Full Text
MATHEMATICS
L. V. OVSYANNIKOV
ON FINDING THE GROUP OF A LINEAR DIFFERENTIAL EQUATION OF SECOND ORDER
(Presented by Academician I. N. Vekua on 28 XII 1959)
In the present paper we report some results connected with the problem of finding the group of transformations preserving a given linear differential equation with partial derivatives of second order for an arbitrary number \(n\) of independent variables \((n>1)\). After deriving the determining equations of the infinitesimal transformations, we establish an invariant form of these equations, containing only the Riemannian space associated with the given equation, a vector of this space, and an invariant. A final solution of the problem has not yet been obtained; here we give only some general facts.
Let us note that the original general formulation of such problems—the problems of group classification of systems of differential equations—belongs to S. Lie \((^{1})\). In particular, he obtained a classification of linear equations of second order for the case \(n=2\), though in noninvariant form. This formulation differs from that recently proposed in the work of I. M. Gelfand \((^{2})\), where the unknowns are equations admitting a given group.
- Let \(u=u(x^1,x^2,\ldots,x^n)\), \(u_i=\partial u/\partial x^i\), \(u_{ij}=\partial^2u/\partial x^i\partial x^j\). We introduce notation for systems of values of the quantities indicated in parentheses:
\(x=(x^1,x^2,\ldots,x^n)\), \(v=(x,u)\), \(p=(v,u_1,u_2,\ldots,u_n)\), \(q=(p,u_{11},u_{12},\ldots,u_{nn})\).
Consider the equation
\[ F(q)\equiv a^{ij}u_{ij}+b^i u_i+cu=0 \quad (a^{ij}=a^{ji}), \tag{1} \]
where \(a^{ij}\), \(b^i\), \(c\) are prescribed analytic functions of \(x\), and the tensor convention of summation over a repeated index is used (as below) (all indices take the values \(1,2,\ldots,n\)).
The problem is posed as follows: to find all transformations of the form
\[ p'=f(p), \tag{2} \]
belonging to certain one-parameter Lie groups and preserving equation (1) in the sense that \(F(q')=0\) whenever \(F(q)=0\).
These transformations generate a local Lie group \((^{3})\) of transformations, denoted below by \(G_F\). In this case one says that equation (1) admits the transformations (2), or, respectively, the group \(G_F\).
We observe that \(G_F\) always contains transformations distinct from the identity. Such transformations are
\[ x'=x,\qquad u'=\alpha u+u_0(x), \tag{3} \]
where \(\alpha\) is an arbitrary parameter, and \(u_0(x)\) is any fixed solution of equation (1). The set of all transformations of the form (3) is itself a group and therefore a subgroup of the group \(G_F\). We shall denote this subgroup by \(T\).
- A one-parameter group of transformations of the form (2) is completely characterized by its infinitesimal operator
\[ Y=\xi^i(p)\frac{\partial}{\partial x^i}+\eta(p)\frac{\partial}{\partial u}+\zeta_i(p)\frac{\partial}{\partial u_i}, \tag{4} \]
which can be prolonged to the second derivatives \(u_{ij}\) and yield an operator acting on functions of \(q\):
\[ \widetilde Y=\xi^i\frac{\partial}{\partial x^i}+\eta\frac{\partial}{\partial u}+\zeta_i\frac{\partial}{\partial u_i} +\omega_{ij}\frac{\partial}{\partial u_{ij}} . \tag{5} \]
Here the quantities \(\zeta_i\) and \(\omega_{ij}\) are computed by the known prolongation formulas
\[ \zeta_i=D_i(\eta)-u_jD_i(\xi^j),\qquad \omega_{ij}=D_i(\zeta_j)-u_{jl}D_i(\xi^l), \tag{6} \]
where the differential operations \(D_i\) are defined by the formula
\[ D_i=\frac{\partial}{\partial x^i}+u_i\frac{\partial}{\partial u} +u_{ij}\frac{\partial}{\partial u_j}\qquad (i=1,2,\ldots,n). \tag{7} \]
In terms of the operator (4), the problem posed is formulated as follows: find all operators \(Y\) such that \(\widetilde YF(q)=0\) whenever \(F(q)=0\). In more detailed notation this invariance condition has the form
\[ \left[ a^{ij}\omega_{ij}+b^i\zeta_i+c\eta +\xi^k\left( \frac{\partial a^{ij}}{\partial x^k}u_{ij} +\frac{\partial b^i}{\partial x^k}u_i +\frac{\partial c}{\partial x^k}u \right) \right]_{F(q)=0}=0. \tag{8} \]
According to the statement of the problem, the quantities \(\zeta_i\) must not depend on \(u_{ij}\). Therefore the aggregate of the terms in the expressions for \(\zeta_i\) according to formulas (6), (7) that contain \(u_{ij}\) must vanish under the condition \(F(q)=0\).
For \(n>1\) the following holds:
Lemma 1. In order that the quantities \(\zeta_i\) not depend on \(u_{ij}\) when \(F(q)=0\), it is necessary and sufficient that there exist a function \(\varphi=\varphi(p)\) such that
\[ \xi^i=-\frac{\partial\varphi}{\partial u_i},\qquad \eta=\varphi-u_i\frac{\partial\varphi}{\partial u_i},\qquad \zeta_i=\frac{\partial\varphi}{\partial x^i}+u_i\frac{\partial\varphi}{\partial u} \qquad (i=1,2,\ldots,n). \tag{9} \]
- We shall call an equation \(F'(q')=0\) equivalent to equation (1) if it is obtained from (1) as a result of a change of variables (substitution) of the form
\[ x'=x'(x),\qquad u(x)=\psi(x)u'(x') \tag{10} \]
and subsequent division by \(\psi(x)\). If in (10) \(x'\equiv x\), then the corresponding equation \(F'(q')=0\) will be called equivalent in the function to equation (1).
Under the change of variables (10), the group \(G_F\) is transformed into a similar group \(G_{F'}\). Therefore the problem of finding \(G_F\) is equivalent to the problem of finding any \(G_{F'}\).
Lemma 2. Except for the case when (1) is equivalent to the equation \(u_{12}=0\), the function \(\varphi(p)\) is linear in the variables \(u,u_1,u_2,\ldots,u_n\).
In what follows we shall set aside the case of the equation \(u_{12}=0\). Then from Lemma 2, in view of (9), it follows that \(\xi^i=\xi^i(x)\) \((i=1,2,\ldots,n)\), \(\eta=\sigma(x)u+\tau(x)\). Therefore the transformations \(G_F\) can be represented as transformations of the form \(v'=f(v)\). Moreover, as a more detailed consideration shows, the function \(\tau(x)\) must be a solution of equation (1). Hence, in conjunction with the analytic sign of a normal divisor \({}^{4}\), it follows
Lemma 3. The subgroup \(T\) is a normal divisor in \(G_F\). The transformations of the factor group \(G_F/T\) are characterized by infinitesimal operators of the form
\[ X=\xi^i(x)\frac{\partial}{\partial x^i}+\sigma(x)u\frac{\partial}{\partial u}, \tag{11} \]
where \(\sigma(x)\) is determined up to a constant summand.
By Lemmas 2 and 3 and formulas (6), (7), (9), condition (8), whose left-hand side turns out to be a linear function of the quantities \(u_{ij}\), “splits” and gives rise to the following system of equations:
\[ a^{ik}\frac{\partial \xi^j}{\partial x^k} +a^{jk}\frac{\partial \xi^i}{\partial x^k} -\frac{\partial a^{ij}}{\partial x^k}\xi^k =\mu a^{ij} \quad (i,j=1,2,\ldots,n); \tag{12} \]
\[ 2a^{ik}\frac{\partial \sigma}{\partial x^k} =a^{kl}\frac{\partial^2 \xi^i}{\partial x^k\partial x^l} +b^k\frac{\partial \xi^i}{\partial x^k} -\frac{\partial b^i}{\partial x^k}\xi^k -\mu b^i \quad (i=1,2,\ldots,n); \tag{13} \]
\[ a^{ij}\frac{\partial^2 \sigma}{\partial x^i\partial x^j} +b^i\frac{\partial \sigma}{\partial x^i} = -\frac{\partial c}{\partial x^i}\xi^i-\mu c, \tag{14} \]
where \(\mu=\mu(x)\) is an auxiliary function to be eliminated from this system. Equations (12)—(14) are the determining equations of the infinitesimal transformations of the group \(G_F/T\). With the aid of these equations the following theorem is easily proved (the converse of which is trivial):
Theorem 1. If equation (1) admits at least one (nonzero) operator (11), then it is equivalent to an equation all of whose coefficients do not depend on one of the coordinates.
4. In what follows we shall restrict ourselves to the assumption that the total rank of the matrix \(\|a^{ij}\|\) is equal to \(n\). Then, observing that under a change of independent variables (10) the coefficients \(a^{ij}\) transform as components of a contravariant tensor, we may introduce into consideration the covariant components of this tensor \(a_{ij}\) \((a_{ij}a^{jk}=\delta_i^k)\). Thus with equation (1) there is associated a Riemannian space \(V_n\) with fundamental form \(a_{ij}dx^i dx^j\). We shall proceed to find forms of the determining equations (12)—(14) that are invariant with respect to \(V_n\).
Since equations (12) are nothing other than the determining equations of conformal transformations of \(V_n\) (5), the preceding considerations lead to the following theorem.
Theorem 2. The group \(G_F/T\) of a nondegenerate equation (1) is a subgroup of the group of conformal transformations of the Riemannian space \(V_n\) associated with this equation.
It follows from this, in particular, that for \(n>2\) the group \(G_F/T\) has order not exceeding \((n+1)(n+2)/2\).
The coefficients \(b^i\) of equation (1) are not components of a vector. However, it is easy to verify that the quantities
\[ a^i=b^i+a^{kl}\Gamma^i_{kl} \quad (i=1,2,\ldots,n), \tag{15} \]
where \(\Gamma^i_{kl}\) are the Christoffel symbols of the second kind, computed for the associated \(V_n\), already form a contravariant vector.
Equations (13) may be regarded as equations serving to determine the function \(\sigma(x)\). If, by means of the covariant components \(a_i\) of the vector \(a^i\), one forms the skew-symmetric tensor \(K_{ij}=a_{i,j}-a_{j,i}\) (commas denote covariant derivatives with respect to the tensor \(a_{ij}\)), then the integrability conditions for equations (13) can be given the following invariant form:
\[ (K_{il}\xi^l)_{,j}=(K_{jl}\xi^l)_{,i} \quad (i,j=1,2,\ldots,n). \tag{16} \]
Equation (14), by eliminating \(\sigma\) with the aid of (13), is reduced to the following invariant form:
\[ H_{,i}\xi^i+\mu H=0, \tag{17} \]
where \(H\) is the invariant defined by the formula
\[ H=-2c+a^i{}_{,i}+\frac12 a^i a_i+\frac{n-2}{2(n-1)}R, \tag{18} \]
where \(R\) is the scalar curvature of the associated \(V_n\).
The analytic nature of the quantities \(K_{ij}\) and \(H\) is clarified by the following proposition:
Lemma 4. In order that two equations of the form (1) with the same \(V_n\) be equivalent with respect to a function, it is necessary and sufficient that these equations have the same tensor \(K_{ij}\) and invariant \(H\).
Thus, the collection of quantities \(K_{ij}\) and \(H\) is a generalization of the Laplace invariants known for the case \(n=2\) \((^6)\).
- If one regards the group of conformal transformations of \(V_n\) as an already studied object, then there next arises the problem of investigating the restrictions of geometric or group-theoretic character imposed on the group \(G_F/T\) by the additional conditions (16) and (17).
An important property of \(H\) is given by the following lemma:
Lemma 5. In passing from equation (1) to the equation \(g(x)F(q)=0\), the invariant \(H\) is multiplied by \(g(x)\).
This property shows that it is necessary to distinguish the cases \(H=0\) and \(H\ne0\). We shall consider here only the latter.
If \(H\ne0\), then, by Lemma 5, without loss of generality one may take \(H=1\). Then from (17) it follows that \(\mu=0\), which proves the following theorem:
Theorem 3. If \(H\ne0\), then \(G_F/T\) is a subgroup of the group of motions of some Riemannian space conformally associated with \(V_n\).
As a particular consequence of this we obtain that, for \(n>2\), the order of the group \(G_F/T\) of an equation for which \(H\ne0\) does not exceed \(n(n+1)/2\).
Of interest is the question of the cases in which the group \(G_F/T\) may have the greatest possible order. We shall agree in general to call a standard form of equation (1), admitting \(G_F/T\) with the given properties, the simplest of the possible forms of this equation obtained by passing to an equivalent equation and multiplying by some factor.
Let us note that if, for \(H\ne0\), equation (1) admits \(G_F/T\) of maximal order, then \(V'_n\), associated with the equation \((1/H)F(q)=0\), has constant curvature \(K_0\) \((^5)\). Hence, after some computations, one obtains:
Theorem 4. For \(n>2\), a standard form of equation (1) admitting \(G_F/T\) of order \(n(n+1)/2\) is either \((K_0=0)\)
\[ \Delta u+u=0, \]
or \((K_0\ne0)\)
\[ \Delta u+\frac{n(n-2)}{(1+r^2)^2}u=0, \]
where \(\Delta\) is the Laplace operator, \(r^2=(x^1)^2+(x^2)^2+\cdots+(x^n)^2\).
Institute of Hydrodynamics
Siberian Branch
Academy of Sciences of the USSR
Received
23 XII 1959
REFERENCES
\(^1\) S. Lie, Arch. f. Math., 6, Heft 3 (1881).
\(^2\) I. M. Gel'fand, DAN, 70, No. 1 (1950).
\(^3\) L. S. Pontryagin, Continuous Groups, Moscow, 1954.
\(^4\) N. G. Chebotarev, Theory of Lie Groups, Moscow–Leningrad, 1949.
\(^5\) L. P. Eisenhart, Riemannian Geometry, IL, 1948.
\(^6\) G. Darboux, Leçons sur la théorie générale des surfaces, Paris, 1889.