6.

7.

V. N. Tulovskii and M. A. Shubin, "The asymptotic distribution of the eigenvalues of pseudodifferential opera tors in l~n, w Mat. Sb., 92, 571-588 (1973). V. I. Feigin, WAsymptotic distribution of eigenvalues for hypoelliptic sys tems in Rn," Mat. Sb. (N. S.), 9_99, No. 4, 594-614 (1976).

SPECTRAL PROPERTIES

OPERATOR PENCILS

A. I. Miloslavskii

OF A CLASS OF QUADRATIC

UDC 517.43

A ser ies of dynamical problems of sys tems ca r ry ing a moving distributed load leads to the study of stability of solutions of the equation

~--~-~ B~+Aw = 0, (i) d t 2 - - d t

where w = w(t) is a function with values in a Hilbert space H. Concerning the opera tors in Eq. (1) we will as - sume the following.

a) The operator A has the form A = A+ + A1, where the operator A+ is positive self-adjoint and its in- ve rse A+ I is completely continuous. The symmet r i c operator A 1 is subordinate to A+ in the sense that

A1 = DAI+/2, where D is a bounded operator .

b) The operator B is skew-symmet r i c and subordinate to A+ in the sense that B = CAI+/2, where C is a bounded operator .

For Eq. (1) we consider the Cauchy problem

d~_ t=o (2) w (0) =: wo, = wl.

Definition 1. By a solution of the Cauehy problem (1), (2) on an interval (a, b), where - ~ _< a _< 0 -< b -< ~, we mean a twice continuously differentiable function w (t) ~ D (A) such that A1/2w(t) is differentiable and w satisfies relations (1) and (2).

THEOREM 1. For any w0 ~D (A), w, ~D (A~) there exists a unique solution of the Cauehy problem (1), (2).

Definition 2. By the stability of the solutions of Eqs. (I), (2) on the positive semiaxis (on the axis) we understand the inequality

I~-f~ 2+ (A+w, w) ~ C(llwlll~ + (A+wo, Wo))

on the semiaxis (on the axis) with a constant C independent of t, w0, and wl.

In seeking the solutions of Eq. (1) in the fo rm w (t) = e ~t z, :: ~ H, we ar r ive at the problem of eigenvalues of the quadratic operator pencil L(~):

L (~) x = (A -q- )~ B -q- k~I) x = 0. (3)

THEOREM 2. The spec t rum g(L) of the pencil L(k) consists of isolated points of finite a lgebraic multi- plicity; these points either lie on the imaginary axis or are distributed symmet r ica l ly with respec t to this axis and are in the disk I hl -< 0.5[IDll.

Let {kj} be the eigenvalues of L(k) among which there are no eigenvalues symmet r i c with respec t to the imaginary axis. We denote by p(h) the algebraic multiplicity of a not purely imaginary eigenvalue X. If h is a

purely imaginary eigenvalue, then by p (h) we denote p (~)=~ [di/2], where dl, d 2 , . . . , d r a re the nonsimple

Ukrainian Correspondence Polytechnic Institute. Translated from Funktsional'nyi Analiz i Ego Prilo- zheniya, Vol. 15, No. 2, pp. 81-82, April?-June, 1981. Original article submitted June 2, 1980.

142 0016-2663/81/1502- 0142 $07.50 © 1981 Plenum l:'ublishing Corporat ion

elementary divisors corresponding to X.* Then we have the inequality

where z is the number of nonpositive eigenvalues of A, counted with multiplicity.

The system of eigenvectors and associated vectors of L(~) forms a bivariate Riesz basis in H @ //,

equipped with the norm Jl • []+,

II x II-,- ~ = It z~. II ~ + 11 A+V% I1 "% z = (X l , x2), x i ~ H , x 2 (~ D ( A ~ ) .

Definition 3. The pencil L(X) is said to be stable if all of its eigenvalues are purely imaginary and their algebraic multiplicity is equal to 1.

THEOREM 3. The stability of solutions of Eqs. (1), (2) (whether on the semiaxis or on the axis) is equiv- alent to the stability of the pencil L(k).

THEOREM 4 (Generalization of the Thompson-Tare Theorem). Suppose the following conditions are satisfied:

I) The operator A is self-adjoint and semibounded from below; the operator A -I exists and A -I ~ op (for the definition of the ideal ~p of completely continuous operators, cf. [1]).

2) The operator B can be writ ten in the form B = G(A + ~I) °~, where 0 -< ~ < 1/2, G is abounded0pera tor and the constant ? is chosen so that A + ?I is positive.

3) The complex Hilbert space H is the complexification of a real Hilbert space and the operators A and B are real.

Then the pencil L(I) has an even (odd) number of positive eigenvalues, considering algebraic multiplici- t ies, according as the operator A has an even (odd) number of negative eigenvalues, considering multiplicities.

If conditions a), b), and 3) are satisfied, A is invertible and has an odd number of negative eigenvalues, then L(I) has at least one positive eigenvalue and the solutions of Eqs. (1), (2) are unstable on the positive semiaxis.

THEOREM 5. Suppose conditions a), b), and 3) are satisfied and A is invertible. Furthermore, assume

that the pencil L(~) has positive eigenvalue X 0. Then we have the inequality

I ~ I (t + II C*A-I clJ)-~< ~ < I ~ I,

where At(;%, j is that negative eigenvalue of A which is the smal les t (largest) in absolute value and A, C = ( I - P)BP, where P is the orthogonal projection corresponding to the negative portion of the spec t rum of A.

The proofs of Theorems 2, 3, and 4 use resul ts of [1, 4, and 5].

We i l lustrate the abs t rac t theorems by the example of an equation derived by Feodos 'ev [2]. The equa- tion of smal l t r ansversa l vibrations of a pipeline secured with joints, ca r ry ing the flow of an ideal incompres - sible fluid, has the fo rm

O~w v~ O~w °~ o*x _ a~W _ O, (4) ~ + ~ ~- ~PV a-Z~ ~- KK -

in dimensionless variables (v and fl are positive constants). It is easy to see that Eqs. (4), (5) can be written in the form (1), where A acts according to the rule

.4 y = y i v + v~y. (6)

on functions u ~ tv~ [0, l] sat isfying the condition

The operator B acts according to the rule

g (0) = y(1) = u"(o) = y- 0) = 0. (7)

BU = 2~vU'. (8)

*We have in mind the elementary divisors of the linear operator corresponding to L(~) and acting in the space H@H.

143

It is easy to verify that conditions a), b) and the hypotheses of Theorems 4 and 5 are satisfied.

Movchan [3] has proved that for small velocities of the fluid (0 -< v < ,~) the solutions of Eqs. (4), (5) are stable and for v = ~n, n ~- 1, 2, . . . they are unstable on the positive semiaxis.

In par t icular , Theorems 2, 3, and 4 imply that the spec t rum of the pencil (3), (6)-(8) is symmet r i c with respec t to the real and imaginary axes and for ~n < v < ~(n + 1), u = 0, 1, 2, . . . the number of eigenvalues of the pencil lying in the open right half-space does not exceed n and the not purely imaginary eigenvalues lie in a disk of radius r =0.5v 2. For ~ ( 2 n - 1 ) < v < 2 ~ n , a = l , 2 , . . . the solutions of Eqs. (4), (5) are unstable on the positive semiaxis , since the spec t rum of the pencil (3), (6)-(8) contains an odd number of positive eigen- values. For ~ < v < 2~ the spec t rum of the pencil (3), (6)-(8) consists of purely imaginary algebraical ly simple numbers and two real numbers ±k 0. Such a distribution of the spec t rum means instability of the "divergence" type.

The author thanks V. I. Yudovich and N. D. Kopachevskii for discussing the resul ts of this paper.

1.

2o

3. 4. 5.

LITERATURE CITED

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonself-Adjoint Operators, Amer. Math. Soc. (1969). V. I. Feodos'ev, Inzh. Sb., iO, 169-170 (1951). A. A. Movchan, Prik. Mat. Mekh., 4, 760-762 (1965). L. S. Pontryagin, Izv. Akad. Nauk SSSR, Ser. Mat., 8, 243-280 (1944). T. Ya. Azizov and I. S. Iokhvidov, Mat. Issled., 6, No. I, 158-161 (1971).

S O M E R E M A R K S ON T H E I N T E G R A B I L I T Y OF THE

E Q U A T I O N S OF M O T I O N OF A R I G I D B O D Y IN AN IDEAL FLUID

A. M. P e r e l o m o v UDC 517.9

The c lass ica l problem of the motion of a rigid body in an ideal fluid (MRBIF) has been studied intensively in the last century and the beginning of our century (cf., e.g. , [1-5]).*

Recently, S. P. Novikov showed that the equations of motion for this problem are the equations for the geodesics of the r ight- invar iant metr ic on the s ix -pa rame te r Lie group E (3), the group of motions of three- dimensional Euclidean space. He also established that under passage to the limit (compression) f rom the group S0(4) to the group E(3), the Euler equations for SO(4) of the top [8] go into the equations of MRBIF for the so- called Clebsch case [2]. Here one can get integrals of motion for the case of [2], although the L - M pair of [8] diverges in the limit.

In the present note we give an L - M pair for the n-dimensional generalization of the Clebsch case, which is closely connected with the L - M pair of Moser [7] for the geodesic flow on an (n - 1)-dimensional ellipsoid. We note that our L - M pair is connected not with the group E (n), but withthe group GL(n, R). We note also that the equations of MRBIF can be integrated with the help of an additional quadratic integral of motion only in the Clebsch [2] and Steklov [3] cases .

1. We give a group- theoret ic descript ion of the equations of MRBIF. Let G = E (3) be the group of motions of three-dimensional Euclidean space R 3, generated by rotations and t ranslat ions, 2¢ be its Lie algebra with the s tandard basis xj, Yk, J, k = 1, 2, 3,

[xj, xk] = ej~x,~, [z~, yk] = ejk~y~, lye, y~l ~ 0, (1)

where eij m is the s tandard totally skew-symmet r i c tensor. A point of the space ~ , the dual space to ~ , will be given by coordinates Ij, Pk-

*S. p . Novikov and V. L. Golo turned my attention to this problem and also indicated the l i terature on the p re s - ent question.

Institute of Theoretical and Experimental Physics . Translated f rom Fuuktsional 'nyi Analiz i Ego Pr i lo - zheniya, Vol. 15, No. 2, pp. 83-85, Apr i l -June, 1981. Original ar t ic le submitted May 5, 1980.

144 0016-2663/81/1502- 0144 $07.50 © 1981 Plenum Publishing Corporation