ Vβσ

Vτ ∂ bε γ

= Cσατ Vγτ

∂ Vβσ ∂ bε

+Cσατ Vβσ Cσατ Vβτ

∂ Vετ µ +Cλτ µ Vγλ Vε ∂ bγ

∂ Vεσ µ +Cσατ Cλτ µ Vβσ Vγλ Vε : ∂ bγ

Whence, setting b = 0 one obtains the relation σ α Cεγ Cσ β = ωεγβ

where

ωεγβ =

ωβ εγ ;

∂ 2Vβα

σ α ∂ Vε +C σ β ∂ bγ ∂ bε ∂ bγ

!

:

b=0

Making the circular permutation of indices ε ; γ ; β twice in the above relation, one obtains two more similar relations, namely Cβσε Cσαγ = ωβ εγ

ωγβ ε ;

σ α Cγβ Cσ ε = ωγβ ε

ωεγβ :

Summing up the three obtained relations, one arrives at the second Jacobi relation (2.3.21). Conversely, consider given Cβαγ satisfying Eqs. (2.3.21). Then, one can construct the system of Eqs. (1.4.8), whose solution is obviously unique and depends on the choice of the vector e: Let the solution be Wβα = Wβα (t; e): Let us set

Vβα (b) = Wβα (1; b)

and demonstrate that these Vβα together with the given Cβαγ satisfy the Maurer-Cartan equations (2.3.17). To this end, we introduce the functions hβα γ (t) =

∂ Wβα ∂ eγ

∂ Wγα ∂ eβ

Cσατ Wβσ Wγτ :

It is clear that hβα γ (0) = 0: Differentiating the above functions with respect to t and using Eqs. (2.3.20) and the relations (2.3.21), one obtains dhβα γ dt

= Cσατ eτ hσβ γ :

Thus, the functions hβα γ satisfy the system of linear homogeneous differential equations with the zero initial conditions. Therefore, hβα γ (t) 0;

2.3 Local Lie group

and, in particular,

73

hβα γ (1) 0:

The latter are Eqs. (2.3.17) for the functions Vβα (e): Letting e = 0 in Eqs. (2.3.20) one obtains the equations dWβα = δβα dt whose solution is Wβα (t; 0) = δβα t: Hence

Vβα (0) = δβα :

Using Theorem 2.9 we conclude that the system of Eqs. (2.3.13) is completely integrable with the obtained Vβα (b): According to Theorem 2.8, there exists a local Lie group Gr where these Vβα are auxiliary functions. The given Cβαγ are the structure constants for the constructed Gr because they are expressed through Vβα by the formulae (2.3.22) following from Eqs. (2.3.17). This completes the proof of Theorem 2.10.

2.3.8 Lie algebra of a local Lie group The proof of Theorems 2.8—2.10 suggests also an algorithm for constructing a group Gr by structure constants. It consists in integration of the system (2.3.20), construction of the functions Vβα and in solving the system (2.3.11). The algorithm shows that in order to restore Gr by its structure constants one has to integrate ordinary differential equations at most. The algorithm has two more important peculiarities. First, it furnishes the group Gr in a definite system of coordinates ∑a : This ∑a appears to be a canonical system of the first kind. Lemma 2.3 shows that in order to prove this fact it suffices to verify that the resulting Vβα (b) satisfy the relations (2.3.19). Indeed, if we set uα (t) = eβ Wβα (t; e)

teα ;

then using Eqs. (2.3.20) we readily obtain the equations duα = Cσατ eτ uσ ; dt

uα (0) = 0;

Uniqueness of the solution uα (t) of the above initial value problems gives the equation uα (t) 0; which provides that (2.3.19) holds by construction of the function Vβα : The other peculiarity is that the resulting coordinates ∑a are analytic. In other words, the multiplication law ϕ (a; b) is a holomorphic function of the variables

74


aα ; bβ at the point a = b = 0: Indeed, since the functions Wβα (t; e) solve the (nonhomogeneous) system of linear equations (2.3.20) with constant (with respect to t) coefficients, they are determined and analytic with respect to t when ∞ < t < +∞: Further, the right-hand sides in Eqs. (2.3.20) are analytic functions of the coordinates e1 ; : : : ; er of the vector e: Therefore, the solution is holomorphic with respect to e; at least in the vicinity of the point e = 0: Due to the equation Vβα (b) = Wβα (1; b) the functions Vβα (b); and hence Aβα (b) are holomorphic. Finally, the solution of the completely integrable system (2.3.11) with the holomorphic right-hand sides is holomorphic, which was to be proved. Thus, analytic coordinates exist in any local Lie group Gr : In fact, the three fundamental theorems of Lie show that investigation of groups Gr is reduced to investigation of tensors of the third order Cβαγ : Of course, this reduction takes place with the accuracy to the local isomorphism of groups Gr : Two local Lie groups Gr and Gr are said to be locally isomorphic if elements of some vicinities of the unit element in Gr and Gr can be set into one-to-one correspondence ga $ g¯a so that g0 $ g¯0 ; ga gb $ g¯a g¯b ; ga 1 $ g¯a 1 : Obviously, it is necessary and sufficient for the local isomorphism Gr and Gr that there exist such systems of coordinates ∑a and ∑a in Gr and Gr ; respectively, where the multiplication laws for elements of Gr and Gr coincide:

ϕ (a; b) = ϕ¯ (a; b): The criterion of local isomorphism can be also formulated in terms of structure α α constants Cβαγ and Cβ γ : Namely if Cβ γ = Cβαγ ; then Gr and Gr are locally isomorphic. Conversely, in locally isomorphic Gr and Gr one can choose systems of coordinates α so that the structure constants calculated in them are equal to each other, Cβαγ = Cβ γ : Comparing the property of a local Lie group Gr to be determined by the set of its structure constants with the corresponding property of Lie algebras (Theorem 2.1), as well as the properties of these constants, one arrives at the following important notion. Definition 2.11. A Lie algebra Lr is called the Lie algebra of a local Lie group Gr if there exist a basis in Lr and a system of coordinates in Gr such that the structure constants Cβαγ and Cβαγ of the algebra Lr and of the group Gr ; respectively, coincide, i.e., Cβαγ = Cβαγ : Since any set of constants Cβαγ satisfying Eqs. (2.3.21) determines a Lie algebra Lr and a local Lie group Gr ; Definition 2.11 establishes a one-to-one correspondence (up to isomorphism) between Lie algebras Lr and local Lie groups Gr : Let us discuss how to realize the Lie algebra Lr of a given local Lie group Gr as a Lie algebra of operators. Consider a space E r of points a with determined auxiliary

2.4 Subgroup, normal subgroup and factor group

75

functions Aβα (a) of the given Gr and construct the operators Xα = Aσα (a)

∂ ∂ aσ

(α = 1; : : : ; r):

(2.3.23)

Computing the commutators of the operators (2.3.23) according to Definition 1.7: ! σ ∂ Aβσ ∂ τ τ ∂ Aα [Xα ; Xβ ] = Aα τ Aβ τ ∂a ∂a ∂ aσ and invoking the Maurer-Cartan equations in the form (2.3.18), one obtains σ (Xα ; Xβ ) = Cαβ Xσ ; σ are the structure constants of the given G : The resulting relations where Cαβ r demonstrate that the linear span fX g of the operators (2.3.23),

X = eα Xα ; is a Lie algebra of operators, namely the algebra Lr ; whose structure constants are equal to the numbers Cβαγ in the basis (2.3.23). The linear independence of the operators (2.3.23) follows from the fact that they take the form Xα =

∂ ∂ aα

at the point a = 0: Thus, the Lie algebra of operators Lr spanned by the operators (2.3.23) is the Lie algebra of the given Gr : The operators (2.3.23) are sometimes termed the shift operators on the group Gr :

2.4 Subgroup, normal subgroup and factor group In this section we will discuss details of the correspondence between Lie algebras Lr and local Lie groups Gr established by Definition 2.11.

2.4.1 Lemma on commutator It is convenient to use another realization of the Lie algebra Lr of the group Gr : Namely we consider the Lie algebra Lr of the directing vectors e of subgroups G 1 of the group Gr : This algebra Lr is the linear space of vectors e (e1 ; : : : ; er ); where the operation of commutation is determined by means of the structure constants Cβαγ of the group Gr by the formulae

76

2 Lie algebras and local Lie groups β γ

[e1 ; e2 ]α = Cβαγ e1 e2

(α = 1; : : : ; r):

(2.4.1)

One can readily verify that the introduced operation of commutation satisfies all the axioms of Definition 1.7 due to the properties (2.3.21) of the structure constants. Let us verify that structure constants of the derived Lr are equal to constants Cβαγ in the β

β

basis feα g determined as follows: the vector eα has the coordinates eα = δα : This follows from Eq. (2.2.2), written for [eβ ; eγ ]; namely [eβ ; eγ ]α = Cσατ eβσ eτγ = Cσατ δβσ δγτ = Cβαγ = Cβσγ δσα = Cβσγ eασ ; whence

[eβ ; eγ ] = Cβσγ eσ ;

which was to be proved. Let us make another preliminary observation. The formulae (2.3.4) entail that ∂ ϕ α Aαγ (a) = = δγα + rβαγ aβ + O(jaj2 ); ∂ bγ b=0 whence

∂ Aαγ ∂ aβ

a=0

= rβαγ :

(2.4.2)

Therefore, equations (2.3.18) provide Cβαγ = rβαγ

α rγβ

(2.4.3)

when a = 0: Consider in Gr two subgroups G1 ; g1 (t) and g2 (t) with the directing vectors e1 and e2 ; respectively. Let us construct a new curve in Gr : p p p p g(t) (2.4.4) ˆ = g1 ( t)g2 ( t)g1 1 ( t)g2 1 ( t); where t 0: Lemma 2.4. The curve g(t); ˆ determined by Eq. (2.4.4) has the directing vector e = [e1 ; e2 ]; where the commutator is defined by Eqs. (2.4.1). Proof. Equations (2.3.9) and (2.4.2) entail that 1 aα (t) = eα t + rβαγ eβ eγ t 2 + O(t 3) 2 along the subgroup Gp 1 with the p directing vector e: Therefore, if b(t) are the coordinates of the curve g1 ( t)g2 ( t); then p p p p 3 β p γ p bα (t) = ϕ α (a1 ( t); a2 ( t)) = aα1 ( t) + aα2 ( t) + rβαγ a1 ( t)a2 ( t) + O(t 2 ) p 1 3 β γ β γ β γ = (eα1 + eα2 ) t + rβαγ (e1 e1 + e2 e2 )t + rβαγ e1 e2t + O(t 2 ): 2


77

Likewise, p 1 3 β γ β γ β γ (eα1 + eα2 ) t + rβαγ (e1 ; e1 + e2 e2 )t + rβαγ e1 e2t + O(t 2 ) 2 p p for the coordinates c(t) of the curve g1 1 ( t)g2 1 ( t): Consequently, the coordinates a(t) ˆ of the curve g(t) ˆ from (2.4.4) that are equal to ϕ (b(t)c(t)) are given by cα (t) =

aˆα (t) = ϕ α (b(t); c(t)) = bα + cα + rβαγ bβ cγ + o(jbj2 + jcj2 ) β γ

β γ

β γ

= [rβαγ (e1 e1 + e2 e2 ) + 2rβαγ e1 e2 = (rβαγ

β γ

β

β

γ

γ

rβαγ (e1 + e2 )(e1 + e2 )]t + O(t 2 ) β γ

3

α rγβ )e1 e2t + O(t 2 ) = Cβαγ e1 e2t + O(t 2 ): 3

3

The relations (2.4.3) are used in the latter transition. Whence, differentiating with respect to t and setting t = 0; one obtains Eqs. (2.4.1) for the vector d aˆ e= dt t=0: The lemma is proved. Lemma 2.4 is termed a lemma on commutator.

2.4.2 Subgroup Definition 2.12. A subset Gs Gr (0 s r) is called a subgroup of the local Lie group Gr if Gs is closed with respect to the multiplication in Gr and if the coordinates of elements ga 2 Gs are given by means of thrice continuously differentiable functions aα = f α (a¯1 ; : : : ; a¯s ) (α = 1; : : : ; r) (2.4.5) for which f α ja=0 = 0 and the rank of the matrix ¯ α ∂f ∂ a¯β a=0 ¯ equals to s: Let Gr be a local Lie group and let Lr be its Lie algebra realized as a Lie algebra of the directing vectors of subgroups G1 according to Eqs. (2.4.1). Theorem 2.11. If Gs is a subgroup of Gr ; then the set feg of the directing vectors e for various curves passing in Gs generates a subalgebra Ls Lr : Conversely, if Ls is a subalgebra in Lr ; then the set fg(t)g of elements of various subgroups of G1 with directing vectors from Ls generates a subgroup Gs Gr : Proof. Let e1 and e2 be the directing vectors of the curves g1 (t) and g2 (t) Gs ; respectively. Then the curve g1 (t)g2 (t) has the directing vector e1 + e2 ; and the

78


curve g1 (α t) has the vector α e1 : Therefore, feg is a linear space, namely a subspace Ls Lr : Further, the curve g(t) ˆ Gs constructed according to Eq. (2.4.4) has the directing vector e = [e1 ; e2 ]: Therefore, for any e1 ; e2 2 Ls one also has [e1 ; e2 ] 2 Ls ; i.e. Ls is a subalgebra in Lr according to Definition 2.3. In order to prove the inverse, we consider Gr in canonical coordinates of the first kind. Let us choose a basis in Lr so that the basis vectors eα 0 (α 0 = 1; : : : ; s) provide a basis in the subalgebra Ls : Let us agree, that the Greek indices with one prime α 0 ; β 0 ; σ 0 ; : : : ; and with two primes α 00 ; β 00 ; σ 00 ; : : : ; run the values 1; : : : ; s; and s + 1; : : :; r; respectively. Then, the coordinates of the vectors e 2 Ls are such, 00 that eα = 0: Since Ls is a subalgebra in Lr ; the equations Cβαγ = [eβ ; eγ ]α entail that

Cβα0γ 0 = [eβ 0 ; eγ 0 ]α = 0: 00

00

(2.4.6)

Let Gs be the set fg(t)g of elements of subgroups G1 with the directing vectors from Ls : Let us introduce into Gr a canonical system of coordinates of the first kind ∑a resulting from construction of Gr by using structure constants described in x2.3. Note that the elements of Gs in the coordinate system ∑a are characterized by 00 the equation aα = 0: Indeed, since ∑a is canonical, the elements of G1 have the 00 00 coordinates aα = eα t; whence aα = eα t = 0 if e 2 Ls : Now let us demonstrate that by virtue of Eq. (2.4.6) one has Vβα0 (b0 ) 0; 00

b0 = (b1 ; : : : ; bs ; 0; : : : ; 0)

(2.4.7)

in ∑a : To this end, take a vector e 2 Ls and single out in the system of Eqs. (2.3.20) a subsystem with α = α 00 ; β = β 0 which has the form dWβα0

00

= Cσα 00r 0 er Wβσ0 ; 00

dt

0

00

Wβα0 (0) = 0 00

(see Eq. (2.4.6)). Due to homogeneity of the equations the solution of the above 00 subsystem is Wβα0 (t) = 0; which entails Eqs. (2.4.7). In order to prove that Gs is closed with respect to multiplication in Gr ; it is 00 sufficient to verify that ϕ α (a0 ; b0 ) = 0: This follows from the fact that, by virtue of Eqs. (2.4.7) the Lie equations (2.3.11) for γ = γ 00 ; β = β 0 and a = a0 ; b = b0 have the form 00 ∂ ϕ α (a0 ; b0 ) γ 00 Vα (ϕ ) = 0; ϕ α (a0 ; 0) = 0: 0 β ∂b Therefore, we satisfy the complete system (2.3.11) by setting ϕ α (a0 ; b0 ) = 0: Indeed, the previous equations in this case are satisfied identically by virtue of Eqs. (2.4.7), whereas the remaining equations become a completely integrable system 00

γ0

Vα 0 (ϕ 0 )

∂ ϕ α (a0 ; b0 ) γ0 = Vβ 0 (b0 ); 0 β ∂b 0

ϕ α (a0 ; 0) = aα : 0

0


79

The latter statement follows from the fact that, by virtue of Eq. (2.4.6), Cβα0γ 0 provide a system of structure constants for Ls and hence, satisfy the Jacobi relations (2.3.21). Theorem 2.11 is proved. 0

2.4.3 Normal subgroup Definition 2.13. A subgroup Gs Gr is called a normal subgroup of the group Gr if gGs g 1 = Gs for any g 2 Gr : Theorem 2.12. If Gs is a normal subgroup of Gr ; then the corresponding subalgebra Ls is an ideal in Lr and vice versa. Proof. If Gs is a normal subgroup of Gr ; then hgh 1g

1

2 Gs

for any g 2 Gs and any h 2 Gr : Therefore, Lemma 2.4 entails that if c¯ 2 Ls and e 2 Lr ; then one also has [e; ¯ e] 2 Ls : According to Definition 2.3 it means that Ls is an ideal in Lr : Conversely, let Ls be an ideal in Lr : By virtue of Theorem 2.11 it corresponds to a subgroup Gs Gr : Let us prove that g(t) ˆ = ga g(t)ga 1 2 Gs for any one-parameter subgroup g(t) 2 Gs and any ga 2 Gr : Since the curve g(t) ˆ is also a subgroup G1 ; it is sufficient to prove that its directing vector eˆ belongs to Ls : Introducing in Gr a system of coordinates ∑a as we did in the proof of Theorem 2.11 and assuming that g(t) has a directing vector e; one finds out that g(t) ˆ is written in the coordinate from as follows: eˆα t = ϕ α (ϕ (a; et); a): Whence, differentiating with respect to t and letting t = 0; one obtains eˆα = Vσα ( a)Aσγ (a)eγ :

(2.4.8)

Further, choosing the basis in Lr in the same way as in the proof of Theorem 2.11 and invoking that Ls is an ideal, one obtains Cβαγ 0 = [eβ ; eγ 0 ]α = 0 00

00

instead of Eq. (2.4.6). Proceeding as in the proof of Theorem 2.11 one obtains the equations 00 Vβα0 (b) = 0 (2.4.9)

80


instead of Eq. (2.4.9). Then, the Lie equations (2.3.11) provide ! 00 ∂ϕα γ 00 Vα 00 (ϕ ) =0 0 ∂ bβ 00

when γ = γ ; β = β 0 : Whence

∂ϕα 0 = 0; ∂ bβ which means that in the given case one also has 00

Aβα 0 (a) = 0: 00

(2.4.10)

By virtue of Eqs. (2.4.9) and (2.4.10), the formulae (2.4.8) written for e 2 Ls provide eˆα = Vσα ( a)Aγσ0 eγ = Vσα00 ( a)Aγσ0 (a)eγ = 0 00

00

0

00

00

0

i.e., eˆ 2 Ls ; which was to be proved. The center of the group Gr is the set of the elements from Gr that are permutable with any element of Gr : Note that the center of the group Gr corresponds to the center Z of the Lie algebra Lr of the group Gr : We leave it as an exercise for the reader to prove the theorem.

2.4.4 Factor group Let Gs be a normal subgroup in Gr : One can introduce the equivalence relation in Gr by the following rule: gs gb

if

gc gb 1 2 Gs :

Accordingly, Gr splits into classes of equivalent elements. Let a system of coordinates ∑a in Gr be chosen as in the proof of Theorem 2.11. Lemma 2.5. The equivalence relation gc gb is identical with the equations c00 = b00 : Proof. We have already shown that

∂ ϕ α (a; b) =0 0 ∂ bβ 00

while proving Theorem 2.12. Since ∑a is canonical system of the first kind, one has

ϕ α (a; b) = ϕ α ( b; a);


whence

81

∂ ϕ α (a; b) =0 0 ∂ aβ 00

as well. Hence, the equations

ϕ α (a; b) = ϕ α (a00 ; b00 ) 00

00

hold. If c00 = b00 then

ϕ α (c; b) = ϕ α (b00 ; b00 ) = 0; 00

00

whence gc gb 1 2 Gs : Conversely, let gc = ga gb ; where ga 2 Gs : Then a00 = 0 and cα = ϕ α (a0 ; b) = ϕ α (0; b00 ) = bα ; 00

00

00

00

i.e. c00 = b00 ; which was to be proved. Let h(g) be the class of equivalent elements containing g: One can determine an operation of multiplication of classes by the rule h(g1 )h(g2 ) = h(g1 ; g2 ): One can easily verify that this operation does not depend on the choice of “representatives” g1 ; g2 of the classes h(g1 ); h(g2 ) and that it satisfies the group axioms. 00 Let us introduce coordinates into the set of classes by taking the numbers aα as coordinates of the class h(ga ): Lemma 2.5 demonstrates that the correspondence h(ga ) $ a00 is one-to-one. The multiplication law for classes in these coordinates is given by the functions ϕ 00 (a00 ; b00 ) satisfying the smoothness requirement for the multiplication law in the local Lie group. Hence, the set of classes h(ga ) is a local Lie group. Definition 2.14. The set of classes of equivalent elements of the group Gr generated by its normal subgroup Gs is called the factor group of the group Gr by its normal subgroup Gs and is denoted by the symbol Gr =Gs : Definition 2.4 given in Lie algebras is an equivalent to Definition 2.14. If Gs is a normal subgroup in Gr and if Ls is a corresponding ideal in the Lie algebra Lr of the group Gr ; then one can construct the factor group Gr =Gs and the quotient algebra Lr =Ls : Theorem 2.13. The quotient algebra Lr =Ls is the Lie algebra of the factor group Gr =Gs : Proof. The quotient algebra Lr =Ls is the set of vectors representing the classes of equivalent vectors e 2 Lr with respect to the ideal Ls : e e¯ if

e

e¯ 2 Ls :

82


Choosing the basis in Lr in the same way as in the proof of Theorem 2.11, one obtains the equivalence criterion in the coordinate form: e00 = e¯ 00 or, to be more exact, 00 00 eα = e¯α : Hence, Lr =Ls can be considered as the set of vectors of the form e00 : By virtue of Lemma 2.5 and construction of the factor group Gr =Gs ; the directing vectors of oneparameter subgroups from Gr =Gs also have the from e00 : Therefore, the operation of commutation (2.4.1) in the Lie algebra of the group Gr =Gs is given by β 00 γ 00

[e1 ; e2 ]α = Cβα00γ 00 e1 e2 : 00

00

The proof is completed by the observation that Cβα00γ 00 are structure constants of the quotient algebra Lr =Ls : 00

2.5 Inner automorphisms of a group and of its Lie algebra 2.5.1 Inner automorphism Definition 2.15. An inner automorphism of a group Gr is a mapping of Gr onto itself given by the formula g ! h 1 gh; where h 2 Gr : If h = ga ; then the corresponding inner automorphism is denoted by the symbol Γa ; so that Γa (g) = ga 1 gga : The operation of multiplication in the set GA of inner automorphisms of the group Gr is defined by ΓaΓb = Γϕ (a;b): (2.5.1) It means that the automorphism ΓaΓb acts upon elements g 2 Gr as follows:

ΓaΓb (g) = (ga gb ) 1 g(ga gb ) = gb 1 (ga 1 gga )gb = Γb (Γa (g)): It is manifest that the automorphisms Γa and Γb coincide if and only if the element ga gb 1 belongs to the center Z of the group Gr : Therefore, there is a one-to-one correspondence between inner automorphisms and elements of the factor group Gr =Z. Moreover, the multiplication law in GA will be the same as in Gr =Z due to (2.5.1). It follows that the set GA is a local Lie group with the multiplication (2.5.1) and that the group GA is isomorphic to the factor group Gr =Z:

2.5 Inner automorphisms of a group and of its Lie algebra

83

Let us demonstrate that the group GA can be represented as a group of linear transformations in the set feg of the directing vectors of subgroups G1 of the group Gr : With this purpose in mind, let us introduce a canonical system of coordinates ∑a of the first kind in Gr and consider a subgroup G1 with the directing vector e : g(t) = get : As it was mentioned in the proof of Theorem 2.12, the curve g(t) ˆ = Γa g(t) = ga 1 g(t)ga is also a subgroup G1 with the directing vector (2.4.8). The formula (2.4.8) determines a linear transformation la in Lr given by the matrix la : lβα (a) = Vσα (a)Aβσ ( a):

(2.5.2)

2.5.2 Lie algebra of GA and adjoint algebra of Lr It is evident that the automorphisms Γa and Γb are identical if and only if the transformations la and lb are identical and that the product ΓaΓb corresponds to the product la lb = lϕ (a;b) ; given by the matrix lβα (ϕ (a; b)) = lσα (b)lβσ (a): It follows that the set L of matrices lβα (a) with the variable a is a local Lie group that is isomorphic to the group of inner automorphisms GA : Thus, we have the following statement. Theorem 2.14. The Lie algebra of the group GA is isomorphic to the adjoint algebra of the Lie algebra Lr : Proof. Let us consider an automorphism Γa along a one-parameter subgroup a = ut with the directing vector u: Then, Γut is also a subgroup G1 in the group GA ; and hence the matrix lβα (ut) is a one-parameter subgroup of matrices in the group L : Therefore lβα (u(t + s)) = lσα (us)lβσ (ut): Differentiating with respect to s and letting s = 0 one obtains dlβα (ut) dt

∂ lσα (a) = uγ l σ (ut): ∂ aγ a=0 β

The constant is calculated by using (2.5.2) and the equation Vσα (a)Aσβ (a) = δβα :

84


α ∂ lσα (a) ∂ Vτ (a) τ ∂ Aτσ ( a) α = A ( a) +V (a) σ τ ∂ aγ a=0 ∂ aγ ∂ aγ a=0 " β Vτ (a)Vvα (a)Aτσ (

= =

=

a)

∂ Aασ (a) ∂ Aασ ( a) + ∂ aγ ∂ aγ ∂ Aα (a) 2 σ γ = ∂a a=0

∂ Aβv (a) ∂ aγ

∂ Aτ ( +Vτα (a) σ γ

# a)

∂a

a=0

a=0

α 2rγσ :

The latter equality follows from Eq. (2.4.2). The equations a

1

=

a and

ϕ α (a; a) = rβαγ aβ aγ + o(jaj2) = 0; i.e. rβαγ aβ aγ = 0; yield that the constants rβαγ are skew-symmetric with respect to the lower indices in the canonical system of coordinates. Therefore, equation (2.4.3) provides α 2rγσ = Cσαγ : Thus, the matrix (2.5.2) satisfies the system of equations dlβα (u;t) dt

= Cσαγ uγ lβσ (ut);

lβα (0) = δβα

(2.5.3)

along the subgroup G1 : a = ut: We have already constructed inner automorphisms of the Lie algebra Lr given by Eqs. (2.2.8) with the matrices fβα (t) in x2.2. Substitution of expressions (2.2.8) into equations of the subgroup G1 ; (2.2.7) yields the following system of equations for these matrices: d fβα (t) = Cσαγ eγ fβσ (t); fβα (0) = δβα ; dt coinciding with (2.5.3) when u = e: Hence, lβα (ut) = fβα (t) when u = e and one obtains that the group Z is isomorphic to the group of inner automorphisms of the Lie algebra Lr : Recall that isomorphic groups have isomorphic Lie algebras. Furthermore, a Lie algebra of a group of automorphisms of the Lie algebra Lr is a Lie algebra fEg of operators E = eβ Eβ ; generated by the operators (2.2.6). The latter Lie algebra is isomorphic to the adjoint algebra LD of the Lie algebra Lr : This completes the proof of Theorem 2.14.

2.6 Local Lie group of transformations

85

2.6 Local Lie group of transformations 2.6.1 Introduction We return now to transformations of the space E N (x) discussed in Chapter 1 and assume that a family of such transformations fTa g is given and depends on r parameters a (a1 ; : : : ; ar ) : Ta : x0i = f i (x; a) = f i (x1 ; : : : ; xN ; a1 ; : : : ; ar ):

(2.6.1)

Definition 2.16. The family fTa g is called a local Lie group GNr of point transformations of the space E N if fTa g is a local Lie group Gr with an usual multiplication of transformations and if the functions f i (x; a) in (2.6.1) are twice continuously differentiable with respect to the variables x; a: If ϕ (a; b) is the multiplication law of elements in Gr ; then multiplication of transformations in GNr is carried out by the formulae Tb Ta = Tϕ (a;b) : f i ( f (x; a); b) = f i (x; ϕ (a; b)):

(2.6.2)

Since the group GNr is a group Gr given by the multiplication rule ϕ (a; b); all notions and facts concerning Gr refer to GNr as well. However, due to the special form of the multiplication law in GNr some new notions and facts arise. Let us introduce the auxiliary functions ∂ f i (x; a) ξαi (x) = (i = 1; : : : ; N; α = 1; : : : ; r) (2.6.3) ∂ aα a=0 and construct the following operators with these functions: Xα = ξαi (x)

∂ ∂ xi

(α = 1; : : : ; r):

(2.6.4)

The operators Xα are referred to as basis operators of the group GNr :

2.6.2 Lie’s first theorem Theorem 2.15. The functions x0i = f i (x; a) satisfy the system of equations

∂ x0i = ξσi (x0 )Vασ (a); ∂ aα

x0i ja=0 = xi ;

(2.6.5) β

where Vβα (a) are auxiliary functions of the group Gr and the linear operators Xα (2.6.4) are linearly independent. Conversely, let us suppose that a local Lie group

86


Gr with auxiliary functions Vβα (a) and linearly independent operators Xα (2.6.4) are given. If the system of equations (2.6.5) has a unique solution for any x 2 E N ; then substitution of the solution of the system (2.6.5) into the formulae (2.6.1) determines a local Lie group of transformations GNr isomorphic to the group Gr : Proof. Let ∆ a be a (small) shift of the point a: By virtue of Eqs. (2.6.1) and (2.6.2), the formula of multiplication of transformations Ta+∆ a = (Ta+∆ a Ta 1 )Ta is written in the coordinate form as follows: f i (x; a + ∆ a) = f i (x0 ; ϕ (a

1

; a + ∆ a)):

Making the Taylor expansion of the right-hand and the left-hand sides, using the definition (2.6.3), and comparing the principal parts when ∆ a ! 0; one obtains Eqs. (2.6.5). The initial conditions of (2.6.5) follow directly from Definition 2.16, since a unit element of the group GNr is an identical transformation of E N : In order to prove that the operators (2.6.4) are linearly independent, let us assume that ∑a is a canonical system of the first kind and that eα0 Xα = 0 for some vector e0 ; i.e. eα0 ξαi 0 (i = 1; : : : ; N): One has x0i = f i (x; e0t) along the subgroup G1 with equations aα = eα0 t; and Equations (2.6.5), (2.3.19) yield dx0i ∂ x0i = eα0 α = ξσi (x0 )Vασ (e0t)eα0 = ξσi (x0 )eσ0 = 0: (2.6.6) dt ∂a Equations (2.6.6) show that x0i = xi along the whole G1 ; i.e., all transformations G1 are identical. It is possible only for e0 = 0; which was to be proved. Let us prove the converse statement of Theorem 2.15. Let the functions obtained as a solution of the system (2.6.5) have the form (2.6.1), thus determining the family fTa g of transformations of E N : Let us demonstrate that these Ta satisfy Eqs. (2.6.2). To this end assume that x0 = f (x; a); x00 = f (x0 ; b)

and

y = f (x; ϕ (a; b)):

Using Eqs. (2.6.5) and (2.3.13), as well as the property (2.3.12), one obtains that

∂ x00i = ξσi (x00 )Vασ (b); ∂ bα

x00i jb=0 = x0i ;

∂ yi ∂ yi ∂ ϕ β β = = ξσi (y)Vβσ (ϕ )Aτ (ϕ )Vατ (b) = ξσi (y)Vασ (b); ∂ bα ∂ ϕ β ∂ bα


87

y0 jb=0 = x0i : One can see that x00i and yi satisfy one and the same system of equations (2.6.5) as functions of the point b and the same initial conditions. Uniqueness of the solution guarantees that x00i = yi for any x; a; b; which is Eq. (2.6.2). This proves the group property of the family fTa g and shows that the mapping Gr ! fTa g given by the formula ψ (a) = Ta is at least homomorphic. One has only to demonstrate that ψ is an isomorphism. Indeed, one would have x0i = xi for all a = e0t along the subgroup G1 from the kernel of the homomorphism ψ with the directing vector e0 ; and hence, eσ0 ξσi (x) 0 according to (2.6.6), i.e. eσ0 Xσ = 0: Since the operators (2.6.4) are linearly independent by assumption, the above equation yields e0 = 0; so that the kernel ψ consists of one point a = 0: This proves that ψ is an isomorphism and Theorem 2.15 is proved. Corollary 2.3. The functions x0i = f i (x; et) satisfy the equations dx0i = eα ξαi (x0 ); dt

x0i jt=0 = xi

(2.6.7)

along the subgroup G1 with the directing vector e: In fact, these equations have already been written out in Eqs. (2.6.6). The structure constants Cβαγ of the group Gr are also called the structure constants of the group of transformations GNr :

2.6.3 Lie’s second theorem Theorem 2.16. The linear span of the operators Xα (2.6.4) is a Lie algebra of operγ ators, the structure constants of which coincide with the structure constants Cαβ of the group Gr ; so that σ [Xα ; Xβ ] = Cαβ Xσ : (2.6.8) Conversely, if an r-dimensional Lie algebra of operators with the basis (2.6.4) is given in E N ; then there exists a local Lie group of transformations GNr ; whose basis operators coincide with the given (2.6.4). Proof. The solvability of the system (2.6.5) with any initial conditions guarantees that it is completely integrable. Note that the system has the form (2.3.14). Writing the test (2.3.15) for complete integrability and using the Maurer-Cartan equations (2.3.17), one can verify that the criterion for complete integrability of the system (2.6.5) is given by Eqs. (2.6.8). Note that this discovers the equivalence of complete integrability of Eqs. (2.6.5) with validity of Eqs. (2.6.8).

88


Conversely, given a Lie algebra of operators spanned by linearly independent operators (2.6.4) for which equations (2.6.8) hold. Let us construct a local Lie group Gr by the structure constants Cβαγ taken from (2.6.8) and certainly satisfying the Jacobi relations (2.3.21). This is possible due to Theorem 2.11. Let us write the system (2.6.5) with the auxiliary functions Vβα (b) in this Gr : The system is completely integrable by virtue of (2.6.8). Application of Theorem 2.15 yields the group GNr isomorphic to Gr ; constructed earlier. Transformations of the GNr satisfy Eqs. (2.6.5) which entail that the operators (2.6.4) coincide with the basis operators of the resulting group GNr : This completes the proof of Theorem 2.16. In what follows, the Lie algebra of operators spanned by the basis operators (2.6.4) of the group GNr is denoted by LNr : It is clear that LNr is a Lie algebra of the group GNr in the meaning of Definition 2.11. Let us point out one application of the above results to group transformations admitted by a system of differential equations (S). The system (S) will be said to admit the group GNr if it admits any subgroup G1 2 GNr in the meaning of Definition 1.6 (Chapter 1). Then, the following corollary of Theorem 2.16 can be formulated: if (S) admits a Lie algebra of operators LNr ; then (S) admits a local Lie group GNr corresponding to this LNr in the meaning of (2.3.17). The proof is trivial. Examples of local Lie groups of transformations are provided by the so-called parametric groups of the local Lie group Gr : Let us consider the multiplication law c = ϕ (a; b) of elements G as transformation of the point a to the point c of the space E r (a) depending on parameters b: The associativity of multiplication of elements Gr : (gc ga )gb = gc (ga gb ); written in a coordinate form provides the equations

ϕ i (ϕ (c; a); b) = ϕ i (c; ϕ (a; b)) similar to Eqs. (2.6.2). Hence, the transformations a!c:

cα = ϕ α (a; b)

form a local Lie group of transformations of Grr : It is known as the first parametric group of the group Gr : The formulae (2.6.3) for this group of transformations coincide with the formulae (2.3.5), and the basis operators of its Lie algebra Lrr coincide with the operators (2.3.23). Likewise, one can define the second parametric group of the group Gr as the group of transformations of the space E r (b); namely the transformations b ! c given by the same formulae ϕ (a; b) = c:

2.6.4 Canonical coordinates of the second kind As it has already been mentioned, the proofs of theorems of existence of the groups Gr or GNr contain an algorithm for constructing multiplication laws (in Gr ) or trans-


89

formations (in GNr ) as well. The corresponding groups are obtained in canonical coordinates of the first kind. However, the algorithm is inconvenient in applications because it appears to be very cumbersome and is hardly ever used in fact. The algorithm of constructing Gr and GNr based on finding a “basis” set of subgroups G1 is more applicable in practice. The algorithm consists in the following. Let us suppose that we know a set of r subgroups G1 of the group Gr with the property that the directing vectors of the subgroups G1 are linearly independent in a certain system of coordinates ∑a : The subgroups can be written by the formulae a = aα (a¯α ) or gα (a¯α ) = ga α (a¯α ) (α = 1; : : : ; r);

(2.6.9)

where a¯α is the parameter of the subgroup with the number α : Now let us compose a single product of all gα (a¯α ): It will be some element ga 2 Gr in the coordinates ∑a : ga = g1 (a¯1 )g2 (a¯2 ) gr (a¯r ): (2.6.10) We claim that every element ga 2 Gr is uniquely represented in the form (2.6.10) when the parameters a¯ α vary independently of each other. In other words, we state that the values of parameters a¯ α can be taken as new coordinates in Gr thus providing a new system of coordinates ∑a¯ : Indeed, equation (2.6.10) shows that equations (2.3.2) hold for a and a; ¯ and one has only to verify that the Jacobian α ∂ a ∂ a¯β a=0 ¯ does not vanish. Since all a¯ α are independent, one can assume while differentiating with respect to a¯ β ; that all a¯α are equal to zero except for a¯ β : Then equation (2.6.10) takes the form ga = gβ (a¯β ) or, by virtue of Eq. (2.6.9), ga = ga β (a¯β ) so that the derivative ∂ aα ∂ a¯β a=0 ¯ is equal to the coordinate eβα of the directing vector eβ of the subgroup G1 with the number β according to definition of the directing vector. Therefore, the considered Jacobian equals to jeβα j and does not vanish due to linear independence of the vectors eβ (β = 1; : : : ; r): The coordinates introduced in Gr according to the formula (2.6.10), are called canonical coordinates of the second kind. Likewise, one can introduce canonical coordinates of the second kind in GNr : In this case, one multiplies the transformations Ta α of subgroups G1 with the parameters aα : The result is written in the form

90


T = Ta1 Ta2 Tar

(2.6.11)

instead of (2.6.10). Example 2.5. Let us consider the Lie algebra of operators L22 with the basis X1 =

∂ ; ∂x

X2 = x

∂ ∂ +y ∂x ∂y

and construct the corresponding group G22 in canonical coordinates of the second kind. The subgroup G1 with the operator X1 is the group G1 of translations and, as we already know from Chapter 1, has the form Ta : x0 = x + a; y0 = y: The subgroup G1 with the operator X2 is the group G1 of dilations and has the form Tb : x0 = bx; y0 = by: The general form of the transformation T(a;b) 2 G22 is provided by multiplication T(a;b) = Ta Tb :

x0 = bx + a;

y0 = by:

However, one can multiply in the reverse order as well: T(a;b) = Tb Ta :

x0 = b(x + a);

y0 = by;

which obviously leads to another system of coordinates in G2 : eNre ; which is an In conclusion, let us mention the notion of the prolonged group G eNre is the Lie evident generalization of Definition 1.5. The Lie algebra of the group G e where X are the operators of the Lie algebra algebra of the prolonged operators X; N of the group Gr :

Chapter 3

Group invariant solutions of differential equations

3.1 Invariants of the group GN r 3.1.1 Invariance criterion Let GNr be a local Lie group of point transformations x0 = Ta x in the space E N (x) and let ∂ Xα = ξαi (x) i (α = 1; : : : ; r) (3.1.1) ∂x be a basis of its Lie algebra LNr of operators. Definition 3.1. A function I(x); which is not identically constant, is called an invariant of the group GNr if I(Ta x) = I(x) for all transformations Ta 2 GNr : Comparing this definition with Definition 1.3 and invoking the corollary of Theorem 2.7, one can see that the function I(x) is an invariant of the group GNr if and only if it is an invariant of every subgroup G1 GNr : Theorem 1.4 gives the necessary and sufficient condition for the group G1 to have I(x) as is its invariant. By virtue of this theorem, I(x) is an invariant of GNr if and only if XI(x) = 0 for any operator X 2 LNr : Since

X = eα Xα ;

one obtains finally the criterion for the invariance of the function I(x) with respect to the group GNr in the form Xα I(x) = 0 (α = 1; : : : ; r); where Xα are basis operators (3.1.1) of the Lie algebra LNr :

(3.1.2)

92

3 Group invariant solutions of differential equations

Thus, an invariant I(x) of the group GNr has to be a solution of the system of linear differential equations (3.1.2). The questions then arise about the existence and the multitude of the solutions of the system (3.1.2). First of all, it is clear that if I 1 (x); : : : ; I s (x) are some solutions of the system (3.1.2), then any function F(I 1 (x); : : : ; I s (x)) is also a solution: Xα F(I) =

∂F Xα I σ = 0: ∂ Iσ

Thus, the questions of functional dependence and independence of functions arise. These questions belong to the classical analysis. Let us recall some facts from this field.

3.1.2 Functional independence Functions f 1 (x); : : : ; f s (x) are said to be functionally dependent if there exists a not identically vanishing function F(z1 ; : : : ; zs ) such that the function F( f 1 (x); : : : ; f s (x)) vanishes identically with respect to the independent variables x = (x1 ; : : : ; xN ) 2 E N ; F( f 1 (x); : : : ; f s (x)) 0: If such a function F(z1 ; : : : ; zs ) does not exist, i.e. if the identity F( f 1 (x); : : : ; f s (x)) 0 with respect to x implies the identity F(z1 ; : : : ; zs ) 0 with respect to the variables z; then the functions f 1 (x); : : : ; f s (x) are said to be functionally independent. In what follows, functions f σ (x) are supposed to be once continuously differentiable. Lemma 3.1. The functions f σ (x) (σ = 1; : : : ; s) are functionally independent if and only if the general rank R(J) of the Jacobi matrix s ∂f J= ∂ xi is equal to s; R(J) = s: If R = R(J) < s; then there exist s R functionally independent functions Fµ (z1 ; : : : ; zs ); such that Fµ ( f 1 (x); : : : ; f s (x)) 0 (µ = 1; : : : ; s R):

3.1 Invariants of the group GNr

93

This lemma is a well-known result of classical analysis and is not to be proved here. Let us consider a few examples. If the number of functions f σ (x) is such that s > N; then the functions are always functionally dependent. Further, if at least one of the functions f σ (x) is identically constant, then the functions are functionally dependent as well. If f σ (x) are functionally independent, then E N contains a system of coordinates (y); where the functions f σ (x) are reduced to y1 ; : : : ; ys :

3.1.3 Linearly unconnected operators Let us turn to investigate a system of equations in the form (3.1.2). The operators Xα are of the form (3.1.1) and not supposed to be the basic operators of LNr so far. Definition 3.2. Operators Xα (α = 1; : : : ; r) are said to be linearly connected if there exist functions ϕ α (x) (α = 1; : : : ; r); not all identically zero, such that

ϕ α Xα 0; and unconnected otherwise. Note that if Xα are linearly unconnected, then r N: In what follows, we will use the notion of a commutator of two operators introduced in Definition 1.7. Definition 3.3. Operators Xα (α = 1; : : : ; s) are said to compose a complete system if they are linearly unconnected and if their commutators [Xα ; Xβ ] satisfy the equations σ [Xα ; Xβ ] = ϕαβ (x)Xσ σ (x): A complete system of operators is said to be Jacobian with some functions ϕαβ σ (x) 0: if all ϕαβ

A system of equations Xα f ξαi (x)

∂f = 0 (α = 1; : : : ; s) ∂ xi

(3.1.3)

is said to be complete (Jacobian) if the operators Xα compose a complete (Jacobian) system of operators. Lemma 3.2. If the system (3.1.3) is complete, then s N: When s < N there exist exactly N s functionally independent solutions of the system, and any of its solutions is their function. Proof. Note that the property of the system (3.1.3) to be complete or Jacobian does not depend on the choice of a system of coordinates in E N : This follows from Lemma 1.4, see x1.6. Further, if s > N; then there are functions ϕ α (x) such that ϕ α Xα 0; since in this case the matrix

94


ξαi (x)

has the number of columns N; less then the number of rows s; so the rows are to be linearly dependent for any x 2 E N : Let us introduce the notion of equivalent systems of operators (or equations of the form (3.1.3)). A system of operators fXα0 g is said to be equivalent to a system of operators fXα g if Xα0 are independent linear combinations (with variable coefficients) β of operators Xα ; specifically, if there exist functions ωα (x) such that β

jωα (x)j 6= 0 and

β

Xα0 = ωα (x)Xβ :

It is evident that equivalent systems of equations (3.1.3) have the same solutions. Let us demonstrate that any complete system is equivalent to some Jacobian system. Indeed, the completeness of the system fXα g entails that the general rank of the matrix ξαi (x) equals to s: If we assume, without loss of generality, that a non-vanishing minor of the order s of the matrix is composed by the first s columns, then one obtains an β equivalent system fXα0 g with ωα (x); equal to elements of the inverse matrix of the 0i minor. The matrix ξα of the resulting equivalent system has the form 0 1 B B0 B B B. B .. @ 0

0

0

ξ10s+1

1

0

ξ20s+1

.. .

.. .

1

ξs0s+1

.. . 0

ξ10N

1

C ξ20N C C C: .. C . C A

ξs0N

The system fXα0 g is Jacobian. Indeed, the equation σ [Xα0 ; Xβ0 ] = ϕαβ Xσ0

and definition of the commutator entail the identities σ ϕαβ = Xα0 ξβ0σ

= Xα0 δβσ

Xβ0 ξα0σ Xβ0 δασ = 0

for σ = 1; : : : ; s: Hence, the system (3.1.3) can be considered to be Jacobian from the very beginning without loss of generality.


95

3.1.4 Integration of Jacobian systems We will carry out a procedure of s steps to reduce the system (3.1.3) to the simplest form. The first step consists in finding by means of Theorem 1.3 such a system of coordinates (y) in E N ; where yi = yi (x) (i = 1; : : : ; N); in which the operator X1 becomes the translation operator with respect to y1 : X1 = Y1 =

∂ ∂ y1

In order to construct the system (y); one has to find solutions yi (x) of the equations 0

X1 y1 (x) = 1;

X1 yi (x) = 0

(i0 = 2; : : : ; N):

The operators Xα written in the variables (y) provide a system of operators fYα g with ∂ Y1 = ∂ y1 As it has already been mentioned, the system fYα g is Jacobian again and if Yα 0 = ηαi 0 (y)

∂ ∂ yi

(α 0 = 2; : : : ; N);

then [Y1 ;Yα 0 ] = 0 provides that

∂ ηαi 0 = 0: ∂ y1

Thus, the coordinates ηαi 0 are independent of the variable y1 : Since the differentiation with respect to y1 is absent in the coordinates of the commutator [Yα 0 ;Yβ 0 ] with the numbers 2; : : : ; N; it follows from the above facts that the system of operators Y10 =

∂ ; ∂ y1

0

Yα0 0 = ηαi 0 (y0 )

∂ 0 ∂ yi

(α 0 = 2; : : : ; s; i0 = 2; : : : ; N);

which is equivalent to the system fXα g; is Jacobian again. Moreover, the operators fYα0 0 g (α 0 = 2; : : : ; s) act in the space E N 1 (y0 ) of the points y0 = (y2 ; : : : ; yN ) and compose a Jacobian system. The construction of the system fYα0 0 g completes the first step. Since fYα0 0 g has all the properties of fXα g; one can make the second, : : : ; s-th step of our procedure and as a result arrive at a system of coordinates (z) and a system of operators fZα g having the form Z1 =

∂ ∂ ; : : : ; Zs = s ∂ z1 ∂z

96

The system becomes


∂f = 0 (σ = 1; : : : ; s): ∂ zσ

If s < N; solutions to the latter system are functions zs+1 ; : : : ; zN ; which are obviously functionally independent and any solution has the form f = f (zs+1 ; : : : ; zN ): Returning to the initial coordinates (x); one obtains solutions of the system (3.1.3), f τ = zτ (x)

(τ = s + 1; : : :; N);

with the properties mentioned in Lemma 3.2. This proves the lemma. In addition note that when s = N the complete system has no functionally independent solutions at all, for it can be satisfied only by a constant. Note that the above proof of Lemma 3.2 contains, in fact, the algorithm for constructing functionally independent solutions of the system (3.1.3). The basic element of the algorithm is the transition fXα g ! fYα0 g; requiring only integration of ordinary differential equations.

3.1.5 Computation of invariance Let us turn back to the problem of invariants of the group GNr with the basis set of operators (3.1.1) of its Lie algebra LNr : Let us introduce a matrix of coordinates of operators Xα : M = ξαi (x) ; (3.1.4) where α is the number of the row, i is the number of the column, and α = 1; : : : ; r; i = 1; : : : ; N: The general rank of the matrix M is denoted by R; so that R = R(M): The following result is formulated in this notation. Theorem 3.1. The group GNr has invariants if and only if R < N: If this inequality is satisfied, then there exist t = N R functionally independent invariants I τ (x) (τ = 1; : : : ;t) of the group GNr such that any of its invariants is their function. Proof. Since Xα span a Lie algebra LNr ; the system of operators (3.1.1) is closed with respect to the operation of commutation, but the operators can be linearly connected. The maximum number of linearly unconnected operators Xα equals to the general rank R of the matrix M (3.1.4). Consider the system (3.1.2) and eliminate operators expressed via R unconnected operators from it. Then one obtains a complete system of R operators. If R = N; there are no invariants. If R < N; the statement follows from Lemma 3.2. Theorem 3.1 is proved. Note that the algorithm described in the proof of Lemma 3.2 is efficient in finding invariants in practice. Let us consider an example of its realization.


97

Example 3.1. L33 : X1 = z

∂ ∂y

y

∂ ; ∂z

X2 =

z

∂ ∂ +x ; ∂x ∂z

X3 = y

∂ ∂x

x

∂ ∂y

These operators are linearly connected, namely xX1 + yX2 + zX3 = 0: Since this is the only connection, we have R = 2 and, according to Theorem 3.1, there is one invariant (t = N R = 3 2 = 1): Let us find it. First, p let us determine invariants of the operator X1 which are obviously x and ρ = y2 + z2 : Let us turn to new variables x; ρ ; z: According to the formula (2.3.7) from 1.2, one obtains Y1 =

y

∂ ; ∂z

Y2 =

∂ xz ∂ + ; ∂x ρ ∂ρ

z

Y3 = y

∂ ∂x

xy ∂ ; ρ ∂ρ

or, turning to equivalent operators, Y10 =

∂ ; ∂x

Y20 = ρ

∂ ∂x

x

∂ ; ∂ρ

and the operator Y3 is eliminated as it is linearly connected with Y2 : Y3 =

y Y2 : z

Invariant of the operator Y20 on the plane (x; ρ ) is I = x2 + ρ 2 : This is the desired invariant of the corresponding G33 : Turning back to the initial coordinates, one finally obtains I = x2 + y2 + z2 : Let us point out some other notions connected with existence of invariants of GNr : A group GNr is said to be transitive if it has no invariants. A transitive group is characterized by the relations r R = N: If r = R the group is said to be simply transitive, and if r > R it is termed multiply transitive. If the group GNr has invariants, it is said to be intransitive. By virtue of Theorem 3.1, a criterion of intransitiveness is the inequality R < N: These terms are connected with properties of a group to contain transformations Ta mapping every point x 2 N to any other point x0 : The notion of a differential invariant of the group GNr is a direct extension of eNre : Therefore, we do not dwell on it here Definition 3.1 to the prolonged group G e and only mention that due to an unbounded increase of dimension of the space EeN ;

98


any group GNr has differential invariants (its order can be higher than one) when successive prolongations are fulfilled and ranks Re are limited by the number r:

3.2 Invariant manifolds 3.2.1 Invariant manifolds criterion Definition 3.4. A manifold M E N is called an invariant manifold of a group G Nr if Ta x 2 M for every point x 2 M and for all transformations Ta 2 GNr : As in x1.3, we consider manifolds M given regularly by the equations σ ∂ψ σ M : ψ (x) = 0 (σ = 1; : : : ; s); R = s: ∂ xi M

(3.2.1)

Theorem 3.2. The manifold M regularly given by Eqs. (3.2.1) is an invariant manifold of the group GNr if and only if the equations X ψ σ (x) M = 0 (σ = 1; : : : ; s) (3.2.2) hold for all operators X 2 LNr : Proof. It is evident that M is an invariant manifold of the group GNr if and only if M is an invariant manifold of every subgroup G1 GNr : Therefore, Theorem 3.2 is a corollary of Theorem 1.5.

3.2.2 Induced group and its Lie algebra Substantially new facts connected with GNr when r > 1 begin from the following definition where the matrix M (3.1.4) and its general rank R are meant. Definition 3.5. A manifold M E N is termed a nonsingular manifold of the group GNr ; if R(MjM ) = R: Otherwise, i.e. when the rank of M decreases on M as compared to its general rank, the manifold M is said to be a singular manifold of the group GNr : Let x¯ be the points of an invariant manifold M of the group GNr : Then, by definition, x¯0 = Ta x¯ 2 M :

3.2 Invariant manifolds

99

Hence, a family fTa g of point transformations of M into itself is determined on M : Manifestly, this family generates a local Lie group denoted further by GNr jM : Definition 3.6. The group GNr jM is called the group induced by the group GNr in its invariant manifold M : Let us construct the Lie algebra of the induced group GNr jM : To this end, we use the fact that M is given regularly by Eqs. (3.2.1) and introduce the system of coordinates (y) in E N by the formulae yσ = ψ σ (x) (σ = 1 : : : ; s);

yτ = xτ

(τ = s + 1; : : : ; N):

(3.2.3)

Then ys+1 ; : : : ; yN can be taken as coordinates of the point y¯ = x¯ on M : Turning to variables y in the operator ∂ X = ξ i (x) i ∂x and taking the result on M ; one obtains operators of the induced group GNr jM in the form

∂ XjM = ξ¯ τ (y) ¯ ; ∂ yτ

ξ¯ τ (y) ¯ = ξ τ jM

(τ = s + 1; : : :; N)

(3.2.4)

due to the criterion (3.2.2) of invariance of M : The operator (3.2.4) is called an operator, induced by the operator X on the invariant manifold M : Manifestly, the operation of transition from X to XjM is linear. Let us demonstrate that it preserves the commutator. To this end, invoke Theorem 1.10, according to which a manifold M that is invariant with respect to Xα ; Xβ ; is invariant with respect to the commutator [Xα ; Xβ ] as well. Therefore, using (3.2.4), one obtains ∂ ∂ [Xα ; Xβ ] M = [Xα ; Xβ ](yi ) i = [Xα ; Xβ ](yτ )jM τ ∂y M ∂y = [Xα (Xβ yτ ) =

∂ ξ¯ τ ξ¯αθ θ ∂y

∂ Xβ (Xα yτ )] M τ ∂y ! ¯τ ∂ ξ ∂ ξ¯βθ θ = [Xα jM ; Xβ jM ]; ∂y ∂ yτ

which was to be proved. Thus, the operators X jM ; induced by all operators X 2 LNr ; compose a Lie algebra denoted by LNr jM and called the Lie algebra induced by the Lie algebra LNr on its invariant manifold M : The induced Lie algebra LNr jM of the operators (3.2.4) is actually the Lie algebra of the induced group GNr jM : The above fact means that there is a homomorphism ψ of the algebra LNr to the Lie algebra LNr jM given by the formula

ψ (X ) = X jM :

100


The kernel of this homomorphism consists of operators X corresponding to the subgroups G1 GNr ; transformations of which leave points of the manifold M unaltered. These are operators whose coordinates satisfy the system of equations

ξ i (x) ¯ = 0 (i = 1; : : : ; N) for all x¯ 2 M :

3.2.3 Theorem on representation of nonsingular invariant manifolds Let us consider the problem of constructing invariant manifolds of the group GNr : If GNr is intransitive and I τ (x) (τ = 1; : : : ;t) is the complete set of its invariants, then any manifold M ; given by a system of equations of the form

Φ σ (I 1 (x); : : : ; It (x)) = 0 (σ = 1; : : : ; s);

(3.2.5)

is an invariant manifold. This follows directly from Definitions 3.4 and 3.1. Let us demonstrate that this procedure of generating invariant manifolds of the group GNr is the most general in a sense. Theorem 3.3. The group GNr has nonsingular invariant manifolds if and only if R < N: If this inequality is satisfied and if I τ (x)

(τ = 1; : : : ;t = N

R)

is a complete set of functionally independent invariants of the group GNr ; then any of its nonsingular invariant manifolds can be given by a system of equations of the form (3.2.5). Proof. Let M be regularly given by Eqs. (3.2.1). The invariance conditions (3.2.2) written for the basis operators Xα ; σ i ∂ψ ξα = 0 (α = 1; : : : ; r); ∂ xi M show that there are linear dependencies between columns of the matrix MjM so that its rank is less than N: Since M is a nonsingular manifold of GNr it follows that R < N by virtue of Definition 3.5. Let us assume that R < N: Consider a nonsingular invariant manifold M of the group GNr given regularly by Eqs. (3.2.1). Note that the transformations of the induced group GNr jM act in the space of N S dimensions. Therefore, GNr jM has a complete set of N s R functionally independent invariants. Now let us take the

3.2 Invariant manifolds

101

invariants I τ (x) (τ = 1; : : : ;t) and consider them on M : These are functions I τ (x) ¯ which are invariants of the induced group GNr jM ; so that we have t = N R as its invariants. Assume that there are N R s0 functionally independent invariants among them; the number can not be higher than the number of functionally independent invariants of the group CrN jM ; so that N

R

s0 N

s

R;

whence s s0 : Moreover, by virtue of Lemma 3.1, there exist s0 functionally independent functions Φ σ (z1 ; : : : ; zt ); such that

Φ σ (I 1 (x); ¯ : : : ; It (x)) ¯ 0 (σ = 1; : : : ; s0 ):

(a)

Let us define a manifold M 0 E N given by the equations M0 :

Φ σ (I 1 (x); : : : ; It (x)) = 0 (σ = 1; : : : ; s0 )

(b)

with these functions Φ σ : The manifold M 0 contains the given invariant manifold M because Equations (b) are satisfied identically for the points x¯ 2 M by virtue of (a). Further, the dimension of M 0 is N s0 and therefore, the inclusion M 0 M provides the inequality N s0 N s; whence s s0 : Thus, s = s0 : The equality of dimensions of the manifolds M and M 0 and the inclusion M M 0 entail that M0 M: Since Equations (b) of the manifold M 0 have the required form (3.2.5), Theorem 3.3 is proved. Theorem 3.3 can also be termed a theorem on representation of nonsingular invariant manifolds of the group GNr : Singular invariant manifolds may have no representation of the form (3.2.5). In order to find them one has to compose a manifold M given by the system of equations obtained by nullifying all minors of the maximum order of the matrix M (3.1.4). This M should also be checked for “invariance” by means of Theorem 3.2. Let us introduce a numerical characteristics of the invariant manifold of the group GNr which will be of importance in what follows. The dimension of the manifold M is denoted by dim M : Definition 3.7. The number

ρ =t

(N

dimM ) = dim M

R

102


where R = R(M ); is called the rank of the invariant manifold M of the group GNr : In other words, the rank of the invariant manifold M of the group GNr is the dimension of M in the space of invariants of the group. Here the space of invariants is understood as the space E t (I); where the coordinates of the point are values of the invariants I 1 ; : : : ; It of the group GNr :

3.2.4 Differential invariant manifolds Differential invariant manifolds of the group GNr are determined likewise. Namely they are invariant manifolds with respect to transformations of the prolonged group eNre : They will be manifolds in the prolonged space E Ne and will be denoted by G f: Naturally, the notion of nonsingular M fis connected with the general rank of M e the matrix M whose entries are the coordinates of the prolonged operators Xeα : All f can be represented by a system of equations of the form (3.2.5) nonsingular M but, generally speaking, the left-hand sides will contain differential invariants of the group GNr : Let us describe one more procedure for obtaining differential invariant manifolds of the group GNr : Using the operators of total differentiation (see x1.4) Di =

∂ ∂ + pli l i ∂x ∂u

(i = 1; : : : ; n);

one can formulate the procedure in the following theorem. Theorem 3.4. If I(x; u) is an invariant of the group GNr and if the manifold, determined by the system of equations Di I(x; u) = 0 (i = 1; : : : ; n);

(3.2.6)

e

is given regularly in E N ; then it is a differential invariant manifold of the group GNr : Proof. Let X be an operator arbitrarily taken from LNr ; and let Xe be its prolongation. Applying Lemma 1.3 to the function I(x; u) and noting that XI(x; u) 0; one obtains the identities e i I(x; u) = Di (ξ j )D j I(x; u): XD Their right-hand sides vanish on the manifold (3.2.6), and hence e i I(x; u) = 0 XD on this manifold. Therefore, the manifold (3.2.6) is invariant by virtue of the criterion (3.2.2) of Theorem 3.2 which obviously holds for differential invariant manifolds as well. All the above reasoning, notions and results can extended directly to higher prolongations of a group, and hence to differential invariants and differential invariant manifolds of higher orders.

3.3 Invariant solutions of differential equations

103

3.3 Invariant solutions of differential equations 3.3.1 Definition of invariant solutions Consider a system of differential equations (S) in the notation accepted in x1.5. We assume that (S) admits a local Lie group H of transformations of the space E N (x; u): A solution uk = ϕ k (x) (k = 1; : : : ; m) of the system (S) is considered as a manifold

Φ E N (x; u): This manifold is also called a solution of the system (S). Definition 3.8. A solution Φ is called an invariant H-solution of the system (S) if Φ is an invariant manifold of the group H admitted by the system (S). In what follows we consider only those solutions of Φ that are nonsingular manifolds of the group H: Moreover, it will be assumed that the manifold S is a nonsingular differential invariant manifold of the group H: If an invariant H-solution of Φ is a nonsingular invariant manifold of the group H; then this manifold has a definite rank ρ which is also termed the rank of the invariant H-solution of Φ : Let us find necessary conditions for existence of invariant H-solutions. Let Xα = ξαi (x; u)

∂ ∂ + ηαk (x; u) k i ∂x ∂u

(α = 1; : : : ; r)

be a basis of the Lie algebra of the group H: Consider the matrix M = ξαi ; ηαk ;

(3.3.1)

(3.3.2)

where α are numbers of rows and R = R(M) is the general rank of the matrix. Definition 3.8 and Theorem 3.3 provide a necessary condition for existence of invariant H-solutions. Namely, the group H should be intransitive, i.e. that it should have invariants. Hence, R < N: Let I τ = I τ (x; u)

(τ = 1; : : : ;t = N

R)

(3.3.3)

be a complete set of functionally independent invariants of the group H: According to Theorem 3.3, equations of a nonsingular invariant H-solution Φ can be written in the form

Φ:

Φ k (I 1 ; : : : ; It ) = 0 (k = 1; : : : ; m);

where the functions Φ k (I) are functionally independent so that

(3.3.4)

104


R

∂Φk ∂ Iτ

= m:

Moreover, by definition of solutions of the system (S), all variables uk

(k = 1; : : : ; m)

should be determined as functions of x from Eqs. (3.3.4). Therefore, the Jacobian should not vanish: ∂Φk ∂ ul 6= 0; i.e.

R

∂Φk ∂ ul

= m:

Note that the equation

∂Φk ∂ ul

=

∂Φk ∂ Iτ

τ ∂I ∂ ul

and the known fact that the rank of a product of matrices is not higher than the smallest rank of factors guarantee the following equation: τ ∂I R = m: ∂ ul Note that the general rank R of the matrix τ ∂I ∂ ul is independent of the specific form of the invariants (3.3.3), provided that they are functionally independent. Thus, we have the following necessary conditions for existence of invariant Hsolutions: τ ∂ I (x; u) R(M) < N; R = m: (3.3.5) ∂ uk It will be proved further that when the conditions (3.3.5) hold, invariant Hsolutions exist, at least “potentially”. We will need the following result. ˜ u; p) be a differential invariant of the group H and let Φ k (x; u) Lemma 3.3. Let I(x; be invariants of the group H satisfying the condition ∂ Φk ∂ ul 6= 0: If the equations


Di Φ k (x; u) = 0

105

(i = 1; : : : ; n; k = 1; : : : ; m)

are satisfied identically upon the substitution pki = ψik (x; u); then the function

˜ u; ψ (x; u)) I (x; u) = I(x;

is an invariant of the group H: Proof. Acting by any operator X from the Lie algebra L of the group H on I (x; u); one obtains ∂ I˜ ∂ I˜ ∂ I˜ XI = ξ i i + η k k + k X ψik : ∂x ∂u ∂ pi p=φ (x;u) e the equation ˜ u; p) is an invariant of the prolonged group H; Since I(x;

∂ I˜ ∂ I˜ ∂ I˜ XeI˜ = ξ i i + η k k + ζik k = 0 ∂x ∂u ∂ pi is satisfied identically with respect to the variables x; u; p: This equation becomes an identity with respect to the variables x; u upon the substitution pki = ψik (x; u): Hence, in order to prove the equation XI = 0 (which is the statement of the lemma by virtue of the criterion (3.5.2)) it is sufficient to prove the identity

ζik (x; u; ψ (x; u)) = X ψik (x; u): To this end, we let

(c)

F = Φ k (x; u)

in the identity (1.4.18) in Lemma 1.3 of Chapter 1. Then, by virtue of the conditions of Lemma 3.3, one obtains the identity e i Φ k j p=ψ (x;u) = 0; XD which is written in detail as X

k ∂ Φk ∂ Φk l ∂Φ l + ψ X + ζ (x; u; ψ (x; u)) = 0: i i ∂ xi ∂ ul ∂ ul

On the other hand,

∂ Φk ∂ Φk + ψil 0 i ∂x ∂ ul by the condition of our lemma. Applying the operator X to this identity, one obtains X

k k ∂ Φk l ∂Φ l ∂Φ + ψ X + (X ψ ) = 0: i i ∂ xi ∂ ul ∂ ul

106


Comparing these two identities, one obtains [ζil (x; u; ψ (x; u))

X ψil (x; u)]

∂Φk = 0; ∂ ul

whence, by virtue of the assumed inequality ∂Φk ∂ ul 6= 0; one obtains (c) and Lemma 3.3 is proved.

3.3.2 The system (S=H) Theorem 3.5. If the system (S) admits a group H satisfying the conditions (3.3.5), then there exists a system (S=H) which connects only the invariants I τ (τ = 1; : : : ;t); the functions Φ k (I) (k = 1; : : : ; m) and derivatives of Φ k with respect to I τ and which has the following property. The functions Φ k (I) provide a solution of the system (S=H) for any invariant H-solution Φ written in the form (3.3.4). Conversely, any solution of the system (S=H); for which k ∂ Φ R = m; ∂ Iτ provides an invariant H-solution Φ in the implicit form (3.3.4). Proof. For the sake of simplicity, let us suppose that the system (S) is of the first order and give the algorithm for constructing the system (S=H): It is very simple: write Eqs. (1.2.2) with indefinite functions Φ k (I); apply the operators of total differentiation Di to them and obtain the system of equations

∂ Φ k ∂ Iτ ∂ Φ k ∂ Iτ l + τ p = 0; ∂ I τ ∂ xi ∂ I ∂ ul i

(3.3.6)

whence all pki (i = 1; : : : ; n; k = 1; : : : ; m) are obtained. The latter operation can be carried out by virtue of (3.3.5). Substituting the resulting expressions for pki into equations (S) we obtain the system (S=H): Let us demonstrate that the system (S=H) thus constructed, in fact, connects only invariants of the group H: To this end, write the differential invariant manifold e of S (which as we agreed above, is supposed to be a nonsingular manifold for H) the group H in an equivalent form via differential invariants of H; namely S:

˜ = 0: Ω µ (I; I)

(3.3.7)


107

The elimination of the variables pki from the equations S; as is described in the algorithm, consists of substitution of the expressions pki = ψik (x; u) obtained from ˜ u; p): According to Lemma 3.3, the Eqs. (3.3.6), into differential invariants I˜ = I(x; latter become invariants of the group H; i.e. they are functions of the invariants I τ (τ = 1; : : : ;t): The expressions for pki derived from Eqs. (3.3.6) in fact contain derivatives ∂ Φ k =∂ I τ : Hence equations of the system (S=H) connect only the invariants I τ and the derivatives ∂ Φ k =∂ I τ : If equations (3.3.4) are equations of a given invariant H-solution Φ ; then the functions Φ k (I) have a specific form. Upon the mentioned substitution of expressions for pki into (3.3.7), equations (3.3.7) are satisfied identically (by definition of a solution) with respect to the variables x; u; and hence, with respect to the variables I: Conversely, if ∂Φk R =m ∂ Iτ for some solution of the system (S=H); then equations (3.3.4) provide functions uk = ϕ k (x) by virtue of (3.3.5). For these functions their derivatives coincide with those derived from the system (3.3.6) and the latter turn the equations S into identities by construction. Hence, these functions furnish a solution of the system (S), which is obviously its invariant H-solution. Theorem 3.5 is proved for the case when the system (S) is of the first order. Alternations to be made in the proof for higher-order systems are evident. At first sight the system (S=H) seems to be more complicated than the original system (S), because the unknown functions in (S=H) depend on t = N R = n + m R independent variables and this number can be greater than the number n of independent variables in the system (S). However, the solutions of the system (S=H) are manifolds in a t-dimensional space of the point I and we are interested only in such solutions that provide m independent equations (3.3.4), i.e. that have the dimension t m in this t-dimensional space. Therefore, the system (S=H) contains actually only t m independent variables. Since this number is equal to the rank of the invariant manifold (3.3.4) of the group H; one obtains finally that the system (S=H) is a system with ρ =t m=n R (3.3.8) independent variables. In other words, the number of independent variables equals to the rank of invariant H-solutions. In this meaning, the system (S=H) is simpler than the original system (S), since one can see from (3.3.8) that ρ < n always. In practice, one often comes across the case when the variables x; u in invariants of the group H “split” in the following sense. Invariants I τ (x; u) can be chosen so that they divide into invariants depending on x; u : I k (x; u) (k = 1; : : : ; m) and invariants depending only on x :

108


I m+r (x)

(r = 1; : : : ; ρ ):

The condition (3.3.5) is written in this case as follows: k ∂ I ∂ ul 6= 0:

(3.3.9)

If the condition (3.3.9) is satisfied, equations (3.3.4) can always be reduced to the form I k (x; u) = V k (I m+1 (x); : : : ; It (x)); (3.3.10) so that V k (k = 1; : : : ; m) will be the unknown functions. In this case, it is convenient to introduce special notation for the latter ρ invariants and to write invariant Hsolutions in the form I k (x; u) = vk (y1 ; : : : ; yρ );

yr = I m+r (x) (k = 1; : : : ; m; r = 1; : : : ; ρ ) (3.3.11)

Then, the system (S=H) is just a system with respect to the functions v(y) of ρ independent variables y1 ; : : : ; yρ : Theorem 3.5 says nothing about existence of invariant H-solutions. It provides only the fact of their “potential” existence in the meaning of reducibility of the system (S) to the system (S=H): Indeed, the system (S=H) can have no solutions and even be just inconsistent. Let us consider a simple example. Let the system (S) consist of one equation for z = z(x; y) : x

∂z ∂z +y = 1: ∂x ∂y

One can readily verify that this equation admits the group H1 with the operator X =x

∂ ∂ +y ∂x ∂y

This group has the following invariants in the space E 3 (x; y; z) : I 1 = z;

I2 =

x y

One can see that the variables x; y and z are separated here, so that any invariant H1 -solution should have the form x z=v : y However, x

∂z ∂z +y 0 ∂x ∂y


109

for such z; with any function v(λ ): Therefore, the system (S=H) has the form 0 = 1; i.e. it is inconsistent.

3.3.3 Examples from one-dimensional gas dynamics Consider several examples of invariant H-solutions for a system of equations of one-dimensional gas dynamics (see Eqs. (1.5.23) in Chapter 1): ut + uux +

1 px = 0; ρ (3.3.12)

S : ρt + uρx + ρ ux = 0; pt + upx + γ pux = 0:

The Lie algebra admitted by the system (3.3.12) has been calculated in Chapter 1. We will consider here the case γ 6= 3: In this case the system (3.3.12) admits the Lie algebra L56 spanned by the operators X1 =

∂ ; ∂t

X2 =

∂ ∂ X5 = x + u ∂x ∂u

∂ ; ∂x

X3 = t

∂ 2ρ ; ∂ρ

∂ ∂ + ; ∂x ∂u

X4 = t

∂ ∂ +x ; ∂t ∂x

∂ ∂ X6 = ρ +p ∂ρ ∂p

(3.3.13)

Hence, the system (3.3.12) admits a local Lie group G56 and any subgroup of this group. We will denote the subgroup H; whose Lie algebra has a basis X ;Y; : : : ; by the symbol H < X;Y; : : : > : The system (3.3.12) has n = 2 independent variables (t; x), m = 3 unknown functions (u; ρ ; p). In this case N = 2 + 3 = 5: The formula (3.3.8) for the rank of an invariant H-solution takes the form ρˆ = 2 R: Since R > 0 it follows that possible invariant H-solutions are either of the rank ρˆ = 0; R = 2 or of the rank ρˆ = 1; R = 1 (the rank of the solution is denoted by ρˆ not to mixed it with the unknown function ρ in the system (3.3.12)). Examining the operators (3.3.13) thoroughly, one can see that R = 1 only for subgroups with one operator, i.e. for one-parameter subgroups. Example 3.2. The subgroup H < X1 > has invariants I 1 = u;

I2 = ρ ;

I 3 = p;

I 4 = x:

The variables here are “separated” so that the H-solutions have the form u = U(x);

ρ = R(x);

p = P(x);

where U; R; P are unknown functions which by virtue of Theorem 3.5 should satisfy the system (S=H): Substituting the above expressions for u; ρ ; p and their derivatives

110


with respect to t; x into Eqs. (3.3.12), one obtains the system (S=H) in the form 1 UU 0 + P0 = 0; R

UP0 + γ PU 0 = 0:

UR0 + RU 0 = 0;

This is a system of ordinary differential equations for U; R; P which is easily integrable. Any solution of the system provides an invariant H < X1 >-solution. Here and in what follows we do not solve the occurring systems (S=H) up to the end, for it is not important in the present lecture notes, besides, it is not always easy to do it. The examples serve to illustrate only the process of forming invariant H-solutions and the abundance of new possibilities. Example 3.3. The subgroup H < X3 > has invariants x ; t

I1 = u

I2 = ρ ;

I 3 = p;

I 4 = t:

The variables are “separated” and the invariant H-solution has the form u=

x +U(t); t

ρ = R(t);

p = P(t):

The system (S=H) appears to be as follows: tU 0 +U = 0;

tR0 + R = 0;

tP0 + γ P = 0:

Example 3.4. The subgroup H < X4 > has the invariants I2 = ρ ;

I 1 = u;

I 3 = p;

I4 =

x t

The solution has the form u = U(λ );

ρ = R(λ );

p = P(λ );

x λ= t

The system (S=H) has the form 8 > > < (U (U > > : (U

1 λ )U 0 + P0 = 0; R λ )R0 + RU 0 = 0;

λ )P0 + γ PU 0 = 0:

The corresponding H-solutions are known as “self-similar”. Example 3.5. The subgroup H < X6 > has the invariants I 1 = u; Composing the matrix

I2 =

p ; ρ

I 3 = t;

I 4 = x:


111

∂ Iτ ∂ uk

for the case, one can see that its rank equals to two. Therefore, the necessary condition (3.3.5) is not satisfied (for m = 3) and the system (S=H) cannot be constructed. Invariant H-solutions do not exist. Example 3.6. The subgroup H < X1 + X6 > has the invariants I 1 = u;

I2 = ρ e t ;

I 3 = pe t ;

I 4 = t:

The necessary condition (3.3.5) is already fulfilled here and the system (S=H) can be constructed. The number of such examples of invariant H-solutions of the rank ρˆ = 1 for the system (3.3.12) can be increased to infinity. The general form of the one-parameter subgroup is H < eα Xα >; so that its invariants are functions of the constants eα (α = 1; : : : ; 6): However, it is not so easy to find them in such a general form, for one will have to make various assumptions about the constants eα while calculating (it is supposed to verify it as an exercise). Moreover, the same parameters will be included in the system (S=H); which will complicate solution of the latter to a much extent. However, the most important is that the main part of the work will appear to be useless due to predetermined connections between different H-solutions; this circumstance is to be discussed in x3.4. Let us take a quick look at invariant H-solutions of the rank ρˆ = 0: In this case t = m and equations (3.3.4) are equivalent to equations I τ = Cτ (τ = 1; : : : ; m); where Cτ are some constants, which are not determined beforehand. The system (S=H) for this case is just a system of finite equations with respect to unknown Cτ : The rank ρˆ = 0 for the system (3.3.12) provides R = 2; by virtue of which such solutions are to be sought on the subgroups H with two operators. For example, the subgroup H < X1 ; X2 > has the invariants u; ρ ; p: Hence, the H-solution has the form u = C1 ;

ρ = C2 ;

p = C3 ;

where Cα are constants. Then, the system (S=H) is satisfied identically when Ck are arbitrary.

3.3.4 Self-similar solutions The notion of a self-similar solution is often used when investigating problems of physics and mechanics. The term is usually used if the solution is such that “proportional” change of some quantities leaves other quantities under consideration unaltered. To make a more specific formulation, the notion of a dilation operator mentioned in x1.2 is used. One can say that H is a group of dilations if all operators of its Lie algebra are operators of dilation.

112


An invariant H-solution is termed a self-similar solution in a narrow sense if H is a group of dilations. We consider a case when the group H acts in the space E N (x; u) and suppose that the basis of its Lie algebra LNr consists of the following dilation operators: n

Xα = ∑ λαi xi i=1

m ∂ ∂ + µαk uk k ∑ i ∂ x k=1 ∂u

(α = 1; : : : ; r);

(3.3.14)

where λαi ; µαk are constants. One can readily verify that a group of dilations has the following two properties: it is Abelian and its Lie algebra LNr does not contain linearly connected operators. Therefore, for a group of dilations one always has r = R: Considering the matrix M (3.3.2) composed by the coordinates of the operators (3.3.14), one can easily see that it has the general rank at any point where xi 6= 0; uk 6= 0 (i = 1; : : : ; n; k = 1; : : : ; m): Therefore, instead of M; one can consider the matrix M1 obtained from M by letting x1 = = x n = u 1 = = u m = 1 : 0 1 1 λ1 λ1n µ11 µ1m B .. C : .. .. M1 = @ ... (3.3.15) . . A .

λr1

λrn

µr1

µrm

We have R(M1 ) = R(M) = r N = n + m: Let us find invariants of the group H with the operators (3.3.14) and find out conditions when (3.3.5) holds. First, it should be r < N; for otherwise H is transitive and has no invariants. Further, let us demonstrate that a complete set of functionally independent invariants of the group H can be composed of invariants of the form I = (x1 )θ1 (xn )θn (u1 )σ1 (um )σ m : To this end, note that

(3.3.16)

Xα I = (λαi θi + µαk σk )I;

by virtue of which the function I is an invariant of H if and only if the exponents θi ; σk satisfy the system of linear homogeneous equations

λαi θi + µαk σk = 0 (α = 1; : : : ; r):

(3.3.17)

The matrix of this system is M1 (3.3.15) and since its rank equals to r; then the system (3.3.17) has N r linearly independent vector solutions (θ ; σ ); namely the vectors (θ τ ; σ τ ) (τ = 1; : : : ; N r): According to the formula (3.3.16) these vectors provide the invariants I τ : These invariants are functionally independent. Indeed, the general rank of the matrix τ τ ∂ I ∂ Iτ θi τ σkτ τ = ; I ; kI ; ∂ xi ∂ u k xi u


113

which is to be calculated at the point xi = 1; uk = 1; coincides with the rank of the τ τ matrix θi ; σk ; which is equal to N r due to linear independence of the vectors (θ τ ; σ τ ): In order to meet the condition (3.3.5) it is necessary and sufficient that the rank of the matrix (σkτ ) equals to m: Let us demonstrate that the latter is fulfilled if and only if the rank of the matrix (λαi ) equals to r: If R((λαi )) < r; then there are numbers ω α such that ω α λαi = 0: Then, denoting

χ k = ω α µαk one obtains from (3.3.17) that the equations χσk k = 0 are satisfied for any solution of the system (3.3.17). Therefore, R((σkτ )) < m: Conversely, if

R((λαi )) = r;

then the system λαi θi = 0 has exactly n r linearly independent solutions. If we take them as a part of the complete system from N r solutions, then the remaining N r (n r) = m solutions of the system should be such that R((σkl )) = m: Thus, finally, the necessary and sufficient condition of “potential” existence of self-similar H-solutions for the group H with the operators (3.3.14) is R((λαi )) = r n:

(3.3.18)

The notion of a group of dilations is closely connected with the so-called theory of dimensions of physical quantities. In order to reveal this connection, let us construct final transformations of the group H with the operators (3.3.14). Upon the construction, e.g. in canonical coordinates of the second kind, one obtains the transformations λi µk λi µk x0i = a1 1 ar r xi ; u0k = a1 1 ar r uk : (3.3.19) λi

λi

depending on r independent parameters a1 ; : : : ; ar : Let us term the aggregate a1 1 ar r as the “dimension” of the quantity xi and parameters aα as a “unit of dimension”. Similar terms are used for the quantities uk : The “dimensions” are denoted in the theory of dimensions as follows: λi

λi

[xi ] = a1 1 ar r ;

µk

µk

[uk ] = a1 1 ar r :

Then invariants I of the form (3.3.16) will be “dimensionless” quantities. This follows from Eqs. (3.3.17). The theory of dimensions provides the so-called Π-theorem claiming that any dimensionless quantity is a function of invariants I of the form

114


(3.3.16). Here it is a particular case of Theorem 3.1 concerning the group of dilations H with the operators (3.3.14). The notion of a self-similar solution formulated above in a narrow sense is not satisfactory, because it is not invariant with respect to the choice of the system of coordinates in E N : Upon a change of a coordinate system, the dilation operator (3.3.14) is no longer a dilation operator in general. For instance, using the coordinates yi = log xi ;

vk = log uk ;

(3.3.20)

one obtains from Xα the translation operators Yα = λαi

∂ ∂ + µαk k ∂ yi ∂v

(3.3.21)

In this connection, the following definition taking into account the main peculiarity of a dilation group is suggested. Definition 3.9. An invariant H-solution is said to be self-similar in a broad sense if the group H is Abelian. Note that the dilation group has two peculiarities: it is Abelian and it does not contain linearly connected operators. In the above definition we use only the first property. The second property appears to be of no importance from the viewpoint of invariants due to the following statement. Theorem 3.6. An Abelian group H; such that R(M) = R; contains a subgroup HR which is similar to the group of dilations and has the same value R: Proof. The equation R(M) = R guarantees that the Lie algebra of the group H contains R linearly unconnected operators Xα = ξαi (x)

∂ ∂ xi

(α = 1; : : : ; R):

(3.3.22)

Since H is Abelian, one has [Xα ; Xβ ] = 0 for all α ; β : Therefore, there exists a subgroup HR H; for which the operators (3.3.22) provide a basis of its Lie algebra. Since these operators are linearly unconnected, we have R((ξαi )) = R: In order to reduce the group HR to a group of dilations, it suffices to reduce it to a translation group. This follows from transformation of (3.3.14) to (3.3.21) by means of (3.3.20). Let us prove that one can determine functions ϕ i (x) (i = 1; : : : ; n) such that they satisfy the system of equations Xα ϕ i = δαi

(α = 1; : : : ; N; α = 1; : : : ; R)

(3.3.23)

and are functionally independent. If such functions exist, then turning to coordinates yi = ϕ i (x) in E N ; one has the transformed operators


Xα = Yα = Xα (ϕ i )

115

∂ ∂ = α i ∂y ∂y

and the theorem is proved. In order to construct the function ϕ = ϕ i (x) when i R we search it in the implicit form F(x; ϕ ) = 0; where F(x; ϕ ) is such that

∂F 6= 0: ∂ϕ The equations

∂F ∂F ∂ϕ + =0 ∂xj ∂ϕ ∂xj

show that it is sufficient to determine the function F(x; ϕ ) as a solution of the system Zα F = Xα F + δαi satisfying the condition

∂F = 0 (α = 1; : : : ; R) ∂ϕ

(3.3.24)

∂F 6= 0: ∂ϕ

Let us demonstrate that the system (3.3.24) is complete (Definition 3.3). Indeed, firstly, we see that [Zα ; Zβ ] = [Xα ; Xβ ] = 0; and secondly, operators Zα (α = 1; : : : ; R) are linearly unconnected, for any linear connection between them would be a linear connection of the operators Xα (3.3.22), which are linearly unconnected by assumption. Since R N; and the operators Zα act in the space of the point (x1 ; : : : ; xN ; ϕ ); having the dimension N + 1; then, by Lemma 3.2, the system (3.3.24) has at least one independent solution F(x; ϕ ): There is certainly such a solution among solutions of the system (3.3.24), for which

∂F 6= 0: ∂ϕ Otherwise all solutions of the system (3.3.24) would also be solutions of the system Xα F = 0 (α = 1; : : : ; R): This is impossible because the latter system has only N R functionally independent solutions, while (3.3.24) has N + 1 R: Thus we have obtained some functions ϕ i (x) for i R: If i > R; then equations (3.3.23) have the form Xα ϕ i = 0 (α = 1; : : : ; R); so that ϕ i are invariants of the group HR : By virtue of Lemma 3.2, there exist N R functionally independent invariants. We take them as functions ϕ R+1 ; : : : ; ϕ N : Thus, the system of functions ϕ i (x) satisfying Eqs. (3.3.23) is constructed. Let us demonstrate that these ϕ i (x) are functionally independent. Indeed, if

116


i ∂ ϕ ∂ x j 0; then there are functions µi = µi (x) such that

µi

∂ϕi = 0 ( j = 1; : : : ; N): ∂ xj

Multiplying this equation by ξαj and summing over j; one obtains

µi Xα ϕ i = µα = 0 (α = 1; : : : ; R): Hence, the previous equalities have the form

µR+1

∂ ϕ R+1 ∂ϕN + + µN = 0 ( j = 1; : : : ; N) j ∂x ∂xj

and mean that the functions ϕ R+1 ; : : : ; ϕ N are functionally dependent (if not all µR+1 ; : : : ; µN are zero). Since this contradicts the choice of the functions ϕ R+1 ; : : : ; ϕ N it follows that the equation i ∂ϕ ∂xj 0 is impossible. Theorem 3.6 is proved. Corollary 3.1. For any self-similar H-solution in a broad sense there is such a subgroup HR H and such a system of coordinates in the space E N ; where this solution is a self-similar HR -solution in a narrow sense. Corollary 3.2. Any self-similar H-solution in a broad sense can be derived as a solution independent of some independent variables in a certain system of coordinates in E N : The latter follows from the possibility of reducing the group HR to a translation group. Note that since a one-parameter group H1 is always Abelian, then all invariant H1 -solutions of the rank ρˆ = 1 for the equations of gas dynamics (3.3.12) are selfsimilar in a broad sense.

3.4 Classification of invariant solutions 3.4.1 Invariant solutions for similar subalgebras It has been demonstrated in the previous section that if the system (S) admits a group G then one can search for particular solutions of the system (S); namely invariant H-solutions for any subgroup H G: An important numerical characteristics of

3.4 Classification of invariant solutions

117

invariant H-solutions is their rank ρ : It denotes the number of independent variables in the system (S=H): Therefore, the main difference (classification) of invariant H-solutions is made with respect to ranks of these solutions. For the rank ρ ; one has the formula (3.3.8),

ρ =n

R;

where R > 0: Accordingly, the rank can have the values ρ = 0; 1; : : : ; n 1: When ρ is fixed, various H-solutions are obtained on different subgroups having one and the same value of the rank R of the matrix M from coordinates of operators of these subgroups. Thus, the problem of enumerating all subgroups H G with a given R arises. Note that the number R is connected with the fact that elements of the group G are transformations in E N : If one takes some other group G0 ; isomorphic to G; it can appear to have no naturally determined number R; though G0 and G have the same subgroups as a matter of fact. Therefore, from the group view-point it is more convenient to operate with the number r; which is the order of the group H; instead of R: Since the difference r R equals to the number of independent linear connections between operators of the basis of the Lie algebra of the group H; then one can easily sort subgroups with respect to the values of R: Thus, one arrives at a purely group problem on enumerating subgroups H of a given order of the local Lie group G: The set of such subgroups is infinite. However, from the viewpoint of invariant H-solutions it is not necessary to know all such subgroups as it follows from the following statements. Definition 3.10. Subgroups H 0 and H of the group G are said to be similar (in the group G) if there exists an element (transformation) T 2 G such that H 0 = T HT 1 : Lemma 3.4. Let two subgroups H 0 and H be similar and let H 0 = T HT

1

:

Let Φ be any invariant H-solution. Then

Φ0 = T Φ is an invariant H 0 -solution. Proof. The property of solution of Φ to be an invariant H-solution can be expressed by the formula H Φ = Φ : Therefore, one has H 0 Φ 0 = T HT

1

T Φ = T HΦ = T Φ = Φ 0;

which was to be proved. If invariant H-solutions Φ are known, then one can consider solutions Φ 0 = T Φ to be known as well. The latter solutions result form Φ upon applying a simple transformation T (which is assumed to be known in its turn) without integration of any system of differential equations.

118


Hence, solving the above group problem, it is sufficient to know the subgroup H G up to similarity in the meaning of Definition 3.10. Note that the relation of similarity of groups H 0 and H; expressed by the formula H 0 = T HT

1

;

is an equivalence relation. Therefore, the whole set of subgroups H of a given order is separated into classes of similar subgroups. Accordingly, the group theoretic problem on enumerating subgroups of a given order of the group G should be considered as a problem on enumerating classes of similar subgroups. The one-to-one correspondence between subgroups of a local Lie group G and subalgebras of its Lie algebra L established in x2.4 makes it sufficient to solve this problem for Lie algebras. Note that the similarity transformation of subgroups expressed by the formula H 0 = T HT 1 is nothing else but an inner automorphism of the group G (see Definition 2.15). Hence, the class of subgroups similar to a given subgroup H is obtained by subjecting it to various inner automorphisms Γa 2 GA ; where GA is the group of inner automorphisms of the group G: In the Lie algebra L it corresponds to transformations of a subalgebra L(H) L by means of the group of inner automorphisms of the Lie algebra L:

3.4.2 Classes of similar subalgebras In a finite-dimensional case, L = Lr ; the group of inner automorphisms, denoted β here by A(L); is realized in the form of a group of matrices (lα (a)) by the formulae (2.5.2) of Chapter 2. Let Xα (α = 1; : : : ; r) be a basis of the Lie algebra LNr of operators of the group N Gr : Let us write an arbitrary operator X 2 LNr in the form of an expansion with respect to the basis with the coordinates e (e1 ; : : : ; er ) : X = eα Xα : β

In the notation of x2.5, the automorphism la with the matrix (lα (a)) acts on basis “vectors” Xα of the Lie algebra LNr by the formula β

Xα0 = lα (a)Xβ

(α = 1; : : : ; r):

(3.4.1)

Hence, any operator X 2 LNr is transformed by this automorphism by the formula β

X 0 = eα lα (a)Xβ ; which means that in a fixed basis Xα the same automorphism can be considered as transformation of the vector l to l 0 with the coordinates β

l 0β = lα (a)eα

(β = 1; : : : ; r):

(3.4.2)

3.4 Classification of invariant solutions

119

By virtue of the formula ∂ lβα (a) ∂ aγ

a=0

= Cβαγ ;

resulting from Eqs. (2.5.3) of Chapter 2, the operators of the group of transformations (3.4.1), derived by the formulae (2.6.3) of Chapter 2, have the form γ

Eα = Cαβ Xγ

∂ ∂ Xβ

γ

or by virtue of the correlation [Xα ; Xβ ] = Cαβ Xγ ; finally Eα = [Xα ; Xβ ]

∂ ∂ Xβ

(α = 1; : : : ; r):

(3.4.3)

According to Theorem 2.14, these operators span the adjoint algebra of the Lie algebra LNr : Application of the operators (3.4.3) for representing the adjoint algebra is particularly convenient because the coordinates of the operators Eα are taken directly from the table of commutators of basis operators of the Lie algebra LNr : One can easily restore the finite automorphisms lα by the operators Eα ; e.g. in canonical coordinates of the second kind. Every subalgebra LNS LNr ; as a linear subspace in LNr can be given by the system of equations l α = ξσα t σ (α = 1; : : : ; r; σ = 1; : : : ; s); (3.4.4) where t σ (σ = 1; : : : ; s) are arbitrary parameters, and ξσα are fixed constants, charβ acterizing a subalgebra. Under the action of an automorphism lα ; this subalgebra transforms into a similar subalgebra, the corresponding equations of which have the form β l α = [lβα (a)ξσ ]t σ (α = 1; : : : ; r) (3.4.5) by virtue of (3.4.2). The formula (3.4.5) contains the whole class of subalgebras similar to the subalgebra (3.4.4). The formulae (3.4.4) and (3.4.5) look especially simple in the case of onedimensional subalgebras. Since there will be only one parameter t; then considering l α to be determined up to an arbitrary common factor, one can assume t = 1: Then (3.4.5) becomes (3.4.2). Example 3.7. Consider the Lie algebra L23 of operators with the basis X1 =

∂ ; ∂x

X2 =

∂ ; ∂y

X3 = x

∂ ∂ +y ∂x ∂y

and find all classes of similar one-dimensional subalgebras L1 : The table of commutators of operators Xα has the form

120


X1

X2

X3

X1

0

0

X1

X2

0

0

X2

X3

X1

X2

0

The formulae (3.4.3) give the following operators Eα of the group of inner automorphisms: E1 = X1

∂ ; ∂ X3

E2 = X2

∂ ; ∂ X3

E3 =

X1

∂ ∂ X1

X2

∂ ∂ X2

Let us find one-parameter groups of automorphisms Aα (t) for every operator Eα by means of integrating Eqs. (2.6.7) of Chapter 2. For example, for E1 these equations are written dX10 = 0; dt

dX30 = X10 ; dt

dX20 = 0; dt

Xα0 jt=0 = Xα :

Whence, one obtains the automorphism A1 (t) :

X10 = X1 ;

X20 = X2 ;

X30 = tX1 + X3 :

A2 (t) :

X10 = X1 ;

X20 = X2 ;

X30 = tX2 + X3 ;

A3 (t) :

X10 = e t X1 ;

Likewise, X20 = e t X2 ;

X30 = X3 :

Let us construct the general inner automorphism in canonical coordinates of the second kind (a; b; c) by the formula la;b;c = A3 ( log a)A2 (b)A1 (c); (with an identity when a = 1; b = c = 0) whence, X10 = aX1 ;

la;b;c :

X20 = aX2 ;

X30 = bX1 + cX2 + X3 :

β

We write the matrix lα of the automorphism, treating the vector X(X1 ; X2 ; X3 ) as the row-vector and obtain 0 1 a 0 b A = @0 a c A : 0 0 1 According to (3.4.2), the corresponding transformation of the vector e is obtained by the formula e0 = Ae; where the vector e (e1 ; e2 ; e3 ) is to be considered as a columnvector. Therefore, the formulae (3.4.2) provide e01 = ae1 + be3 ;

e02 = ae2 + ce3;

e03 = e3 :

3.5 Partially invariant solutions

121

Thus, the adjoint group of the Lie algebra L23 is constructed. Now we take the subalgebra L1 ; consisting of the operator X = eα Xα and consider the following possibilities. (a) e3 6= 0; the automorphism A can be chosen so that e01 = e02 = 0: To this end, it is sufficient to let e1 e2 b = a 3; c = a 3 e e Moreover, one can let e3 = 1: This provides a class of subalgebras L1 ; similar to the subalgebra < X3 > : (b) e3 = 0; here by means of the parameter a; one can have (e01 )2 + (e02 )2 = 1 or

e01 = cos ϕ ;

e02 = sin ϕ :

This provides a one-parameter family of classes of subalgebras similar to the subalgebras < cos ϕ X1 + sin ϕ X2 > depending on the parameter ϕ : The final result of the classification is as follows. Any subalgebra L1 L23 is similar to one of subalgebras < X3 >;

< cos ϕ X1 + sin ϕ X2 >;

while these subalgebras are not similar to each other for any ϕ from the interval 0 ϕ < 2π : If properly developed, this method of constructing classes of similar subalgebras of a finite-dimensional Lie algebra can be used for subalgebras of the second, third and higher orders.

3.5 Partially invariant solutions 3.5.1 Partially invariant manifolds Let GNr be a local Lie group of point transformations in the space E N (x): This section is devoted to expansion of the notion of an invariant H-solution. This is done on the basis of the following definition. Definition 3.11. A manifold N E N is said to be a partially invariant manifold of the group GNr if N lies in an invariant manifold M of the group GNr ; i.e. N M: In what follows, M will be taken as the smallest invariant manifold of the group GNr containing N : It is determined as an intersection of all invariant manifolds

122


of the group containing N : Moreover, we naturally assume that M 6= E N (even locally), for otherwise the given definition is meaningless. The smallest invariant manifold M of the group GNr containing the given manifold N can be constructed as follows. Take any transformation Ta 2 GNr and transform N by means of Ta : This yields the manifold Na = Ta N : The union

S a

(Na ) of all manifolds Na obtained when Ta runs the whole group GNr

obviously contains the given N : It is invariant with respect to all transformations Ta 2 GNr and is the smallest invariant manifold of the group GNr containing N : Hence, M can be determined by the formula [

M=

Ta 2 GNr :

(Ta N );

(3.5.1)

a

We make an additional assumption that nonsingular manifolds N ; M of the group GNr are considered. Let us find out how the dimensions of the manifolds N and M are connected with each other. Let R be the general rank of the matrix M = (ξαi (x)) whose entries are the coordinates of basis operators of the Lie algebra LNr of the group GNr : We transform a point x0 2 N by an arbitrary Ta 2 GNr and obtain the manifold of points x0 = Ta x0 depending on the parameters a: The tangent element to this manifold is given by the infinitesimal transformation x0i = xi0 + ξαi (x0 )aα

(i = 1; : : : ; N)

and has the dimension equal to the dimension of the space Λ of vectors λ with the coordinates λ 0 = ξαi (x0 )aα ; resulting when the vector a(a1 ; : : : ; ar ) runs an rdimensional space. It is known from linear algebra that the dimension of Λ is equal to the rank of the matrix (ξαi (x0 )): The latter equals to R due to nonsingularity of N : Thus, the dimension of the manifold of the points x0 = Ta x0 with any Ta 2 GNr equals to R; due to which the dimension of M determined by the formula (3.5.1) exceeds the dimension of N by R at most: dim M dim N + R:

(3.5.2)

The invariant manifold M ; being nonsingular for GNr ; has the definite rank ρ (Definition 3.7) given by the formula

ρ = dim M

R:

(3.5.3)


123

3.5.2 Defect of invariance Definition 3.12. The rank of the partially invariant manifold N is the rank of the smallest invariant manifold M containing it. The defect of invariance of a partially invariant manifold N is the number

δ = dim M

dimN :

(3.5.4)

The defect of invariance is an important numerical characteristics of a partially invariant manifold demonstrating “to what extent” it is not invariant. Equations (3.5.3) and (3.5.4) provide the following correlation between the basic numerical characteristics of N and of the group GNr :

ρ = δ + dim N

R:

(3.5.5)

Let N be regularly given by the system of equations

ψ σ (x) = 0 (σ = 1; : : : ; s):

(3.5.6)

Then dim N = N

s:

Let us find some inequalities for the invariance defect δ : We introduce the number t = N R; equal to the number of invariants in a complete set of invariants of the group GNr : In this notation one has dim N

R=t

s:

Since ρ 0; then it follows from (3.5.5) that δ s t: Further, equations (3.5.2) and (3.5.4) provide that δ R: At last, the inequality ρ t 1 and (3.5.5) provide

δ t

1+R

N+s = s

1:

Thus, the invariance defect δ satisfies the inequalities maxfs

t; 0g δ minfR; s

1g:

(3.5.7)

By means of the invariance defect one can formulate the following necessary condition for partial invariance of a manifold N : Let Xα (α = 1; : : : ; r) be a basis of the Lie algebra LNr of the group GNr : Theorem 3.7. If the partially invariant manifold N is regularly given by Eqs. (3.5.6) and has an invariance defect δ ; then the general rank of the matrix

∆ = (Xα ψ σ (x)) at the points N is equal to δ :

124


Proof. There are functions ψ¯ σ (x; a) such that the manifold Na = Ta N is given by the equations ψ¯ σ (x; a) = 0 (σ = 1; : : : ; s): (3.5.8) Moreover, these functions can be chosen so that

ψ¯ σ (x; 0) = ψ σ (x) (σ = 1; : : : ; s): On the other hand, the manifold Na is a locus of the point x0 = f (x; a) when the point x runs through the manifold N : Therefore, the equations

ψ¯ σ ( f (x; a); a) = 0 (σ = 1; : : : ; s) hold on N identically with respect to the parameters a: Differentiating with respect to aα and using the Lie equations (2.6.5) of Chapter 2, one obtains

∂ ψ¯ σ i 0 β ξ (x )Vα (a) ∂ x0i β

∂ ψ¯ σ = 0: ∂ aα

Letting a = 0 here and invoking the choice of functions ψ¯ σ (x; a) we see that the following equations hold on N : ∂ ψ¯ σ (x; a) Xα ψ σ (x) = : (3.5.9) ∂ aα a=0 Now let us consider the following procedure of constructing the manifold M (3.5.1) by means of Eqs. (3.5.8). Let us take the matrix σ ∂ ψ¯ (3.5.10) ∂ aα and denote its general rank by ν : Then, one can express ν parameters a from Eqs. (3.5.8), e.g. a1 ; : : : ; aν via variables x and the remaining parameters a¯ (aν +1 ; : : : ; ar ): To this end one has to use ν equations (3.5.8). Substituting the resulting expressions into the remaining s ν equations, one arrives at equations containing no parameters a: Obviously, these are equations of the smallest invariant manifold M containing N : Whence, dim M = N and by virtue of Definition 3.12

(s

ν) = N

s + ν = dim N + ν

ν = δ:

Thus, the general rank of the matrix (3.5.10) is equal to δ ; due to which the statement of the theorem follows from (3.5.9). The condition of Theorem 3.7 may appear to be sufficient, i.e. the equation R(∆ ) = δ


125

can guarantee that the defect of N given by Eqs. (3.5.6) equals to δ ; but we cannot prove it. Note only that when δ = 0; the manifold N is invariant and then the criterion of Theorem 3.7 transforms into the criterion of Theorem 3.2, and the latter is necessary and sufficient.

3.5.3 Construction of partially invariant solutions Consider a system of differential equations (S) again in the space E N (x; u) and discuss its solutions considered as manifolds Φ in E N : Definition 3.13. A solution Φ of the system (S) admitting a group H is said to be a partially invariant H-solution with the rank ρ and the invariance defect δ ; if Φ is a partially invariant manifold of the group H and has the rank ρ and the invariance defect δ : It is clear that a partially invariant H-solution with the invariance defect δ = 0 is just an invariant H-solution. Since the system (S) has n independent variables xi and m unknown functions uk (n + m = N); the dimension of the manifold Φ is fixed and equals to dim Φ = n: In this case it is reasonable to rewrite the correlations (3.5.3) — (3.5.7) keeping in mind that now s = m: Moreover, we are interested only in partially invariant H-solutions with the rank

ρ < n; and hence, by virtue of (3.5.5),

δ < R:

Furthermore, let us introduce a notation for the number of equations of the invariant manifold M ; namely µ = n + m dimM : Thus, one has the relations t = n+m 0 ρ < n;

R;

ρ = δ + n R; µ = m δ ; n; 0g δ minfR 1; m

maxfR

1g

(3.5.11)

for the system (S) in E n+m : In particular, the inequalities (3.5.11) show that in the case m = 1; i.e. if there is only one unknown function in (S) (e.g. one equation), only the value δ = 0 is possible. Hence, there can be no partially invariant H-solutions other than invariant H-solutions in the given case.

126


Let us turn to the problem on the algorithm for finding partially invariant Hsolutions. Unfortunately, we do not have complete representation of a partially invariant manifold and therefore we can determine only the invariant part of the solution, i.e. the manifold M : Let H be a group with a given number R: One can write the inequalities (3.5.11) and select some value of δ ; thus defining the numbers ρ and µ according to (3.5.11). Let us assume that a complete set of invariants I τ (τ = 1; : : : ;t) of the group H is known. We shall look for manifolds M ; where partially invariant H-solutions of the rank ρ and of the invariance defect δ can lie, giving them by the system of µ equations of the form M : Ψ ν (I 1 ; : : : ; It ) = 0 (ν = 1; : : : ; µ )

(3.5.12)

with unknown functions Ψ ν (I): Unlike the case of invariant solutions, we cannot require that Equations (3.5.12) provide all variables uk as functions of x: Therefore, the variables uk (k = 1; : : : ; m) are to be divided into main, e.g. u1 ; : : : ; uµ and parametric, uµ +1 ; : : : ; um ; so that equations (3.5.12) could be solved with respect to the main variables u: Let us denote the parametric variables u by u; ¯ and their derivatives by p: ¯ Likewise, equations resulting from application of the operators of total differentiation Di to Eqs. (3.5.12), DiΨ ν = 0

(i = 1; : : : ; n);

(3.5.13)

can provide only the main derivatives p; expressing them via parametric derivatives ¯ p: For the sake of simplicity we assume that the system (S) is of the first order. Substituting the expressions u = (x; u); ¯

p = p(x; u; ¯ p) ¯

into Eqs. (S), one can find expressions for some parametric derivatives via the remaining ones, e.g. in the form ¯ p¯ = p¯ (x; u; ¯ p): Since the latter expressions are not derived by differentiation of the form (3.5.13), but from Eqs. (S), compatibility conditions of the form Di p¯ lj = D j p¯il

(3.5.14)

should hold, where derivatives of the second order can appear due to differentiation ¯ If the second derivatives have independent expressions, one has of derivatives p: to write compatibility conditions for them again, etc. If one makes no additional assumptions on the system (S), then it is rather difficult to trace the procedure up to the end in detail. In the theory of differential equations it is known as the process of reducing an “active” system (i.e. that can generate new equations according to (3.5.14)) to a “passive” system (for which conditions of the form (3.5.14) pro-


127

vide no new equations independent of the available ones). One can only claim that the resulting “passive” system consists of the proper passive system P imposed on parametric functions u; ¯ and of a system (S=H) expressing conditions of passiveness of the system (P): The system (S=H) contains no parametric functions or their derivatives and connects only invariants I; functions Ψ (I) and their derivatives up to some order. This statement is proved like the corresponding statement of Theorem 3.5. Thus, as a result of the above procedure, the system (S) is reduced into the system (P) + (S=H) having the following properties. The system (S=H) is a system with respect to functions Ψ (I) from (3.5.12). Taking any solution of (S=H); one can find all parametric functions u(x) ¯ from the system (P) and then all the main functions u(x) from (3.5.12). The resulting u¯ (x) and u(x) provide a solution of the system (S), namely a partially invariant H-solution of the rank ρ and the invariance defect δ : One can make the same remark about the system (S=H) as in the case of invariant H-solutions. Namely, the number of independent variables equals to the rank ρ of the considered partially invariant H-solution. Let us consider the system of equations of gas dynamics (3.3.12) as an example. Before searching for specific partially invariant solutions of this system, let us compose a table of all possible types of such solutions based on the relations (3.5.11). In case of the system (3.3.12), one has n = 2; m = 3; so that (3.5.11) take the form t =5

R;

0 ρ < 2;

ρ = δ +2

R;

µ =3

δ;

2; 0g δ minfR

maxfR

1; 2g:

This leads to the following table, in the last column of which we have a sketch of the invariant manifold M containing the corresponding partially invariant H-solutions. t

δ

ρ

µ

Form of M

1

4

0

1

3

I 1 ; I 2 ; I 3 (I 4 )

2

2

3

0

0

3

I 1 = C1 ; I 2 = C2 ; I 3 = C3

3

2

3

1

1

2

I 1 ; I 2 (I 3 )

4

3

2

1

0

2

I 1 = C1 ; I 2 = C2

5

3

2

2

1

1

I 1 (I 2 )

6

4

1

2

0

1

I 1 = C1

No: R 1

Solutions of the form 1 and 2 are invariant and have already been discussed in x3.3. Let us consider an example of the type 3 : Let us take the subgroup H with the operators X3 and X6 from (3.3.13). One has R = 2; t = 3: One easily obtains the invariants

128


I 1 = tu

x;

I2 =

p ; ρ

I 3 = t:

The manifold M for the type 3 is given by the equations tu

p = ψ (t) ρ

x = χ (t);

or, in the solved form, M : u=

x + ϕ (t); t

p = ψ (t)ρ :

(3.5.15)

Here, u; p are the main variables, and ρ is a parametric one. Substitution of Eqs. (3.5.15) into Eqs. (3.3.12) yields

ϕ ρx + ψ = 0; t ρ ρ ρ1 x 1 x + +ϕ + = 0; ρ t ρ t x ρ ρ1 ψ x ψ +ψ +ϕ + ψ 0 + γ = 0: ρ t ρ t

ϕ0 +

One can find from this system both parametric derivatives if ψ 6= 0: Moreover, one obtains one equation (from the second and the third ones) without ρ ; namely t ψ 0 + (γ

1)ψ = 0:

For parametric derivatives one obtains

ρx = ρ

tϕ 0 + ϕ ; tψ

tϕ 0 + ϕ ρt x = +ϕ ρ t tψ

1 t

(3.5.16)

Now one has to write the condition (3.5.14) of compatibility for these equations, namely ∂ ρx ∂ ρt = ∂t ρ ∂x ρ This yields the equation t

∂ tϕ 0 + ϕ tϕ 0 + ϕ + = 0; ∂ t tψ tψ

which together with the previous one t ψ 0 + (γ

1)ψ = 0

composes the system (S=H): The passive system P is reduced to (3.5.16). Integrating the equations (S=H); one obtains the first integrals

3.6 Reduction of partially invariant solutions

tγ

1

129

tϕ 0 + ϕ = C2 : ψ

ψ = C1 ;

Eliminating ψ from the second equation by using the first equation and then integrating, one obtains t ϕ = C2t 2 γ +C3 where C2 =

C1C2 2 γ

Finally we obtain the following solution of the system (S=H) :

ψ = C1t 1 γ ;

ϕ = C2t 1

γ

+

C3 t

Substituting this solution into (3.5.16), one obtains a totally integrable system

ρx = ρ

C2 ; t

ρt x C2 (C3 +C2t 2 γ ) = C2 2 + ρ t t2

t

Integration of the first equation of this system yields x log ρ = C2 + log ω (t); t whence,

ρ = ω (t)e

C2 xt

;

(3.5.17)

where ω (t) is a certain function which is defined from the second equation of the system. It is clear that the number of such examples can be increased to infinity by selecting subgroups of the second order of the group G56 in different ways. Generally speaking, this does not provide new solutions of the system (3.3.12) other than invariant H1 -solutions. This issue is discussed in the following section.

3.6 Reduction of partially invariant solutions 3.6.1 Statement of the reduction problem The section investigates the situation connected with the fact that any invariant manifold of a group H is at the same time an invariant manifold of any subgroup H 0 H: This follows directly from Definition 3.4, since any transformation Ta 2 H 0 belongs to H: Consequently, any partially invariant H-solution is also a partially invariant H 0 solution if the subgroup H 0 H: However, this transition from H to H 0 H changes. Generally speaking, the rank and the defect of invariance of the smallest invariant

130


manifold M containing the solution Φ : Let us agree to mark all symbols relating to the subgroup H 0 by a prime. Lemma 3.5. The rank of a partially invariant solution Φ does not decrease and its defect of invariance does not increase,

ρ0 ρ;

δ0 δ;

when turning to the subgroup H 0 H: Proof. Let us consider a partially invariant solution Φ as an H-solution and an H 0 solution simultaneously, where H 0 H: Let M and M 0 be the corresponding smallest invariant manifolds containing Φ : Manifestly M 0 M ; since M is an invariant manifold of H 0 and Φ M : Further, let t be the number of invariants in the complete set of invariants for the group H and let µ be the number of equations defining M : Then, ρ = t µ and ρ 0 = t 0 µ 0 ; respectively. Whence,

ρ0

ρ = t0

t

(µ 0

µ ):

(3.6.1)

Manifestly, t 0 t: Further, the increase of the number of invariants of the complete set t 0 t with transition from H to H 0 H cannot be less than the increase of the number of equations of invariant manifolds µ 0 µ : Indeed, otherwise M would not be the smallest invariant manifold of the group H; containing the solutions Φ : Thus, t0 and (3.6.1) provides

t µ0

µ

ρ0 ρ:

The inequality δ 0 δ follows directly from the formula

δ = dim M

dim Φ

and from the inclusion M 0 M : Let us assume now that we are searching for particular solutions of the system of differential equations (S) making progress step by step increasing the ranks of the investigated partially invariant solutions. Since the rank equals to the number of independent variables in the “resolving” system (S=H); then increase of the rank means a more difficult process of finding solutions of the system (S=H); and therefore of partially invariant H-solutions of the system (S): Therefore, if we do not want to find a partially invariant solution Φ more complicated when turning from the group H to its subgroup H 0 ; it should be required that the rank is not increased with the transition. Since by virtue of Lemma 3.5 it cannot decrease either, our requirement means that the rank keeps unaltered ρ 0 = ρ : Further, when ρ 0 = ρ ; the formulae (3.5.11) provide that δ δ 0 = t 0 t; where t and t 0 are numbers of invariants of a complete set of groups H and H 0 respectively. If δ 0 = δ ; then t 0 = t; i.e. the groups H and H 0 H have the same invariants and the transition to the subgroup H 0 provides nothing new.


131

According to the above reasoning we turn to the following formulation of the problem of “reduction” of partially invariant H-solutions. Find out if there exists a subgroup H 0 H; for which a partially invariant H-solution of the rank ρ and of the invariance defect δ is at the same time a partially invariant H 0 -solution of the same rank ρ 0 = ρ ; but of a smaller invariance defect δ 0 < δ : Every time when the answer is positive we say that a reduction of a partially invariant H-solution Φ occurs. Theorems that guarantee the existence of such reduction are called theorems on reduction. The importance of theorems on reduction is that in the above process of searching for partially invariant H-solutions they allow to avoid the abundance of useless work because partially invariant H 0 -solutions are obtained before H-solutions in this process when H 0 H: As a simple example of reduction of a partially invariant H-solution, consider the solution of the system (3.3.12) derived at the end of the previous section. The group H was a group H < X3 ; X6 >; where the operators Xα were given in (3.3.13). The resulting solution was given by the formulae (3.5.15) and (3.5.17). These formulae provide the following equations tu

x = t ϕ (t);

p = ψ (t); ρ

x

ρ eC2 t = ω (t);

where C2 =const. Let us consider the subgroup H 0 < X3 whose operator is written in detail as X3

C2 X6 = t

∂ ∂ + ∂x ∂u

C2ρ

∂ ∂ρ

(3.6.2)

C2 X6 > of the group H;

C2 p

∂ ∂p

It is readily verified that the functions I 1 = tu

x;

I2 =

p ; ρ

x

I 3 = ρ eC2 2 ;

I4 = t

provide a complete set of independent invariants of the group H 0 (i.e. of invariants of the operator X3 C2 X6 ). Since equations (3.6.2) connect only these invariants, the considered partial invariant H-solution (3.6.2), for which

ρˆ = 1;

δ = 1;

is an invariant H 0 -solution with the rank ρˆ 0 = 1 and the invariance defect δ 0 = 0: Thus, any partially invariant H-solution (3.6.2) can be reduced to an invariant H 0 solution with respect to a subgroup H 0 H: Let us emphasize that the situation is non-trivial because the subgroup H 0 varies from solution to solution since it depends on the values of the constant C2 itself. Therefore, the reduction property of partially invariant H-solutions is not purely of a group character, but depends considerably on the structure of the initial system (S).

132


3.6.2 Two auxiliary lemmas Let us prove two lemmas, the first one being a particular case of a more general theorem on the number of essential parameters of a system of functions. Consider a system of functions

ϕ k = ϕ k (x; a) = ϕ k (x; a1 ; : : : ; aτ ) (k = 1; : : : ; m)

(3.6.3)

smoothly depending on the point x 2 E n and on parameters a1 ; : : : ; ar : Let us compose two matrices ∂ϕk ; ∂ ϕ k ∂ 2ϕ k ; (3.6.4) ; ∂ aα ∂ aα ∂ xi∂ aα where α = 1; : : : ; r are the numbers of rows. Lemma 3.6. If the matrices (3.6.4) have the same general rank equal to δ ; then there are functions Bε (a) (ε = 1; : : :; δ ) and functions ϕ¯ k (x; B1 ; : : : ; Bδ ) (k=1;: : :; m); such that the following equations hold identically in the variables x; a :

ϕ k (x; a) = ϕ¯ k (x; B(a)) (k = 1; : : : ; m):

(3.6.5)

Proof. It can be assumed without loss of generality that the rank minor of the first matrix (3.6.4) is in the left upper corner, i.e. it corresponds to the values k = 1; : : : ; δ ; α = 1; : : : ; δ . Let us split the values of the index α = 1; : : : ; r into values σ = 1; : : : ; δ and τ = δ + 1; : : :; r: The conditions of the lemma provide that the last r δ rows of the first matrix (3.6.4) are linear combinations of the first δ rows: k ∂ϕk σ ∂ϕ = λ t ∂ aτ ∂ aσ

(k = 1; : : : ; m; τ = δ + 1; : : :; r);

(3.6.6)

where, generally speaking, λτσ = λτσ (x; a): Since the second matrix (3.6.4) has the same rank as the first one and contains the first matrix as a part, then the relations

∂ 2ϕ k ∂ 2ϕ k σ = λ (x; a) τ ∂ xi ∂ a τ ∂ xi ∂ aσ hold with the same λτσ : By virtue of these relations, differentiation of Eqs. (3.6.6) with respect to xi provides the new relations

∂ λτσ ∂ ϕ k = 0 (k = 1; : : : ; m; i = 1; : : : ; n) ∂ xi ∂ a σ expressing the fact of linear dependence of the first δ rows of the first matrix (3.6.4). But the first δ rows are linearly independent by condition. Hence

∂ λτσ = 0 (i = 1; : : : ; n): ∂ xi


133

The latter equations mean that the coefficients λτσ in Eqs. (3.6.6) are independent of x and can only be functions of parameters a at most: λτσ = λτσ (a): Therefore, equations (3.6.6) can be considered as equations to which functions ϕ k (x; a) satisfy as functions of parameters a with any fixed x: One can readily see that these equations generate a complete system (see definition (3.3.18)). Indeed, the rank of the matrix composed by coordinates of the operators Yτ =

∂ ∂ aτ

λτσ (a)

∂ ∂ aσ

(τ = δ + 1; : : :; r)

is obviously equal to r δ ; and the commutator [Yτ 1 ;Yτ 2 ] of any two of them should be expressed linearly through these operators. Otherwise, we would have some linear dependencies among the first δ rows of the first matrix (3.6.4) which contradicts the assumption. Note that the system (3.6.6) contains r independent variables and r δ equations for a fixed k: Therefore, according to Lemma 3.2, it has r (r δ ) = δ functionally independent solutions, that can be chosen to be functions of variables a only. Let us denote these solutions by Bε (a) (ε = 1; : : : ; δ ): Since every function ϕ k (x; a) satisfies the same system as B(a); then equalities of the form (3.6.5) should hold with the derived B(a) due to Lemma 3.2. Lemma 3.6 is proved. The second lemma deals with some property of the prolongation of the group GNr : We assume as before that a basis of the Lie algebra LNr is given by the operators Xα = ξαi

∂ ∂ + ηαk k i ∂x ∂u

(α = 1; : : : ; r)

(3.6.7)

and that the prolongation of these operators with respect to functions u(x) has the form ∂ Xeα = Xα + ζαk i k : ∂ pi Consider the matrices of coordinates of the operators Xα and Xeα : e = ξαi ; ηαk ; ζ k : M = ξαi ; ηαk ; M αi Lemma 3.7. If the group GNr is intransitive and the general ranks of the matrices M e are equal to each other, then the operators (3.6.7) are linearly unconnected. and M Proof. Let us assume that the operators (3.6.7) are linearly connected on the contrary to the statement. We can assume, without loss of generality, that the first R of the operators Xσ (σ = 1; : : : ; R) are linearly unconnected, and that the last r R operators Xτ are expressed via the first ones by the formulae Xτ = ωτσ Xσ

(τ = R + 1; : : :; r);

(σ = 1; : : : ; R);

(3.6.8)

where ωτσ = ωτσ (x; u): Let us lead this assumption to a contradiction proving that e implies that all ωτσ are constants. Then equality of the ranks of matrices M and M equations (3.6.8) would mean that the operators (3.6.7) are linearly dependent and hence do not provide a basis of LNr :

134


The equation e =R R(M) = R(M) shows that the prolonged operators should also be linearly connected with the same coefficients ωτσ : Xeτ = ωτσ Xeσ (τ = R + 1; : : :; r): (3.6.9) Indeed, otherwise one could find a non-vanishing minor of the order R + 1 in the e Let us write Eqs. (3.6.8) for the coordinates of the operators (3.6.7): matrix M:

ξτi = ωτσ ξσi ;

ητk = ωτσ ησk :

(3.6.10)

Taking into account Eqs. (3.6.10) we write the relations (3.6.9) in the coordinate form as follows: ζτki = ωτσ ζσk i or, by virtue of the formulae (1.4.10) of Chapter 1, Di (ητk )

pkj Di (ξτ ) = ωτσ [Di (ησk ) j

pkj Di (ξσ )]: j

However, equation (3.6.10) provide the equalities Di (ξτj ) = ωτσ Di (ξσj ) + ξσj Di (ωτσ ); Di (ητk ) = ωτσ Di (ησk ) + ησk Di (ωτσ ); by virtue of which the previous relation is written

ησk Di (ωτσ )

pkj ξσ Di (ωτσ ) = 0; j

or

σ σ σ ∂ ωτσ j ∂ ωτ k j ∂ ωτ l k k ∂ ωτ l + η p ξ p ξ p p = 0; (3.6.11) σ σ i j σ ∂ xi ∂ ul ∂ xi ∂ ul i j where i = 1; : : : ; n; k = 1; : : : ; m; τ = R + 1; : : : ; r: Since the functions ξ ; η ; ω are independent of the variables p; and (3.6.11) must be an identity with respect to independent variables x; u; p; equations (3.6.11) are easily “split” with respect to variables p; which leads to three series of equations

ησk

ησk

∂ ωτσ = 0; ∂ xi

δij ησk ξσj

∂ ωτσ ∂ ul

∂ ωτσ = 0: ∂ ul

(a)

δlk ξσj

∂ ωτσ = 0; ∂ xi

(b) (c)

Let us consider the matrix MR = (ξσi ; ησk ) similar to the matrix M; but composed of coordinates of only the first R operators (3.6.7). Let us assume that σ are numbers of columns in the matrix MR ; so that it has R columns and N rows. Since the group


135

GNr is intransitive, then R < N and there is a row in the matrix MR after eliminating which the remaining matrix MR0 still has the rank R: Let it be the row with the number i0 for example (the reasoning for a number k0 is similar). If we let that i = i0 and j 6= i0 in Eqs. (a), (b) and introduce the quantities

∂ ωτσ = zσ0 ; ∂ xi 0 we obtain the system of linear homogeneous equations

ησk zσ0 = 0;

ξσj zσ0 = 0 ( j 6= i0 ):

The matrix of the latter system is exactly the matrix MR0 : Since there are exactly R “unknown” zσ0 ; then R(MR ) = R provides that zσ0 = 0 (σ = 1; : : : ; R): We let now i = j = i0 in Eqs. (b) and denote

∂ ωτσ = vσ : ∂ ul Then, along with Eqs. (c), we obtain the following system of linear homogeneous equations with the matrix MR :

ησk vσ = 0;

ξσj vσ = 0

for vσ (σ = 1; : : : ; R): Whence, vσ = 0 (σ = 1; : : : ; R) as well. Finally, denoting

∂ ωτσ = zσ ∂ xi and taking k = l in (b), one obtains the equations ησk zσ = 0; whence it follows that

zσ = 0

ξσj zσ = 0;

(σ = 1; : : : ; R)

as above. This completes the proof of Lemma 3.7. Note that the requirement of intransitiveness of the group GNr is essential. The following example demonstrates that Lemma 3.7 can appear to be not valid for transitive groups. Consider G23 with the operators X1 =

∂ ; ∂x

X2 =

∂ ; ∂u

X3 = x

∂ ∂ +u ∂x ∂u

136


Prolonging these operators with respect to the function u(x); one obtains that Xeα = Xα

(α = 1; 2; 3):

e = 2: However, the statement of Lemma 3.7 is not satisfied, Therefore, R(M) = R(M) since the operators Xα are linearly connected: X3 = xX1 + uX2 : This happened due to the fact that the group G23 under consideration is transitive.

3.6.3 Theorem on reduction Let us turn back to the problem of reduction of partially invariant H-solutions. It was shown that the system (S) splits into the system (P) and the system (S=H) for such solutions. We reveal reduction for the case when the following property of the systems of equations holds. Property 3.1. The system (S) is of the first order and equations (P) make it possible to find expressions for all derivatives of the first order pki (i = 1; : : : ; n; k = 1; : : : ; m): Theorem 3.8. For any partially invariant H-solution of the rank ρ ; where the Property 3.1 is satisfied, there is a subgroup H 0 H such that the solution is an invariant H 0 -solution of the rank ρ 0 = ρ : Proof. Let Φ be the considered partially invariant H-solution and Φa be the manifold derived from Φ under the action of the transformation Ta 2 H: Let us write equations for Φa in the form

Φa :

uk = ϕ k (x; a) (k = 1; : : : ; m):

(3.6.12)

Consider the Jacobi matrix

∂ ϕ k (x; a) ∂ aα

(3.6.13)

and assume that its general rank is equal to δ : The number δ is the invariance defect of the H-solution Φ : Indeed, if we assume, without loss of generality, that the rank minor of the matrix (3.6.13) is in the upper left corner, then the first δ equations of (3.6.12) allow one to express the parameters aσ (σ = 1; : : : ; δ ) via the variables x; u and the remaining parameters denoted here by a¯ : If one substitutes the resulting expressions into the remaining m δ equations (3.6.12), then one obtains equations without parameters a¯ (otherwise, the rank of the matrix (3.6.13) would be higher than δ ). These are equations of the smallest invariant manifold M containing the solution Φ : The number of resulting equations is equal to µ = m δ and according


137

to definition (3.4.3) the invariance defect of the manifold Φ equals to N (m δ ) n = δ : If t is the number of functionally independent invariants of the complete set for the group H; then the rank of Φ equals to ρ = t (m δ ) = t + δ m: Differentiating Eqs. (3.6.12), one obtains expressions for all derivatives pki =

∂ ϕ k (x; a) ∂ xi

If one substitutes the known expressions aσ = aσ (x; u; a) ¯ into the latter equations then the result will contain no variables a¯ : Indeed, the resulting equations should be equivalent (on solution Φ ) to equations of the passive system (P) by virtue of Property 3.1. Furthermore, the system (P) is invariant with respect to the group H and does not contain the parameters a: It follows that the rank of the matrix ∂ ϕ k (x; a) ∂ 2 ϕ k (x; a) ; ∂ aα ∂ xi∂ aα is equal to the rank of the matrix (3.6.13), i.e. δ : By virtue of Lemma 3.6, one can make a conclusion that there are functions B1 (a); : : : ; Bδ (a) such that the right-hand sides of (3.6.12) depend only on x and the variables B: Therefore, the part of Eqs. (3.6.12) which is not invariant can be reduced to the form

ψ σ (x; u) = Bσ (a) (σ = 1; : : : ; δ ):

(3.6.14)

Note that according to the above assumption this part is the first δ equations of (3.6.12). Consider possible transformations Ta 2 H satisfying the system of equations Bσ (a) = Bσ (0) (σ = 1; : : : ; δ ):

(3.6.15)

Since Φ0 Φ ; all such Ta have the property of leaving the manifold Φ invariant. Further, if Φ is subjected first to the transformation Ta ; and then to the transformation Tb from the set of transformations with the property (3.6.15), then one obtains the transformation Tb Ta = Tϕ (a;b) having the property (3.6.15) again. Therefore, the set of all Ta 2 H with the property (3.6.15) is a group, namely a subgroup H 0 H: Since all Ta 2 H 0 leave the manifold Φ invariant, we conclude that Φ is an invariant H 0 -solution. Note that (3.6.15) is exactly an (r δ )-dimensional manifold in the parametric space of the group H; for otherwise M would not be the smallest manifold containing Φ : Therefore, the order of the group is equal to r0 = r δ : Let us prove the statement about the rank. Let R be the rank of the matrix M composed from the coordinates of basis operators of the group H: Then the number e Since the of invariants is equal to t = N R: Let us consider a prolonged group H: k invariant equations (P) allow to find expressions for all derivatives pi due to Property 3.1, the increase of the number of invariants in transition from H to H 0 is not to be smaller than the number of these variables pki ; i.e. mn: This follows from Theorem 3.3. But this increase can not be greater than mn; for the dimension of the space

138


is increased by this number. Thus, the number of functionally independent invariants of the prolonged group is equal to t˜ = t + mn: Whence, according to Theorem 3.1, e from coordinates of the prolonged basis one obtains that the rank of the matrix M e operators of the group H equals to R = N + mn (t + mn) = R: Since the group H is intransitive (δ < m); Lemma 3.7 shows that the basis operators of the group H are linearly unconnected and hence, the order of the group is r = R: Basis operators of the subgroup H 0 H are obviously linearly unconnected as well. Thus the order of the subgroup H 0 is equal to r0 = R0 : Then, one obtains

ρ0 = n

R0 = n

r0 = n

r+δ = n

R+δ = ρ

for the rank ρ 0 of the invariant H 0 -solution of Φ ; which was to be proved. As an example of application of Theorem 3.8 consider the equations of gas dynamics (3.3.12) and find partially invariant solutions of the type 3 from Table in x3.5.3. Let us make an additional requirement: the desired H-solutions should be irreducible to invariant H 0 -solutions with respect to a subgroup H 0 H: While it is unknown on which groups H to search for the solutions, it is known that the smallest invariant manifold M of the group H containing them is to be given by two equations. Consider a case when these equations can be solved with respect to the main variables u; p : u = f (t; x; ρ ); p = g(t; x; ρ ); (3.6.16) so that the parametric variable is ρ : By virtue of Theorem 3.8, the above requirement is reduced to making it impossible to find both parametric derivatives ρt ; ρx from equations resulting from the substitution of expressions (3.6.16) to Eqs. (3.3.12). Upon substituting (3.6.15) into (3.3.12), one obtains (ρ 2 fρ2

gρ )ρx = ρ ft + ρ ( f

fρ (ρ gρ

γ g)ρx = gt + f gx

ρ f ρ ) f x + gx ; (ρ gρ

γ g) fx ;

(3.6.17)

ρt + ( f + ρ fρ )ρx + ρ fx = 0: It is not possible to find both derivatives ρt ; ρx from these equations only if (the general case) ρ 2 fρ2 = gρ ; ρ gρ = γ g: Let us limit our consideration by the case γ 6= 0; 1; 3: Then the general solution of the latter equations is written in the form g=

a2 γ ρ ; γ

f =

2a

γ

1

ρ

γ 1 2

+ b;

where a = a(t; x); b = b(t; x) are arbitrary functions. Substituting these expressions into the first two equations (3.6.17), one can see that they are reduced to at = ax = bt = bx = 0;

3.7 Some problems

139

so that in fact a =const., b =const. Thus, equations of invariant manifolds (3.6.16) containing a partially invariant H-solution of the system (3.3.12) can be only as follows: 2a γ 2 1 a2 M : u= ρ + b; p = ρ γ ; (3.6.18) γ 1 γ provided that this solution is irreducible to an invariant one. Recall that G56 is the group generated by the operators (3.3.13). One can easily verify that subgroups H G56 having (3.6.18) as an invariant manifold exist indeed. The subgroup H of the maximum order is H4 and has the form 2γ H4 = H X1 ; X2 ; X4 ; X5 bX3 + X6 : γ 1 Solutions of Eqs. (3.3.12), where the variables are connected by the relations (3.6.18), are known in gas dynamics as simple waves.

3.7 Some problems In conclusion of the present lecture notes let us discuss some problems useful for further development of the theory and applications of group properties of differential equations. Based on the fact that a theory is enriched by accumulating examples of its application we point out that it is desirable to have calculated admissible groups for possibly wider classes of partial differential equations. In particular, the following problems, unsolved so far, can be singled out. Problem 3.1. Find the group admitted by an arbitrary linear differential equation with constant coefficients P(D)u = 0; where D is the vector

∂ ∂ ;:::; n ∂ x1 ∂x

;

and P(D) is a polynomial with constant coefficients. Problem 3.2. Find the group admitted by a system of linear partial differential equations of the first order with constant coefficients. Problem 3.3. Make the group classification of equations of magnetohydrodynamics in the three-dimensional case. Problem 3.4. Find the group admitted by Einstein’s equations from the general relativity.

140


For some systems of equations the admissible group is already known, but there is no complete classification of partially invariant solutions. In particular, this is the case with equations of gas dynamics. Problem 3.5. Classify partially invariant solutions of equations of gas dynamics in two-dimensional and three-dimensional cases. At present there are a lot of examples of nonlinear systems of equations admitting an infinite Lie algebra of operators. However, the issue of using an infinite Lie algebra for constructing classes of partial solutions of such systems of equations is not sufficiently investigated. Problem 3.6. Elaborate efficient algorithms of using an admissible infinite Lie algebra for constructing classes of partial solutions of the corresponding equations. Classifying partial solutions, e.g. invariant H-solutions, we face the necessity to construct classes of similar subalgebras of a given Lie algebra. In some cases the problem is investigated, however there are difficulties in calculation in applications. Problem 3.7. Elaborate efficient algorithms of constructing classes of similar subalgebras of a given Lie algebra. Together with solutions derived on the basis of finite invariants of a group GNr ; one can pose a question on finding solutions with application of differential invariants. It is possible that this will enrich the stock of partial solutions provided by group properties of a system (S): This issue is probably not investigated at all. Problem 3.8. Develop a theory of differential invariant and partially differential invariant solutions of differential equations with a known admissible group. When searching for invariant H-solutions of the system (S), one obtains a new system (S=H): A group admitted by the system (S=H) should be somehow connected with the properties of the system (S), in particular, with the group G admitted by the system (S). The only result that can be easily obtained is the following: if H is a normal subgroup in G; then the system (S=H) admits the factor group G=H: However, examples demonstrate that the most general group admitted by the system (S=H) can be considerably wider than the factor group. Problem 3.9. Develop methods for finding the group admitted by the system (S=H) directly in terms of a system (S) and its admitted group H: An important part in enumerating partially invariant solutions of a system (S) is played, as we could already see, by properties of reduction of such solutions to a smaller invariance defect. Only a particular case of such reduction, based on Property 3.1, is mentioned in the lecture notes. The general situation with reduction of partially invariant solutions is not clear. Problem 3.10. Find theorems on reduction for systems (S) and groups H with more general properties than Property 3.1.

3.7 Some problems

141

In applied theories connected with solution of problems for differential equations a given system (S) is often modelled by a simpler system (S ): A thorough investigation of particular cases of such modelling demonstrates that (S ) always appears to be admitting a wider group than (S). Problem 3.11. Find general principles of modelling a given system (S) by a simpler system (S ) whose admitted group is wider than that of (S): Practice shows that calculation of operators admitted by specific systems (S) with a large number of variables requires a lot of almost mechanical work on writing out and elementary transformations of a huge number of equations, in short, a lot of elementary algebraic calculations. This part of work can obviously be “delegated” to computers. Problem 3.12. Develop a computer algorithm of algebraic calculations for maximum simplification of systems of determining equations in calculating operators admitted by a given system (S):

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