Poincare method of proving nonintegrability

we denote it by ~k" E. G. Shuvalov showed that shifts . . . of. invariants give a complete, v• family of functions for the Lie algebra si(2, C ) | k+1 for k = i, 2, 3, 4.

in-

~k

THEOREM 4.4.2 (see [9]).

Let L be the space dual to one of the Lie algebras L12, L 9, Then on L there exists a complete, involutive family of functions which are first integrals of the Euler equations described in parts 3.4-3.9 of Chap. i, i.e., these equations are completely integrable in the Liouville sense.

so(3)OE3, Ah, m, LmGA0,n (corresponding notation was introduced in Sec. 3, Chap. i).

CHAPTER 4 QUESTIONS OF NONINTEGRABILITY IN HAMILTONIAN MECHANICS I.

Poincarg Method of Proving Nonintegrability

I.i. Perturbation Theory and the Investigation of Systems Close to Integrable Systems. In the space of all Hamiltonians open regions are distinguished which sometimes fill out almost the entire space and consist of Hamiltonians f of "general position" for which the corresponding Hamiltonian systems v = sgrad f are not Liouville integrable (or not integrabie in some other more general sense). The picture we have described is not a rigorously proved theorem, since in the formulation presented above too many objects are in need of a correct specification which is not always possible. Nevertheless, results presently known of "negative character," i.e., results asserting the nonintegrability of many concrete types of systems, make it possible to view the principle formulated above as some experimental observation which may serve as a guide in the study of concrete systems. Thus, integrable cases fill a set of "measure zero" in the space of all systems. It is already clear from this that the search for integrable Hamiltonian systems is a very difficult problem, since it is necessary to somehow "guess" or algorithmically discover in the immense set of all possible Hamiltonians those rare cases when some additional symmetries cause the appearance of a sufficient number of integrals. In Chap. 2 regular methods were indicated for constructing functions on homogeneous symplectic manifolds which in practice, as a rule, give integrable Hamiltonian systems which are interesting from the viewpoint of mechanics (see Chap. 3). We shall now demonstrate to the reader that a Hamiltonian "taken at random" most often generates a nonintegrable system. Let (M, m) be a symplectic manifold, let v 0 = s g r a d H 0 be a completely integrable Hamiltonian system, and let T n be one of the compact, connected level surfaces of a collection of first integrals fl, f2,..-,fn which are functionally independent and pairwise in involution in a neighborhood of T n. As we know, T n is diffeomorphic to an n-dimensional torus. In an open neighborhood U of the torus T n we consider curvilinear "action-angle" coordinates sl,..., an, 91 ..... ~n, where ~ are angular coordinates on the torus and s i are coordinates normal to the torus. For brevity we introduce the vector notation s=(s1,.:., an), 9=(~i,-i., ~n). We represent the neighborhood U as a direct product U = D n • T n, where D n = D is an open domain in Rn(s), for example, homeomorphic to a sufficiently small disk. Thus, we have separated the regular coordinates in the neighborhood U into two groups: U(s, 9) =Dn(S) xTn(~) 9 The integral trajectories of the system v 0 = s g r a d H 0 are distributed on the torus T n (and on tori s • T n, where s6Dn), near to it), forming a rectilinear winding (see Fig. i0). As we know (see the Liouville theorem), in the coordinates (s, ~) the Hamiltonian H ~ depends only on the variables s, i.e., H = H(s) in the neighborhood U. We now consider a perturbation of the original Hamiltonian system by means of a perturbation of its Hamiltonian H 0. We consider a family of Hamiltonian systems v~=sgradH(s, ~, s), where H(s, 9, s) is a real analytic function defined on the direct product ! U X (--e0, so) and such that for E = 0 we obtain the original Hamiltonian, i.e., H(s, % 0)=H0(s). The Hamiltonian equations

OH~

~=~-~(s),

v0-----sgrad/40 in the coordinates

where ~(s)=(~1(s) .... ,~n(s))i

(s,~} can be written thus:

As we see, these equations are ex-

plicitly integrable. If the point s = s o is fixed, it determines a torus in the neighborhood U on which the vector ~(s ~ is constant (does n o t depend on ~), and hence the equations of motion can be integrated as follows: s(t)~s0, q(t)=~0-b~(s0)t. The equations of the perturbed Hamiltonian system v = s g r a d H can be written in the following form: s ~

O0~' H ~ _---~, 0H . . . . ~.) + .... where f-/(s,~)=/-/0(s)~-sff1(s,

Assuming that the parameter

2729

Graph

17'~/s)

i Fig. i0

Fig.

ii

is small, we arrive at the problem of integrating the perturbed system in a neighborhood of the completely integrable Hamiltonian system. In some cases Poincar~'s method makes it possible to prove nonintegrability of the perturbed system in a neighborhood of an integrable system. The term of the expansion /-[I(S, q)) plays a basic role in the investigation of the system for nonintegrability; this function is sometimes called the perturbing function. We expand the perturbing function in a multiple Fourier series: ff1(s, 9)-- ~

f-fm(S)e i(m'~) 9

Here

m~Z n

m = (m I ..... m n) is an integral vector running over all nodes of the integral lattice Z n of rank n, i.e., Z n - ~ Z X . . . X Z (n times). (m, ~ ) denotes the usual scalar product of the vectors n

m and ~, lee., (~, ~)-~-~ m ~ k .

Therefore,

the expansion of the function H I has the form

k=l

Hi(s, 9)---- ~

~

~I~+...+~:~

(m~..... ran)

We c o n s i d e r

the

domain (ball)

Dn t r a n s v e r s a l

to

the

Liouville

torus

Tn ( s e e

above).

Definition 1.1.1. T h e P o i n c a r ~ s e t P i n t h e d o m a i n Dn i s t h e s e t o f a l l p o i n t s S G Dn for which there exist n - 1 linearly independent integral vectors a~ .... , a n _ 1 ~ Z n such that the following conditions are satisfied: i) all scalar products

(a~, ~(s))are equal to zero, l~
2) H ~ k ( s ) ~ 0 . Here ~0(s) = (001(s) , ..,00n(S)) is the vector giving the components of the vector field v 0 = s g r a d H 0 on the torus Tn(s). This vector is naturally called the frequency vector. The numbers ml .... ,mn are the frequencies of the uniform motion of the trajectory over the torus T n. A torus with a collection of frequencies ~i,.. ",mn is called a nonresonance torus if the equality klm I + . . . + kn~ n = 0 with integral coefficients kl,...,k n implies that .all k i = O. it is obvious that on nonresonant tori the integral trajectories are dense. In the resonance case the trajectories fill out (in a dense manner) tori of lower dimension. We denote by A(V) the class of functions analytic in a domain F o R ~. A set N c V is called a uniqueness set for the class A(V) if any analytic function equal to zero on the set N vanishes identically everywhere on the domain V. In particular, if two analytic functions coincide on a uniqueness set N, then they coincide on the entire domain V. For example, the set N of points of a one-dimensional interval A ~ on the real axis R ~ is uniqueness set for the class A(A I) if and only if it has a limit point inside the interval A I. We consider the unperturbed Hamiltonian system v0 = s g r a d H 0. (

0 2Ho

in a neighborhood U of an invariant torus T n if del i ~ ) ~ = 0

It is called nondegenerate

in the domain D n, where U z

D n • T n (see above). We recall that the Hamiltonian H 0 may be assumed to depend only on the variables s, i.e., H 0 = H0(s I ..... Sn). Let s o = (s~ ..... s~) be a fixed point in the domain Dn cU. 2730

THEOREM i.i.i (see [41]). We suppose that the unperturbed Hamiltonian system v 0 = sgradH 0 is nondegenerate in a neighborhood U of the invariant torus T n. Suppose, further, that the point s06D n is a noncritical point of the Hamiltonian H0(s) and that in some neighborhood V the Poincar4 set N is a uniqueness set for the class A(V). Then the perturbed Hami!tonian equations v = sgradH,

i.e.

s=

OH ~ . Off h,(s)___ffo(S)_~_el_fl(sI~)~

have

no integral f not depending on the Hamiltonian H which can be represented in the form of a formal power s e r i e s f ( s , ~ ) = ~ / ~ ( s , ~ ) s ~ with coefficients analytic in the domain U = D n x T n. For applications the following variants of Theorem !.i.I are also useful. THEOREM 1.1.2 (see [41]). Suppose Hamiltonian D0(s) of the unperturbed system is n o n degenerate in the domain D n, and suppose the Poincar4 set N is dense in the domain D n. Then the Hamiltonian equations of the perturbed system v = sgradH have no formal integral /(s, ~ ) = ~ / 1 ~ s ~ with smooth coefficients

f~(s, ~P) defined

in a neighborhood U = Dn • T n of the

invariant torus Tn~ which does not depend on the Hamiltonian ff(s, ~). Analogous assertions are valid also for nonautonomous Hamiltonian systems (for details see [41]). 1.2. Nonintef~ability of the Equations of Motion of a Dynamically Nonsymmetric R ~ Bo__q~ with a Fixed Point. The results of the preceding subsection can be applied to prove the assertion formulated in the title of the present subsection. We first formulate the problem more precisely. A. Poincar4 already posed the question of whether there exists one additional integral distinct from the three known integrals in the problem of the motion of a heavy rigid body about a fixed point (see [80]). In [39] Koslov gave a negative answer for Hamiltonians H close to the Hamiltonian H 0 corresponding to the completely integrab!e Eu!er case. We shall consider this question in more detail. The general equations of motion of a heavy solid body were written out above. As was noted, these equations always have three integrals: the total energy fl = H = z/2(K, h-l(K)) + m(r, e), f2 = (K, e), and fs = (e, e). The second integral f2 corresponds to the circumstance that the projection of the kinetic moment onto the vertical is always constant. This integral corresponds to the group of rotations of the body about the vertical axis. Having fixed the values of the two integrals f2 and fs, it is possible to reduce the order of the system by two units, and it becomes a vector field on four-dimensional level surfaces J423~R 6. Moreover, on these surfaces a system is obtained which turns out to be Hamiltonian. Its Hamiltonian is the total energy of the body with a fixed value of the projection of the kinetic moment K onto the vertical (K, e) = const. The Hamiltonian H can be represented in the form H = H 0 + eH1, where H 0 - the kinetic energy - is the Hamiltonian of the completely integrable Euler problem on the inertial motion of a rigid body (i.e., under the condition that the body is fastened at the center of mass), while gH l is the potential energy of a body in a homogeneous gravitational field. Here E is the product of the weight of the body with the distance from the center of mass to the point of suspension (fastening) of the rigid body. We assume that the parameter s is small, i.e., we consider the motion of the rigid body obtained by a small perturbation of the integrable Euler case. From a mechanical point of view this is equivalent to the study of of rapid motions of a rigid body in a moderate force field. THEOREM 1.2.1 (Kozlov; see [39, 41])~ If a heavy rigid body of general form is dynamically nonsymmetric, then its equations of motion v = sgradH, where the Hamiltonian H = H 0 + eH z is a small perturbation of the Hamiltonian H 0 of the integrable Euler case, have no formal additional fourth integral ~ / k e ~ with analytic coefficients on the four=dimensional

level

k>0

surface M 2 ~ w h i c h

does not depend on the function H = H 0 + eH I.

In particular,

this gives a negative answer to Poincar4's question.

krhat happens "far" from the integrable Euler case, i.e., for a rigid body which differs strongly from a body satisfying the Euier conditions, is so far unclear. The situation is that the "real methods" existing today for the analysis of Hamiltonian systems are effective only in a small neighborhood of the rare integrable cases. The "complex" methods (which we do not touch on here due to lack of space) do not make it possible to fully describe the '~rea! cases" of nonintegrability.

2731

1.3. Splitting of Separatrices. Let V n be a smooth manifold which we identify with configuration space (the space of positions) of some Hamiltonian system, and let M 2n = T*V be the cotangent bundle to V n which is a symplectic manifold and is naturally identified with the phase space of the sxstem. Points of T*V have the form (x, 6), where x E V n and $ @ T ~ V n (i.e., is a covector) is a linear function on TxVn. Suppose that on the phase space M 2n there is given a Hamiltonian H = H(x, 6, t) depending, generally speaking, on the time t. It is then possible to consider the extended phase space K ~ * ~ = T * V X R = ( E , t ) with coordinates (x, 6, E, t) on which the equations of motion will again be Hamiltonian equations: 9x ~ ~aQ

6= '

E=~, = t -~-~", OQ

where

Q(x,~,

OQ ~x

'

E, t)=H(x, ~,t)--E~ ~EV, ~@T~V.

Suppose the Hamiltonian H is periodic in t with period 2~ and depends on some parameter ~, i.e., H = H(x, 6, t, g). We suppose that for s = 0 the Hamiltonian H(x, g, t, 0) = H(x, 6) does not depend on the time and satisfies the following four conditions. i) The Hamiltonian H0(x , 6) on the 2n-dimensional manifold T*V n = M2n(x, 6) has at least two critical points (x_, 6-) and (x+, ~+) at which the eigenvalues of the linearized Hamiltonian system:~== ---~, OH0 x9= - ~aHo are real and nonzero.

In particular, the 2~-periodic solutions

(x_, ~_)----(x_(t),~_ (t)) and (x$, ~+)--'X+(i),~+ (t)) have hyperbolic type. It is worth emphasizing that critical points of the Hamiltonian, i.e., points where gradH 0 = 0 (and hence sgradH 0 = 0), play a major role in problems of integrability of Hamiltonian systems. 2) We denote by A+ (respectively, A_) the stable (respectively, unstable) separatrix manifold of the critical point (x+, $+) [respectively, (x_, 6-)], i.e., the manifold consisting of separatrices tending to the point (x+, ~+) as t + +~ [respectively, consisting of separtrices tending to the point (x_, g_) as t + - ~ ] . If the critical points of the Hamiltonian are nondegenerate, then near the point (x+, 6+) [respectively, (x_, 6-)] the separatrix manifolds may be considered disks, and in this case they are called separatrix disks (stable or unstable correspondingly). Along stable separatrix surfaces the trajectories tend to the critical point (with increasing time), while along unstable s@paratrix surfaces the trajectories pass away from the critical point (with increasing time). Condition 2 is that A+ = A_, i.e., the stable separatrix surface A+ of the point (x+, 6+) coincides with the unstable separatrix surface A_ of the point (x_, $ _ ) In particular, from this it follows that H0(x+, 6+) =

H0(x-, $-). 3) We suppose that in configuration space vn there exists a domain D = D n containing both points x+ and x_ and possessing the following property. We consider the domain T * D c T * V ~ M 2n. It contains both critical points (x+, 6+) and (x_, 6-) of the Hamiltonian H (see Fig. ii). Condition 3 requires that the part of the separatrix surface A+ = A_ in the domain T*D can be represented in the form of an n-dimensional graph of the gradient of some analytic function defined in the domain D. This means that the separatrix surface can be represented in the form

~ - as~ where s o is a real-analytic function of x.

Such n-dimensional surfaces

are called Lagrangian surfaces, but we shall not go into a description of their properties here, since for formulating effect of splitting of separatrices of interest to us they are not required. Under the conditions enumerated above on the domain D there arises a naturally 9

0

defined vector field described on D by the following system of differential equations x = ~ •

~( as~ax(x) g i v e s t h e g r a p h d e s c r i b e d a b o v e .

(H0(~(X), x)), w h e r e ~ = ~ ( ~ j = , t h e p o i n t s x+ and x_ t h e as t ~ +~ ( F i g . 1 2 ) .

integral

trajectories

of this

In small neighborhoods of

s y s t e m t e n d t o x_ as t ~ - ~ and t o x+

0 (H0(~(x), 4) Finally, we require that the differential equation X----~

x)) h a v e i n t h e do-

main D n a separatrix x0(t) joining the points x_ and x+, i.e., passing (for increasing t) from x_ to the point x+ (Fig. 12). The original (2n-dimensional) Hamiltonian system thus defines under assumptions 1-4 some n-dimensional system on the domain D n. Its properties to considerable degree reflect the integrability or nonintegrability of the original system. The system generated by the Hamiltonian H0(x, ~) may be considered the unperturbed system. In applications it is most Often completely integrable. The Hamiltonian system sgradH(x, 6, t, s) may thus be treated

2732

Fig. 12 as a perturbation of the system with Hamiltonian H0(x, g). It is therefore possible to apply the Poincar@ method (see Subsec. i.i) to study integrability or nonintegrability of the perturbed system. We can now describe the important effect of splitting of separatrices first discovered by Poincar6 and applied by him to justify nonintegrability of certain systems. We consider a small perturbation H(x, ~, t, e) of the Hamiltonian H0(x, E). For small values of the perturbation parameter e the separatrix surfaces A+ and A_ do not vanish but become perturbed surfaces A S and A~ which, of course, need not coincide, generally speaking. For a more precise description we consider open neighborhoods D$ and D~ of the points x+ and x_, respectively, in the configuration space V n (Fig. 13). In the regions f).XR(t) and D _ X R ( t ) the equations of the separatrix surfaces can then be written ~ =

0S_

and ~='O-~-x'

respectively, where S• t, e) is a function periodic in t (with period 2~) which is analytic for all X6D+ and small values of the parameter e. For ~ = 0 the separatrix surfaces A + = A + and A§ In the general case for small values E ~ 0 the separatrix surfaces A S and A~, considered as subsets in the direct product T*(D§ do not coincide (Poincar@). 0

,

0

-

Defintion 1.3.1. The phenomenon described above is called splitting of the separatrix surfaces (or splitting of separatrices). THEOREM 1.3.1 (Poincar~; see [80]). If H(x, ~, t, e)=Ho(x, ~)+eHl(X, ~, t ) + . . and Hi(x+, ~+, t)=Hl(x-, ~-, t) and if, moreover, f{H0, H1}(xo(t), $(Xo(t))dt=/=~ then for small values of the parameter s ~ 0 the perturbed separatrix surfaces A S and A~ do not coincide. Splitting of separatrices turns out to be an obstruction to integrability of the perturbed Hamiltonian system (considered in a neighborhood of an integrable unperturbed system). We consider a perturbed Hamiltonian system with Hamiltonian H(x, ~, t, e) =H0(x, ~) +eHl(X, ~, t) + ... satisfying all conditions 1-4 (see above). In particular, it is assumed that the unperturbed Hamiltonian system has two hyperbolic positions of equilibrium (x+, ~+) and (x_, $_) joined by a separatrix which is called a double asymptotic solution t-+(x0(0, ~o(t)),tER. THEOREM 1.3.2 (Bolotin; see [41]).

Suppose the following conditions are satisfied:

a) I{H0,{H0,H~}}(x0(t),~0(t),t)dt~0; b) for small values of the parameter s the perturbed system also has a doubly asymptotic solution (separatrix) t + (x~(t), ~s(t)) close to the solution t + (x0(t), g0(t)). Then for small fixed values s ~ 0 in any neighborhood of the closure of the trajectory (xg(t), Ss(t)) the Hamiltonian equations v = sgradH have no complete, independent collection of first integrals in involution. It is precisely the splitting of the separatrices which makes it possible to prove nonintegrability of the equations of motion of a dynamically nonsymmetric rigid body with a fixed point in a neighborhood of the integral Euler case (see Subsec. 1.2 of the present section). In the problem of the rapid motion of a nonsymmetric rigid body the Hamiltonian has the

form H=Ho+effl,

where

Ho-----~(Af<,K),H1=-xoel+Foe2+zoe3,

where r = (r0, Y0, z0) is

the radius vector of the center of mass of the rigid body, e = (e l, e2, e a) and A ~

a20 ; 0 a3/ the numbers al, a2, a3 are the inverse of the principal moments of inertia of the rigid body. For s = 0 we obtain the completely integrable Euler case, since H = H 0. In this unperturbed problem on all noncritical three-dimensional level surfaces MI23 given by the three integrals 2733

.dles)

I_

Fig. 13

Fig. 14

fl = Ho = ci = const > 0, f2 = c2 = const, f3 = c3 = 1 = const there exist two unstable periodic solutions. Namely, if a1
=

K s = 0,

A"~ = K ~

el = = cos (a~Kl) t,

j: r ~ ,

e~ = e o =

e ~ = ~ sin (a=Kl) t,

_+

K~'

~2 = 1 - { c=

~=

From the inequality (K, e)~-<(K, K)(e, e) and the independence of the integrals fl, f2, f3 on the joint level surface M123 it follows that ~2 > 0. Stable and unstable separatrix surfaces of the two periodic (in t) solutions indicated can be defined as the intersections of 3-dimensionai manifold M123 with the hyperplanes K1]/~1+_K2]/as--a2=O. The splitting of these separatrix surfaces was studied in [34, 40]. The behavior of solutions of the perturbed problem was investigated by numerical methods on a computer in the interesting work [133]. From the schemes obtained by computation showing the behavior of the integral trajectories it is clearly evident that the invariant curves of the perturbed problem begin to decompose in a neighborhood of the separatrices. Precisely this phenomenon is responsible for the nonintegrability of the equations of motion of a heavy, nonsymmetric body with a fixed point (near the integrable Euler case). 1.4. Noninte~rability in the General Case of the Kirchhoff Equations of Motion of a Rigid Body in an Ideal Fluid. In the preceding subsection general methods were indicated for proving nonintegrability of systems v = sgradH which are small perturbations of completely integrable systems. We observe that in the autonomous case the condition of splitting of separatrix surfaces situated on some fixed energy level can be written in the form

~{/o, H:} dt 4=O, where f0 is an integral of the unperturbed system, and H = H 0 + ~HI. hoff equations

We consider the Kirch-

K"=[K, 0]'~-" [e, U], e=ie, (o], (o--i-OKOH, U=~--,OH I 1 [H ~-~(AK, K ) ,_t_(BK, e)+y~Ce, e). As we know, these equations describe the motion of a rigid body in an ideal, unbounded fluid. The matrix A = diag (el, a2, as) is diagonal, while the matrices B and C may be assumed symmetric. THEOREM 1.4.1 (Kozlov, D. A. Onishchenko; pairwise distinct. If the Kirchhoff equations pendent on the three classical integrals fl = six-dlmenslonal space R6(K, e), then matrix B

be),

and t h e f o l l o w i n g r e l a t i o n

see [41]). Suppose the numbers a l, a 2, a a are have an additional (fourth) integral not deH, f2 = (K, e), f3 = (e, e) and analytic on is automatically diagonal, B = diag(bl, b2,

b.~-i.b~..l_b~--b ~ , b~--b= n ~ - - ~ - - n - a---7--. . . .

holds:--~-

I f B = O, t h e n an i n d e p e n d e n t

(fourth) integral exists only in the case when the matrix C is diagonal, C = diag (c l, c ~ c3), and the following condition holds: Kirchhoff equations are not integrable,

.c~--e3 t--------Cc'--C~=O. . e,--e,__ Thus, in the general case the

a~

a~

a~

Here it is appropriate to note that the integrable Clebsch case is defined precisely by the condition indicated above: b2--b8 -~ b3--b~ ',b~--b~~0. at

2734

a:

a~

The proof of Theorem 1.4.1 is also

based on the phenomenon of splitting of separatrices. For this it is necessary to represent the Kirchhoff equations as perturbations of integrable equations. We introduce the small parameter ~ having repiaced e by se in the Kirchhoff equations. Then on a four-dimensional level surface of the two integrals M23 = {f2 = (K, e) = c 2, f3 = (e, e) = c3} the Kirchhoff equations turn out to be Hamiltonian with Hamiltonian H = H 0 + sH I + s2H2, are the restrictions of the functions I/2(AK, K), (BK, e), (Ce, e) (respectively) to the level surfaceM2~. Smallness of the perturbation parameter s means that the energy constant f~ = H = c~ is much larger than the constants c 2 and c~. For s = 0 we again obtain the integrable Euler case of the inertial motion of a free rigid body. It is therefore possible to apply the technique of the preceding subsection. We note that in Theorem 1.4.1 nonintegrability of the general Kirchhoff equations was proved not only in a small neighborhood of the integrabie Euler case but also "far" from it, i.e.~ for collections of matrices A, B, C filling out an open dense region in the entire fifteen-dimensional space of the parameters. This is the difference between Theorem 1.4.1 and Theorem 1.2.1, in which nonintegrability of a dynamically nonsymmetric rigid body with a fixed point is proved only in a neighborhood of the integrable Euler case. This is connected with the fact that in the latter case the coordinates of the vector e have the meaning of the direction cosines of a unit vector, and hence (e, e) = i. This prevents multiplication of e by a small parameter s, i.e., it forbids the application of the technique used in the proof of Theorem 1.4.1. 2.

Tool~ical

Obstructions to Complete Integrabilit X

2.1. N o n i n t e ~ o f the Equations of Motion of Natural Mechanical Systems with Two De~rees of Freedom on Surfaces of Large Genusm We consider a natural mechanical system with two degrees of freedom. This means that its configuration space is two-dimensional. We shall assume that it is a two-dimensional, compact, orientable, real-analytic manifold M2o It is known from elementary topology that such a manifold is diffeomorphic to the sphere S 2 to which there are attached g handles (Fig. 14). The number g is usually called the genus of the surface. It is also known that this is the only topological invariant of orientable, closed, connected surfaces, i.e., two surfaces of this type are diffeomorphic if and only if their genera coincide. We consider the cotangent bundle T*M 2 to the manifold M 2. It is well known that the cotangent bundle of an arbitrary smooth manifold T*M n can be made a symplectic 2n-dimensional manifold in a natural way. In the case of the surface M 2, i.e., a system with two degrees of freedom, the cotangent bundle T ~ M 2 has the structure of a four-dimensional, real-analytic, symplectic manifold. The motion of the system is described by the Hamiltonian equations sgradF, where F is the Hamiltonian of the system, which we assume to be a real-analytic function on T*M. We shall take the Hamiltonian in the form F(x, $) = K(x, $) + U(x)~ where K(x, $) for all x~M is a quadratic form in the variables ~ T ~ M , and the function U(x) depends only on x~M. We assume that the functions K(x, ~) and U ( x ) a r e real-analytic on the manifolds T*M and M, respectively. Usually the quadratic function T(x, ~) is identified with the kinetic energy of the system, and the function U(x) is identified with the potential energy of the system (and is called the potential). If the configuration space is not too complex, then such systems often admit complete integrability. To such examples belong, in particular, the inertial motion of a material point on a two-dimensional sphere or two-dimensional torus (given in the standard metrics). As we see, this surface M 2 has small genus, namely: zero (in the case of the sphere) and one (in the case of the torus). If the Riemannian metric on the sphere S 2 and on the torus T 2 is not standard, then, of course, the corresponding Hamiltonian system may not admit complete integration. Nevertheless, it is possible to describe all Riemannian metrics on the sphere and torus for which complete integration is possible; see [45] or the survey [103]. Here we emphasize only that the case of the sphere and torus (in principle) admits complete integration. The fact of integrability depends here only on properties of the Riemannian metric: simple metrics determine integrable system, while complex metrics determine nonintergrable system, in other words, in the present case the obstruction to integrability "rests" in properties of the metric, i.e., it carries metric rather than topological character. It turns out that if configuration space is topologically more complex, i.e., if the number of handles is greater than one, then a purely topological obstruction appears forbidding analytic

2735