Topological obstructions to complete integrability

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

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Hamiltonian systems from having a sufficient number of analytic, commuting, functionally independent integrals on the cotangent bundle. In other words, Hamiltonian systems of this type do not admit complete integration if the configuration space has large genus (greater than one). THEOREM 2.1.1 i(Kozlov; see [41]). Suppose the real-analytic configuration space M 2 has genus greater than one, i.e., it is homeomorphic to a sphere with g handles where g > i. Then the canonical Hamiltonian equations ~ = -dF/dx, i = dF/d$, where F(x, ~) is a real-analytic Hamiltonian on T*M 2, has no first integral which is real-analytic on T*M 2 and is functionally independent of the energy integral (i.e., of the Hamiltonian) F(x, ~). If the assumptions of Theorem 2.1.1 are relaxed and attention is restricted only to smooth manifolds M 2 and smooth Hamiltonians, then, in general, the assertion of Theorem 2.1.1 is false. More precisely, it is easy to construct an example of a smooth Hamiltonian F(x, g) = K + U on T*M 2 such that the canonical Hamiltonian equations on T*M 2 have an additional (second) smooth integral not depending on the function F(x, ~) on some open subset (which is not, by the way, dense in T'M2); see [41]. At the same time it is unclear whether it is possible to construct a natural "mechanical" smooth Hamiltonian for which there exists a second additional smooth integral which is independent almost everywhere, i.e., on an open dense subset in T~M 2, where genus (M 2) > io Another proof of this theorem was later obtained by Kolokol'tsov [45]. Theorem 2.1.1 is a corollary of a more general assertion regarding nonintegrability of equations of motion for fixed, sufficiently large values cf the total energy. We consider the four-dimensional manifold T * ~ 2 and a Hamiltonian system sgrad F where F = X + U. The level surfaces Qh={(X, ~)~T*/H21~(x, ~)=h=const} are given by the equation K + U = h and are three-dimensional, analytic submanifolds of T*M m if the energy constant h is sufficiently large. Namely, it suffices to assume that h > maxU, where the maximum of the potential energy U is taken over the entire manifold M 2. Under this assumption we have K = h - U > 0 on the level surface F = h. Since K is a positive-definite quadratic form on each tangent space, it follows that at each point x@)M 2 the equation K(x, ~) = h + U(x) > 0 defines a circle in the tangent plane T,M 2 (Fig. 15). It is clear that we have here made essential use of the assumption that the energy constant h is sufficiently large. For small h a solution of the equation K = h - U = const may, for example, degenerate, and instead of a circle in the tangent plane we obtain a point. Thus, for sufficiently large h a level surface of the Hamiltonian F turns out to be a three-dimensional, analytic manifold Q which is fibered over the manifold M 2 with fiber a circle. Since the vector field s g r a d F is tangent to the surface Qh = {F = h}, we obtain on O~ an analytic system of differential equations. THEOREM 2.1.2 (see [41]). If the genus of the analytic surface M 2 (the configuration space of the original system) is greater than one, then for all h > m a x U the analytic flow sgrad FIQ on the three-dimensional manifold of constant energy O~ has no additional realanalytic integral. In the smooth case under the assumptions of Theorems 2.1.1 and 2.1.2 it is possible, it turns out, to assert the absence of new (additional) smooth integrals satisfying certain conditions. THEOREM 2.1.3 (see [41]). If the genus of the smooth surface M 2 is greater than one, then for all sufficiently large h, i.e., for h > m a x U , the flow s g r a d F I Q k has no smooth first integral f defined on the smooth level surface Q~ and satisfying the following conditions: i.

The integral f, considered as a smooth function on the level surface of the energy Q~, has only a finite number of critical values (i.e., values to which there correspond critical points of the function f on Q~).

2.

Points x@/H 2 for which the sets {f = c = const} are finite or coincide with the entire fiber - the circle S~ - are dense in M 2. Here S~cTx/Pl 2.

If the surface M 2 and the Hamiltonian F are analytic, then both conditions I and 2 of Theorem 2.1.3 are automatically satisfied (property i needs special proof); therefore, Theorem 2.1.1 follows immediately from Theorem 2.1.3 in the analytic case. More generally, if a compact, orientable surface M 2 is nonhomeomorphic to a sphere or torus, then the equations of motion of the system indicated above have no new integral which is a smooth function on T*M 2, is analytic for fixed x@M on the two-dimensional cotangent planes T~M, and has only a finite

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TxM ~

Fig. 15

Fig. 16 number of distinct critical values. The number of critical points hereby need not be finite. Functions which are polynomials in the velocities give an example of integrals analytic in the momenta $. It is easy to see that a smooth function on a compact% smooth, closed manifold has a finite number of critical values if all its critical points are isolated (there are then a finite number of them) or if they fill out nondegenerate critical submanifolds (then there are also a finite number of them). 2.2. Nonintegrability of Geodesic Flows on Riemanian Surfaces of Large Genus with a Convex Boundary. In this subsection we present results obtained by S. V. Bolotin. Let M 2 be a connected, compact, two-dimensional, analytic Riemannian manifold with boundary 8M homeomorphic to a circle (i.e., the boundary is connected). It is well known from elementary topology that each such manifold is obtained as follows: it is necessary to excise from a sphere with g handles a certain number of sufficiently small, two-dimensional, open, nonintersecting disks (Fig. 16). In other words, such a manifold is homeomorphic to a two-dimensional plane region (the boundary of which, generally speaking, is not connected). We consider the Euler characteristic x(M 2) of such a surface. By defi~ition it is equal to the number 1 - rankH1(M; Z ) (we assume that the surface is connected). From a homotopy point of view the surface M with nonempty boundary 8M is homotopically equivalent to a socalled "bouquet" of several circles, i.e., it is obtained from a finite collection of circles by gluing one to the other at a single point (Fig. 17). The number k of these circles is precisely equal to the rank of the one-dimensional homology group HI(]W; Z). Moreover, this group is always free [in the case of a nonempty boundary it is also isomorphic to Z G . . . O Z (k times) _--~Z~]. For example, by excising one disk from the two-dimensional torus we obtain a torus with one hole which is homotopically equivalent (i.e., continuously contractible in the present case) to a bouquet of two circles, i.e., in the present example k = 2 and x(M) = 1-2 = -i (see Fig. 18). We note that by excising from a surface M with nonempty boundary 8M more and more disks we, of course, change the Euler characteristic of the surface, generally speaking. On M we consider the geodesics defined by a given Riemannian metric. The boundary 8M (generally speaking, nonconnected) is called locally geodesically convex if any two sufficiently nearby points x and y lying on the boundary can be joined by a unique geodesic lying entirely inside the manifold M (Fig. 19). We consider the cotangent bundle T*M and let H be the quadratic form defining a Riemannian metric on M.

At each p o i n t , x ~ M

it gives a positive-definite

form ff(x,~)~gij(x)~i~L i,/

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SI

81_t

SK Fig.

17

TZ\~ z

SIVS~

Fig. 18

,X

,y DM Fig. 19 A geodesic flow has the function H(x, ~) as its first integral. We consider the three-dimen3 X z$)~T*fWllq(x,~)--h=const}. As we know sional level surfaces Q~ of the integral H, i.e., Qh={( (see part 2.1), these surfaces are fibered over the surface M with fiber a circle if h > 0. For simplicity we fix the value h = i equal to one, and consider the three-dimensional manifold Q~. THEOREM 2.2.1 (S. V. Bolotin). Let M 2 be a connected, compact, two-dimensional, realanalytic Riemannian manifold with a locally geodesically convex boundary and such that x(M) < 0. Then a geodesic flow of the Riemannian metric on the three-dimensional manifold of constant energy h = i has no analytic first integral not dependingion the energy integral and in involution with it on T*M. If the boundary of the manifold is empty, then we obtain Theorem 2.1.1o 2.2.1 generalizes Theorem 2.1.1.

Thus, Theorem

2:.3: Nonintegrability of t h e P r o b l e m of n Gravitating Centers for n > 2. Let z!,...,z n be distinct points of the complex plane'C. The Hamiltonian of the problem of n centers has the form 1

ff=~ [p[~-JcV (z); (z, p ) E T * U = U X C ,

where

U--=C\{zl..... zn}

tion of a point

z6U

is configuration space, and V is the potential of gravitational attrac-

by points

zl.....zn: V(z)==--~/Iz--z~[, i=I

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~0.

Since V < 0, for h > 0

the level surface {H = h} of the energy integral H is an analytic hypersurface in phase space. THEOREM 2.3.1 (S. V. Bolotin). Let n > 2. Then for h~>0 the problem of n centers has no analytic first integrals on the level surface of the energy integral {H = h}. Thus, the problem of n centers is integrable in the Liouville sense in the region { H ~ 0 } only in the Kepler and Euler cases. The question of integrability of the problem of n centers in the region {H < 0} remains open. The classical methods of Poincar~'s perturbation theory are apparently not applicable for a proof of nonintegrability of this problem. The proof of Theorem 2.3.1 is based on the existence of an infinite nuz~ber of unstable periodic motions on a level of nonnegative energy. 3.

T~ological

Obstructions to Analytic I n t e ~ b i l i t y

of Geodesic Flows

on Manifolds Which Are Not Simply Connected In this section we present a generalization of V. V. Koziov's theorem on analytic integrability of geodesic flows of Riemannian metrics on two-dimensional surfaces of large genus. This generalization was obtained by I. A. Taimanov and is valid for analytic Riemannian manifolds of arbitrary dimension which possess a "sufficiently large" fundamental group. Let M n be a closed manifold. We consider a geodesic flow on M n which on the cotangent bundle T*M n with the natural symplectic structure is Hamiltonian with Hamiltonian H(x, p) = i/2gij(x)piPj, where x67FJn, Pi are the coordinates in a fiber of the cotangent bundle T*M n , and gij(x) is the Riemannian metric on M n. By Liouville's theorem for a sufficiently good description of the geodesic flow it suffices to produce a collection of functionally independent, involutive first integrals of the flow. Now we already have one integral - this is the Hamiltonian H, and we can therefore reduce the geodesic flow to the level surface H = i which we denote by L. On L there is given the moment mapping F:L--~R~-I, F(q)=(f1(q) ..... In_1(q)) , where {71.... , IN-l, I~=H} is a complete collection of functionally independent first integrals in involution. We define the concept of a geometrically simple collection of first integrals. Definition 3.1. A complete collection of involutive first integrals of a geodesic flow ll,...,In- I, In = H is called geometrically simple, if i) L contains a closed set F such that t

L \ r is open and dense, has a finite number of linearly connected components L \ F =

U Us,

and on L \ F the moment mapping F has maximal rank; 2) F:U~ -> F(U~) is a fibering into ndimensional Hamiltonian tort over the regions F(II~) homeomorphic to (n - l)-dimensional disks T(U~)-------D~-I; 3) for any point qCL there exists a neighborhood W(q) such that W(q)f](L\F) has a finite num.ber of linearly connected components. For such a situation we have THEOREM 3.1. If fundamental group ~I(M n) of a closed Riemnnian manifold M ~ is not almost commutative, i.e., does not contain a commutative subgroup of finite index, then a geodesic flow on M n does not admit a geometrically simple collection of first integrals. THEOREM 3.2. If dimHi(Mn;Q)>dirnM n, then a geodesic flow on M n does not admit a geometrically simple collection of first integrals. We consider the analytic case: M n is an analytic manifold with an analytic metric. In this case w~ith help of results of Gabrielov [18, 76] extending the work of Loyasevich [152] the fact can be proved that a complete collection of involutive analytic integrals is geometrically simple. This follows from the semianalyticity of the set of critical points of F : L - + R ~-I and the constructiveness [76] following from this of the set of critical values~ the complement to which in R ~-I has a finite number of linearly connected components. By augmenting the set of critical values~ C l by an additional constructive set C 2 consisting of "'partitions," we arrange that R~-I\(CIUC2)decomposes into a finite number of disks. Then F=F-I(CIUC2) and condition 3) follows from the closedness of the family of constructive sets relative to taking the complete preimage under a proper analytic mapping. From this we obtain THEOREM 3.3. If the fundamental group ~I(M n) of an analytic closed Riemannian manifold M n is not almost commutative, then a geodesic flow does not possess a complete involutive

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