Euler equations on Lie algebras

distinguish an arbitrary finite collection of nonintersecting and non-self-intersecting smooth circles ~i among which there are precisely m contractible circles (the remaining are noncontractible in MS). In M g 2 x S l the circles ~i determine tori T ~ 2 = ~ $ ~. We c u t M ~ S ~ along all these tori after which we identify these tori by means of certain diffeomorphisms. As a result a new three-dimensional manifold is obtained. It turns out that the surface Q has just such a form. Problem. Find an explicit, convenient corepresentation of the group ~I(Q), where Qa is the surface of Theorem 2.1.3. Give an explicit classification of surfaces of constant energy of integrable systems. How can we obtain a lower bound of the number of solid tori (i.e., stable periodic solutions on QS) in terms of the topological invariants of Q (homology, homotypy) in the general case and not only'for R < 2? Develop a complex-analytic analogue of the Morse theory of integrable systems constructed above. On the analytic manifold M ~ does there exist an integrable foliation into two-dimensional (in the real sense) complex tori? In examples of surfaces of type KZ it is probably possible to obtain such obstructions in explicit form. 3.

Euler Equations on Lie Algebras

3.1. General Euler Equations. Let f be a smooth function on an orbit of the coadjoint Ad*, i.e., fCC~(~(t~ Then the Hamiltonian equations x = sgrad f on the orbit G(t) relative to the Kirillov form on O(t) have the form x==adar(x)(x)~ Here ~df~G**~_G. If fEC~(G*), then on each orbit there arises a Hamiltonian vector field v=ad~r(x)(x ), xEG*, fCC=(G*). These equations can be "glued together" into a single equation on the space G*. Definition 3.1.1. An equation of the form x=addf(x)(X) " * the space G* where G is some Lie algebra.

is called an Euler equation on

An Euler equation on G* possesses the remarkable property that the corresponding vector field is tangent to all orbits of the representation A d ; and on each orbit these equations are Hamiltonian. As an example we write out explicitly the Euler equations for the Lie algebra so(3)={el, e2, ea} with the commutation relations tel,e2]=ea, [el,e~=--e2, [e2,eel=el 9 In this case the Euler equations can be written in the form (for the function

f=~

ai]xixi :

1,7=I

~2 :

- - a23X2 X 1"~ ~ l x 2 x 8 - ~

=

+

( a I1 - - a 83) XlX 3 + a 13 (x~ - - x~), aD + a

The Euler equations when the Hamiltonian f is a quadratic function on G* are of major interest. We shall consider this case in more detail. If there is a linear operator C:G* G, then it is possible to construct a quadratic function f on G*:f(x) = 2-i<x, C(x)>. We shall assume that the linear operator C is self-adjoint, i.e., = . In this case df X = C(x). Thus, each self-adjoint, linear operator C:G* + G defines on G* a system of nonlinear equations x = ad~(x)(X). This system of equations on G* is Hamiltonian on all orbits of the representation Ad~; as the Hamiltonian it is possible to take the function f(x) = 2-1<x, C(x)> restricted to an Orbit of the representation Ad;. We shall rewrite the Euler equations in coordinates. Let e I .... ,en be a basis of the Lie algebra G, let el,...,e n be the dual basis in the dual space G*, x~G*, x~-xle z, and let C:G* + G be the linear operator defining the Euler equations, C(e~)=ai~ i. Then the Euler equations have the following form:

Xs:=ai]C)~x~xk, S = I

.....

dimG,

(1)

where C~j is the structure tensor of the Lie algebra G in the basis el,...,e n. Proposition 3.1.1. Suppose a function FCC~(G *) is constant on orbits of the coadjoint representation of the Lie algebra ~, corresponding to the Lie algebra G. Then F is a first integral of the Euler equations x:ad~f(x)(x). In coordinates this can be written as follows: if the function F satifies the system of partial differential equations

OF C~jx~-o-~]=O, i:1 then

it

is

a first

integral

of the Euler

equations

.....

dimG,

(2)

x = a d ~ r ( x ) ( x ). 2693

This proposition provides a certain analogue of the Jacobi method of integrating canonical systems of equations of classical mechanics. In solving Hamiltonian equations by the Jacobi method we must solve the partial differential equation

OS ( q'-Fq-]=O aS --~7-~l-l_t, (nonlinear

in general). In the method expounded above we also go over from ordinary differential equations to the partial differential equations (2), but, in contrast to the Jacobi method, these are linear equations. There is an algorithm for finding solutions of such equations (see, for example, [81, 127]). Below we shall indicate a special class of linear operators (section operators for an arbitrary linear representation) for which knowledge of a complete collection of functionally independent solutions of Eq. (i) makes it possible to establish a complete collection of first integrals of Eq. (i), and hereby a purely algebraic procedure of shift of the argument is used. Concerning Euler equations on infinite-dimensional Lie algebras, see, for example, [6, 26, 51, 116, 139, 140, 144]. Infinite-dimensional Lie algebras arising in connection with the study of the Euler and Korteweg-de Vries equations are considered in the book [142]. 3.2. The Equations of Motion of aR i g i d Body Fixed at One Point. We recall that the classical Euler equations of motion of a rigid body have the form A~ = (B - c)qr, B~ = (C A)pr, C~ = (A - B)pq, where A, B, C are the moments of inertia of the body relative to the axes Ox, Oy; and Oz, respectively, and p, q, r are the components of the vector of angular velocity (see, for example, [3, 23, 115]). One can become acquainted with Euler equations from a geometric point of view in the works of Tatarinov [85, 86]. LEMMA 3.2.1. The Euler equations for the Lie algebra so(3) of skew-symmetric 3 x 3 real matrices on the space so(3)* are equivalent to the equations of motion of a three-dimensional solid body fixed at one point. LEMMA 3.2.2. The Euler equations of motion of a rigid body on the algebra so(3) are equivalent to the following system having the commutator form ~ ( X ) = [~X, X], XEso(3), ~(X) =

IX +Xl, l=d~ag(~, ~=,~3), ~=2-~(-:A+B+C),~2=2-~(A'B+C), ~3=2-~(A+B--C). Definition 3.2.1. We call the equations ~ ( X ) = [~X, X], X~so(n), ~(X)=XI+IX,

where I = diag (%1, %2 .... ,An) (hi + %j ~ 0 for all i, j) the equations of motion of an n-dimensional rigid body with a fixed point. In the next section the invariant description of the operator the works [63, 65, 66] will be given.

~(X)=]X+X[

found in

3.3. The Equation of Motion of a Free Rigid Body. The motion of a rigid body in threedimensional space R 3 is described by the equations K = [ K , ~]+[e, ~], ~=![e, ~], where K is the kinetic moment of the body, m is the angular velocity of the body, and the vectors e and u are determined by the physical content of the problem and describe the interaction of external forces. These general equations have the following three classical integrals: fl = H, where the total energy H has the form H = 2-I(K, h-l(K)) + m(r, e), and f2 = (K, e) and f~ = (e, e). We consider, for example, the problem of the motion of a heavy rigid body about a fixed point. Then e is a unit vector directed along the vertical axis Oz, and hence the third integral f3 has the form f3 = (e, e) = i. The general equations of motion of the rigid body give a vector field on the six-dimensional Euclidean space R 6=R~(K) X R3(e). It turns out that this vector field is Hamiltonian on the joint level surface of the two integrals f2 = C2 = const and fs = Ca = const relative to a certain natural Poisson bracket. This makes it possible to apply the machinery of investigating general Hamiltonian systems to this system. We consider the joint level surface M2s of the two integrals f2 and f3, i.e., 7H~3= {~2=C2, :3=C3>0}. This surface is a four-dimensional submanifold in the Euclidean space R 6 . Moreover, its topological structure can be easily described. The level surface M2~ is diffeomorphic to the cotangent bundle of the two-dimensional sphere. On the space R6(K,e) we now define the operation {', "}. If K = (K l, K2, K~) and e = (el, e2, e3), then on the generators K i and e i the operation {-, "} is given by Table I. It is easy to see that this operation defines the structure of a Lie algebra on the space C=(R6). It can be verified that it is isomorphic to the Lie algebra of the group of motions of three-dimensional Euclidean space. In other words, it is isomorphic to the semidirect sum of two Lie subalgebras: the three-dimensional Lie subalgebra R~(K), isomorphic to the Lie algebra of the group SO(3) of rotations of the space R~, and the three-dimensional

2694

TABLE 1 K,

K~

K~

e,

]

K|

0

--f~

K~

0

I --e~

K,

K~

0

--K~

K~

--K~

K,

0

et

0

--e~

e~

e~

e~

0

--e,

0

e~

--e~

e~

0

0

9

e~

e~

e~

e~ 1

0

--e~ [

e,

0

0

0

0

0

O

0

0

I I

--e l

Abelian Lie subalgebra R~(e), isomorphic to the Lie algebra of the Lie group of shifts (translations) in Rh The Lie algebra Rs(/() is hereby represented on the subalgebra R~(e) in standard fashion~ The operation {', "} defined above gives on the space C~(R 6) a Poisson bracket {f, g} which is degenerate. It is possible to restrict this Poisson bracket to the joint level surfaces M2~ of the integrals f2 and f~. It can be seen that as a result a nondegenerate Poisson bracket {f, g} is obtained on the space of functions defined on the manifold M23 . THEOREM 3.3.1 (S. P. Novikov~ I. Shmel'tser)__. The equations of motion of a rigid body K = [K, ~] + [e, u], ~=l[e, ~] can be represented on the joint level surface M2~ of the two integrals fz and f3 in the Hamiltonian form f = {f, h}', where h is the restriction of the function H to M2s. Regarding the Hamiltonian property of the Euler equations of the dynamics of a rigid body in an ideal fluid, see also the work [2]. 3.4. Rotation of a Rigid Body about a Fi_xed Point in a Newtonian Field with a Potential ~(xl, x2, x3). Let ~, ~, ~ be three unit vectors of a fixed coordinate system referred to a reference system S rigidly connected with the rigid body. The equations of motion of the rigid body in the system S have the form 9

00-

CU

#U

where U(~, ~, y) is the potential function

U (a, p, ~) =.f

~ (r)

~ ((r, o~), (r, ~), (r, ~)) dr#r2dr3,

T

here T is the rigid body, and rl, r2, r~ are-the coordinates in the reference system S. We shall define the Lie algebra LI2. By definition, it is the semidirect sum so(3)+1~3+ R3-kRs with basis X~, Y~=, i, ], =, ~=I, 2, 3, in which the commutation relations have the following form:

[X~, X A=sukXk, [X,, Y~I=su~/~,

(4)

lrL r~l =o. THEOREM 3.4.1 (see [9]).

Equations (3) are Euler equations of the form ac=ad]r(x)(x) in 3

the space L~2 and have Hamiltonian I-I-~-2-1(M,~)--U(~, ~, ?), 7~/~=~fg~o~. k=l

2695

The p o t e n t i a l

O2U----Obecause

OW , 02U .' t h e t h r e e e q u a t i o n s ( i = 1, 2, 3) 0-~t~-Y~2.-ff

f u n c t i o n U(~, 8, X) s a t i s f i e s

of the Laplace equation A ~ = 0 .

In the simplest case where the potential

is a linear function of the Coordinates, ep_~-alx1-{-a2x2+a3x3, Eqs. (3) are equivalent to the Euler-Poisson equations and, generally speaking, cannot be represented in the form f'~ ad*c(M)(M) by means of some section operator C; they are nevertheless Euler equations in the sense of Definition 3.1.i. 3.5. Dynamics of a Rigid Body with Distributed Electric Charge in an Ideal Incompressible Fluid. In the presence of constant gravitational and electric fields and under the condition of equality of the repulsive force and the force of gravity and zero total charge of the body the equations of motion in a reference system S connected with the body whose center coincides with the center of mass of the rigid body under the condition of vortex-free flow have the form

x,o .+p p=pXo,

x

?=?X~,

x s,

(s)

6=6Xo,

where ~ i s t h e a n g u l a r v e l o c i t y , u i s t h e v e l o c i t y of t h e r i g i d body in t h e f l u i d , p i s t h e t o t a l momentum, M i s t h e t o t a l a n g u l a r momentum ( i n t h e r e f e r e n c e system S ) , m i s t h e mass o f t h e body, g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n , X i s t h e d i r e c t i o n of t h e f o r c e of g r a v i t y , the vector r defines the position of the center of mass of the displaced volume of fluid, E is the electric-field intensity, d is the vector of dipole moment, and ~ is the direction of the electric field. THEOREM 3.5.1 (see [9]). Hamiltonian

Equations (5) are Euler equations in the space L~l and have

8

H ~- 2 -I Z

(aijMi34J-{- 2cuMiPJ "~ bupiPJ)-- mg

(r' y)-- E (d, 6),

i, j = l OH

OH

where aij, bij, cij are arbitrary constant coefficients which ensure that the quadratic form indicated is positive definite. 3.6. Rotation of a Magnetized Rigid Body about a Fixed Point in a Homogeneous Gravitational and Magnetic Field. In this case the equations describing the motion of the rigid body have the form

{~=vx~,

M=M X~+mgr

X ? + h Z X 6,

(6)

~ - a X ~,

where Z is the vector of the magnetic moment of the rigid body, 6 is the direction of the magnetic field, and h is its intensity. THEOREM 3.6.1 (see [9]). tonian

Equations

(6) are Euler equations in the space L~ with Hamil-

H = 2 -I (M, co) - - mg (r, ~,) - - h (Z, 6), 8

M i ~ X fik~k, k=l

where the Lie algebra L 9 has the commutation relations described in part 3.4 with ~, $ = i, 2. For Z = 0 these equations go over into the classical Euler-Poisson equations. 3.7. Dynamics of a Rigid Body with an Ellipsoidal Cavity Filled with Magnetic Fluid Performing Homogeneous Vortical Motion. The equations of motion in the reference system S have the form

/i'/--M X A, K----K X B + u X w ,

where M is the total angular momentum of the rigid body and the fluid relative to a common center of mass, A is the angular velocity of rotation of the rigid body, B is the angular

2696

(7)

velocity of internal rotation of the fluid, K is the vortex vector of fluid velocity, and the vectors u and w are connected with the magnetic field frozen into the fluid. The coordinates of the vectors A and B can oe expressed linearly in terms of the coordinates of the vectors M and K; the coordinates of the vectors u and w are also connected by linear relations. THEOREM 3.7.1 (see [9]). Equations (7) are Euler equations in the dual space to the Lie algebra so(3) 9 E~ and have Hamiltonian

A)+(K, B)+(u, w)),

H=2-'((M,

where E 3 is the Lie algebra of the group of rotations of Euclidean space R 3. 3.8. Dynamics of a Rigid Body with n Ellipsoidal Cavities Filled with Magnetic Fluid Performing Homogeneous Vortical Motion. In this case the dynamics is described by the equations

{

M=-,~ x ~, K ~ = K ~ X B ~ + u ~ X ~ , Ua=Ua XBa,

(8)

where a = l,o..,n, M is the total angular moment (relative to a common center of mass) of the rigid body and the fluid in all ellipsoidal cavities, A is the angular velocity of rotation of the rigid body, and the vectors Ka, Ba, Ua, w a characterize the motion of the fluid and the frozen-in magnetic field in cavity a. The coordinates of the vectors A, Bz,...,B n can be expressed linearly in terms of the coordinates of the vectors M, KI,...,Kn, and the coordinates of the vectors u a and w a are connected by linear relations; the coefficients of these linear connections depend on the components of the inertia tensor of the rigid body and the parameters of the ellipsoidal cavities. We denote by Ak, m the direct sum so(3)G...~so(3)@ E s G . . . G E 3 of k copies of the Lie algebra so(3) and m copies of the Lie algebra E~. THEOREM 3.8.1 (see [9]). Equations (8) are Euler equations in the dual space to the Lie algebra L = AI, n and have Hamiltonian

H -- 2 -~((M, A)) + ~ ((K~, B~) + (u~, ~)). In the absence of a magnetic field in k cavities L reduces to the Lie algebra Ak+z,n_ k.

( u ~ = w~ = 0 ,

~ = 1 .....

k) the

Lie

algebra

3.9. Rotation of a Rigid Body with n Ellipsoidal Cavities Filled with Magnetic Fluid about a Fixed Point in a Newtonian Field with Potential qix!, x2, xs), In this case the dynamics are described by the equations

OU

OU

-t--~b~?X ?, ~.=~xA, I~=I~XA, ~ = v X A ,

(9)

which form an indecomposable combination of the equations in parts 3.4, 3.8. THEOREM 3.9.1 (see [9]). Equations algebra L-~-L12eAo,n and have Hamiltonian

(9) are Euler equations in the dual space to the Lie n

In the absence of a magnetic field in k cavities the Lie algebra L reduces to the Lie algebra

L12| Ak,,,-k. Combining the conditions o f p r o b l e m s 2 . 5 a n d 2 . 8 , we o b t a i n E u l e r e q u a t i o n s connected with the Lie algebra [email protected]. in the absence of an electric field the Lie algebra L reduces to the Lie algebra Lg@A~.n-k ; in the absence of both electric and gravitational fields the Lie algebra L reduces to the Lie algebra E3@Ak, n-k =Akw-k+1. The Lie algebras Lg@A~.n-k and Akm-k+1 are also connected with the equations obtained by combining the conditions of problems 3.6 and 3.8.

3.10. Kirchhoff Equations of Motion of a Rigid Body in a Fluid. We connect the coordinate system with the moving body. Let u i be the components of the velocity of forward motion of the origin of the coordinates, and let ~i be the components of the angular velocity

2697

of rotation of the rigid body. Then the kinetic energy of the fluid and rigid body system has the form T-~2-~(A~j~%+Bue~j)+C~j~j, where Aij, Bij, C ij are constants depending on the shape of the body and on the densities of the body and the fluid; repeated indices are summed from i to 3. Let N = (y~, Y2, Y~), where Yi = 8T/8~i, K = (xl, x=, xa), where x i = 8T/~u i. Then the inertial motion of the rigid body in the ideal fluid is described by the equations

{

@=NXo-I-KXU, dK

(i0)

KX~,

where U = (ul, u2, ua), co = (c01, c02, ~a). The kinetic energy of the rigid body is an arbitrary positive-definite, homogeneous quadratic form in the six variables ui, mi; it is thus determined by the 21 coefficients (Aij, Bij, Cij) = (lij). Equations (i0) in the general case have the three classical Kirchhoff first integrals K 2 = K~ + K~ + K~ = const, NK = KxN x + KyNy KzN z = const,, and, finallyi T = const. THEOREM 3.10.1 (see [99~ i01~ 102]). The system of Euler equations for the Lie algebra E(3) of the Lie group of motions of Euclidean space R a coincides with the equations of inertial motion of a rigid body in an ideal fluid. 3.11. Equations of Inertial Motion of a Rigid Body in an Incompressible, Ideally Conducting Fluid. We consider the classical equations of the magnetohydrodynamics of an incompressible, nonviscous, ideally conducting fluid

I ~ T + (rot ~) X ~-- p- (rot H) • H - - grad II, OH

[~

= rot (~, X H).

As has been shown in [17], these equations have as the simplest finite-dimensional analogue the equations

{M= [~,M]+ lJ,h'],

(11)

on the Lie algebra so(n) of skew-symmetric matrices. Let G be an arbitrary Lie algebra over the field k. We construct a new Lie algebra Q~,~(O), ~, ~6k, containing G as a subalgebra. As a linear space Q=,~(G) is the direct sum G e G . The elements of the first term we denote by a6O, while the elements of the second term we denote by eb, i.e., any element of Q=,o(O) has the form x +ey, x, y6O. On ~=,s(O) we define the product [Xl@TYl, x2@eF2]----[xl,x2] +~[y~, y2]@a([yl, x2[@[Xl, y2]nualyl,yJ). This operation gives the structure of a Lie algebra on~=,~(O). We introduce the notation ~(G) = G + eG, where g2 = 0 for the Lie algebra Q0,0(O). It may be assumed that,Q(O)*-------O*.4-cO~. We denote / s (O)~ by f = x * + e y ~, where X% y*s THEOREM 3.11.1 (see [92, 97]). Let x + e y G e ( O ) , x * + e y * s *. Then ad~+~y(x*@ey*) ~ad]x*+ad~y*+sad~v *, and therefore the Euler equations d=ad~ca )(a), =ca(o) * have the form

X = adxX-r- advY, /~---- ad;Y, where a = (X, Y) and C(a) = (x, y). In the case where G = so(3) the Euler equations for the Lie algebra ~(so(3)) coincide with the finite-dimensi0nal approximations (ii) of the equations of magnetohydrodynamics. 4.

Section Operators

4.1. General Constructions and Definitions. For finite-dimensional Lie algebras there is a simple and natural procedure for constructing dynamical systems which are analogues of the equations of motion of a multidimensional rigid body. These systems are constructed on the basis of the section operators introduced by Fomenko in [105, 106]. Let H be a Lie algebra, let ~ be the corresponding Lie group, let p:H ~ End (V) be a representation of the Lie algebra H in the linear space V, let e:g~-~Aut(V)be the corresponding representation of the group, and let ~(X) be the orbit of the element X % V under the 2698