Morse theory of completely integrable Hamiltonian systems

Fig. 4

Fig. 3

arranged in the simplest possible manner~ In the coordinates ~ b . . , ~ it becomes a field with constant (!) components, Joe., it is completely determined by giving the velocity vector at some one point of the torus. All the remaining velocity vectors are obtained from it by parallel translation (transport) in the coordinates ~I ..... ~ (see Fig. 3). The Liouville theorem acquired great importance in modern geometry and mechanics due to the fact, as it turned out, that many mechanical systems (and their multidimensional analogues) possess a collection of integrals in involution which make it possible to integrate the system in the sense indicated. Definitio N 1.3.1. We shall say that a Hamiltonian system v = sgrad f on a symplectic manifold M 2n is completely integrable in the Liouville sense (or admits complete commutative integration) if for it there exists a collection of n functions fl,...,fn in involution, whereby fl = f and the functions fl,...,fn satisfy the conditions of Liouville's theorem. The problem of Liouville integration of a given system means the inclusion of its Hamiltonian f in a family of functions in involution and such that from them it is possible to select n independent functions where n is half the dimension of the enveloping symplectic manifold. If it is possible to find such a collection of functions, then (under the assumptions of Theorem 1.3.1) the trajectories of the system in question move along tori of half the dimension, giving on them conditionally periodic motion (in the appropriate "action-angle" coordinates). From the point of view of the Lie algebra F(M) of functions on M 2n the problem of integration of aHamiltonian system is equivalent to seeking a commutative subalgebra T(f) containing the Hamiltonian f of the given system and possessing the property that in T(f) it is possible to select an additive basis of n independent functions. The following question arises: on any analytic (algebraic) manifold does there exist a complete involutive collection of analytic (algebraic) functions? Apparently, in this class of manifolds the answer to this question is negative. It is very probable that there exist topological and analytic obstructions which make it impossible to construct a complete involutive collection on an arbitrary algebraic symplectic manifold. For a more detailed study of the problem it is necessary to consider a fibration with singularities arising on a symplectic manifold on which there is at least one complete involutive coilection fl .... ,fnIndeed, we consider the mapping g:M2~-+R ~, where g(x) = (fl(x) ..... fn(X)). By the Liouville theorem almost all the fibers of this mapping are n-dimensional tori, while on the singular fibers there is defined an action of n one-parameter subgroups generated by the flows sgrad fi" In the next section>this program will be realized for a special class of functions "Morse" functions. From this it follows that the search for complete involutive families of functions is a very complex problem. In Chap. 4 we shall demonstrate to the reader that a Hamiltonian "chosen at random" most often generates a nonintegrable system. 2.

Morse Theory of Completely InteBrable Hamiltonian Systems.

Topology

of the Level Surfaces of Constant Energy of Hamiltonian Systems, Obstructions t__o Integrability,

Classification of Rearrangements

of General Position of

Liouville Tori in a Neighborhood of a Bifurcation Diagram In the present section we briefly present the elements of a new theory of "Morse type ~' of integrable Hamiltonian systems constructed by A. T. Fomenko. 2688

2.1~ The Four-Dimensional Case. Recently many new cases of Liouville integrability of important Hamiltonian systems on symplectic manifolds M 2n have been discovered. In connection with this a very current problem is that of finding stable periodic solutions of integrable systems (see Definition 2.1.2). Some cases of the existence of periodic trajectories of Hamiltonian systems are noted in the works [125, 128]. It turns out that for n = 2, proceeding only from information regarding the one-dimensional integral homology group HI(Q; Z) (or from information regarding the fundamental group) of a fixed three-dimensional surface Q3 of constant energy on which the given system is integrable it is sometimes possible to guarantee the existence at least of two stable periodic solutions of the system on this surface Q3 c M 4. These solutions can be effectively found by studying the minima and maxima of an additional (second) integral defined on the surface of constant energy. Thus, this result not only gives the existence of two stable solutions but also makes it possible (in principle) to find them. This assertion follows from the more general classification assertion of A. To Fomenko regarding the canonical representation of such a surface Q3 in the form of a union of elementary manifolds of the four simplest types. It is hereby assumed that the system v possesses on Q3 a second smooth "Morse" integral, i.e., one such that its critical points on the surface Q~ are organized into nondegenerate submanifolds. In connection with this A. T. Fomenko develops a specific theory of "Morse type" of integrable systems which differs from the usual Morse theory and uses the familiar Bott theory of functions (see [124]) with degenerate critical points (such functions could be called Bott functions). Here also some important ideas of Novikov [72, 74], Kozlov [41], and Anosov [I] receive a natural development. It further turns out that nonsingular surfaces of constant energy of integrable Hamiltonian systems possess specific properties which distinguish them among all three-dimensional manifolds. From this, new topological obstructions to the integrability of Hamiltonian systems in the class of Morse functions are obtained. Thus, suppose on M ~ there is given a Hamiltonian system v = sgradH where H is a smooth Hamiltonian. We consider a fixed noncritical surface Q3 of constant energy, i.e., Q = {H = const} and gradH ~ 0 on Q. Suppose the system v is integrable on Q with the help of a second independent smooth integral f which commutes with H. Definition 2.1.1. The integral f is called a Morse (or Bott) integral on Q if its critical points form on Q nondegenerate critical submanifolds, i.e., the Hessian d2f is nondegencrate on subspaces normal to these submanifolds (Fig. 4). The class of such integrals is broader than the class of analytic integrals. The acquired experience of investigating concrete mechanical systems shows that the majority of integrals already found are Morse integrals. Definition 2.1.2. Let ~ be a closed integral trajectory of the system v on Q3 (i.e., a periodic solution). We say that y is stable if some tubular neighborhood of it in Q is completely fibered into two-dimensional tori invariant relative to the system v. An integrable system may not have stable periodic solutions. Example: a geodesic flow of the plane two-dimensional torus. We prove the existence of a simple connection between the following three objects: a) a Morse integral f on Q, b) stable periodic solutions of the system v on Q, c) the integral homology group HI{Q; Z) [or the fundamental group ~I(Q)]. THEOREM 2.1.1 (A. T. Fomenko). Let v = sgradH be a Hamiltonian field on a smooth symplectic four--dimensional manifold M 4 (compact or noncompact) where H is a smooth Hamiltonian. We suppose that the system v is integrable on some nonsingular, compact, three-dimensional level surface Q of the Hamiltonian H by means of a Morse integral f on Q. If the homology group HI(Q; Z) is finite, then v has on Q no fewer than two stable periodic solutions, f hereby achieves a local minimum or maximum on each of these trajectories. This criterion is effective, since verification of the Morse property of the integral f and computation of the rank of HI(Q; Z) are usually not hard. In concrete examples the surfaces Q of constant energy (or reductions of them) are frequently diffeomorphic to the sphere S 3, projective space R P 3 , or S l x S ~ For example, after suitable factorization for the equations of motion of a heavy rigid body in the zone of large velocities it may be assumed that Q ~ S I • S s [41]. In the problem of motion of a four-dimensional rigid body by inertia with a fixed point we have Q ~ S l • S 2. In the integrable (three-dimensional) case of Kovalevskaya it may be assumed that some Q ~ S I • S 2. If the Hamiltonian H has an isolated point of a minimum or maximum on M 4, then all sufficiently nearby level surfaces Q are the spheres S a.

2689

Proposition 2.1.1. Suppose the system v = s g r a d H is integrable by means of a Morse integral f on some surface of constant energy Q homeomorphic to S 3, R P 3, or S I • S 2. Then the system v has on Q at least two stable periodic solutions. In particular, as we see, the integrable system has two stable periodic solutions not ~ only on small spheres surrounding the point of a minimum or maximum of H but also on all "distant" expanding level surfaces as long as they are homeomorphic to S s. The criterion of Theorem 2.1.1 is sharp in the sense that examples are known where the system has on Q = R P s exactly two (and no more) stable periodic solutions. Let R = rank ms(Q) be the least possible number of generators of the fundamental group of the surface Q. THEOREM 2.1.2. Suppose the system v is integrable on some nonsingular compact surface QS of constant energy in M 4 by means of a Morse integral f. If R = i, then v has on Q no fewer than two stable periodic solutions on which f achieves a local minimum or maximum. If rank HI(Q; Z ) ~ 3 , then v may have on Q no stable periodic solutions at all. In the case of an integrable geodesic flow of the plane torus T 2 we have Q = T s, rank H I ( Q ; Z ) = 3 , and all periodic solutions of this system are unstable. From results of Anosov, Klingenberg, and Takens (see [i, 36]) it follows that in the set of all geodesic flows on smooth Riemannian manifolds there exists an open, d e n s e s u b s e t without stable periodic trajectories. Thus, the property of a flow of not having stable trajectories is a property of general position. COROLLARY 2.1.1. We consider a two-dimensional manifold diffeomorphic to the sphere with a Riemannian metric of general position, i.e., without stable closed geodesics. Then the corresponding geodesic flow is nonintegrable in the class of smooth Morse integrals on each individual nonsingular surface of constant energy. Question: can any three-dimensional manifold be a surface of constant energy of an integrable system? COROLLARY 2.1.2. It is far from the case that each three-dimensional, smooth, compact, orientable manifold can play the role of a surface of constant energy of a Hamiltonian system integrable by means of Morse integral (on this surface). Thus, the topology of the surface Q serves as an obstruction to integrability. All these results follow from the general Theorem 2.1.3 (see below). If f is a Morse integral on Q, then with each critical submanifold T of it there is connected a separatrix diagram P(T), i.e., the set of integral trajectories of the field g r a d f entering T and leaving T. We call the integral f orientable if all these separatrix diagrams are orientable. Otherwise we call the integral nonorientable. We consider the following simplest three-dimensional manifolds whose boundaries are two-dimensional tori T 2. !) Solid tori S I x D 2. 2) Cylinders T 2 x D z. 3) Direct products (oriented saddles) N 2 x S I, where N 2 is the disk with two holes. 4) We consider the nontrivial fibration A s + S l with base S I and fiber N 2. The boundary of the manifold A s is formed by two tori T 2. It is clear that A s (we call it a nonoriented saddle) is realized in R 3 as a solid torus from which a second (thin') solid torus passing twice around the axis of the large solid torus (a double coil) is bored out (Fig. 5)~ Space K s of the oriented skew product of the Klein bottle with a segment; its boundary is the torus T 2. THEOR~I 2.1.3 (A. T. Fomenko). (The basic classification theorem in dimension 4.) Let v = sgrad H be a Hamiltonian system integrable on some nonsingular, compact, three-dimensional surface of constant energy QS c M 4 by means of a Morse integral f. Let m be the number of periodic solutions of the system v on the surface Q on which the integral f achieves a local minimum or maximum (then they are stable). Then Q = m(S I x D 2) + p(T = x D I) + q(N 2 x S I) + s(A a) + ZK s, i.e., Q is obtained by gluing together m solid tori, p cylinders, q orientable saddles, s nonorientable saddles, s manifolds K s along some diffeomorphism of the boundary tori. If the integral f is orientable, then s = 0, i.e., there are no nonorientable saddles. 2.2. The General Case. Let v = s g r a d H be a smooth, integrahle system on M 2n, and let F:~I2~-+R" be the moment mapping, i.e., F(x) = (f1(x), .... fn(X)), where fi are commuting smooth integrals and fl = H. A point x6fW is regular if rank dF(x) = n and is critical otherwise. Let N c M be the set of critical points, and let E = F(N) be the set of critical values (the bifurcation diagram). If a~R~\Z, then the compact fiber B a = F - * ( a ) c / W == consists of Liouville

2690

S~x2Z

b

8,

A3

~ff. ~L -'..

d

C

Fig.

5

Under deformation of a outside I the fiber B a is transformed by means of diffeomorms. If the curve 7 along which a moves encounters Z, then the fiber B a is subject to Logical rearrangements. Problem: describe such rearrangements. It turns out that a com~_ solution of the problem exists in the case of general position. If d i m e < n - i, then Fibers B a where ~ C R ~ \ ~ are diffeomorphic. The main case is when d i m e = n - i. We con5 types of (n + !)-dimensional manifolds, i) We call the direct product D 2 • T n-i a lpative solid torus. Its boundary is the torus T n. 2) We call the product T n • D i a ~der. Its boundary is two tori T n. 3) Let N 2 be the two'dimensional disk with two ~. We call the direct product N 2 x T n-I an oriented toric saddle. Its boundary is formed N~

nree tori T n. 4) We consider all nonequivalent fibrations A ~ - ~ T n-1 with base the torus and fiber N 2. They are classified by elements ~Ef-II(Tn-I;~Z2)=Z~-~~ A special case is T n-l for ~ = 0. If ~ ~ 03 then the fibration A~ is nontrivial. We call the manifold 3r ~ ~ 0 a nonoriented toric saddle. It has as boundary two tori T n. 5) Let p:T n + K n two-sheeted covering over the nonorientable manifold K n. We denote by K n+1 the cylinder F he mapping p. It is clear that d i m K n+~ D = n + 1 and 3Kpn+~ = T n . We shall describe five s of rearrangements of the torus T n. - i ) The torus T n realized as the boundary of the ipative solid torus D 2 • T n-1 is contracted onto its "axis" - the torus T n-1 (we write T n-l § 0). 2) Two tori T n and T n - the boundary of the cylinder T n • D I - m o v e toward another and coalesce into one torus T n (i.e., 2T n + T n + 0). 3) The torus T n - the lower dary of the oriented toric saddle N 2 • T n'1 - rises upward and in correspondence with the logy of N 2 • T n-1 decomposes into two tori T n and T n (i.e., T n + 2Tn). 4) The torus T n of the boundaries of A~ - is lifted along A~ and in its "middle" is rearranged again into (doubly coiled) torus (i.e., Tn-+T"). Such rearrangements are parameterized by nonzero nl n 1 ents 5) We realize the torus T n as the boundary of Kpn + l . By deformit inside Kp along the projection p, we finally cover the nonoriented manifold K n in sheeted fashion by the torus T n after which the torus "vanishes." We fix the values of last n - i integrals f2 ..... fn and consider the (n + l)-dimensiona! surface X n+l obtained. estricting fl = H to it, we obtain a smooth function f on X n+l. We shall say that a rengement of Liouville tori forming a nonsingular fiber B a (assumed compact) is a rearrangeof general position if in a neighborhood of the rearranged torus T n the surface X n+i is acts nonsingular, and the restriction f of the energy fl = H to X n+i is a Morse function he sense of part 3.1 in this neighborhood. In terms of the diagram Z this means that path y along w h i c h m o v e s pierces Z transversa!ly at a point c, a neighborhood of which is a smooth (n - l)-dimensional submanifold in ~R~, and the last n - i integrals f_~.....

(~6~-/1(T n+-;1Z2)=Z2-'-

~re independent

on X n+l

in a neighborhood

of t h e

torus T n .

THEOREM 2.2.1 (A. T. Fomenko). Classification theorem of rearrangements of Liouville .) i) if d i m e < n - i, then all nonsingular fibers B a are diffeomorphic. 2) Let dim n - i. Suppose a nondegenerate Liouville torus T n moves along a joint (n + l)-dimen~al nonsingular level surface of the integrals f2,...,fn carried along by the change of value of the energy integral fl = H. This is equivalent to the situation that the point ;(T~)6R ~ m o v e s along a path y in a direction toward Z. Suppose the torus T n is subject to :rangement. This occurs if and only if T n encounters critical points N of the moment mapF [i.e., the path y at a point c transversally pierces the (n - l)-dimensional sheet E]. I all possible types of rearrangements of general position are exhausted by compositions 2691

of the five canonical rearrangements 1-5 indicated. In case 1 (the rearrangement T n + T n-1 § 0) with growth of the energy H the torus T n is converted into the degenerate torus T n-1 after which it vanishes from the surface of constant energy H = const (limiting degeneration). In case 2 (the rearrangement 2T n + T n § 0) with growth of the energy H two tori T~ and T~ coalesce into the single torus T n after which they vanish from the surface H = const. In case 3 (i.e., T n + 2T n) with growth of H the torus "breaks through" the critical level of the energy and decomposes into two tori T~ and T~ on the surface H = const. In case 4 (i.e., T n + T n) with growth of H the torus T n "breaks through" the critical energy level and again becomes the torus T n (nontrivial transformation of a two-sheeted winding). In case 5 the torus T n covers in two-sheeted manner the nonorientable manifold K n after which it vanishes from the surface H = const. By changing the direction of motion of the torus T n, we obtain five reverse processes of bifurcation of the torus T n. The rearrangements known earlier of two-dimensional tori in the Kovalevskaya case and in the Goryachev-Chaplygin case (see the work of Kharlamov [112]) are special cases (and compositions) of the rearrangements described in Theorem 2.2.1. Rearrangements of Liouville tori were also studied by Pogosyan [79]. As H varies the torus T n drifts along the level surface X n+~ of the integrals f2,...,fn. It can happen that T n contracts to the torus T n-~. Such limiting degenerations arise in mechanical systems with dissipation. If a small friction is introduced into an integrable system, then in first approximation it may be assumed that dissipation of the energy is modeled by a decrease in the value of H and hence causes a slow evolution (drift) of Liouville tori along X n+1. Answer to the question: what is the topology of the surface X n+1, is given by the following THEOREM 2.2.2. Suppose M 2n is a smooth symplectic manifold and the system v = sgradH is integrable by means of smooth, independent, commuting integrals H = fl, f2,...,fn 9 Let X n+1 be any fixed compact, nonsingular joint level surface of the last (n - i) integrals. Suppose the restriction of H to X n+1 is a Morse function. Then %n~i:___ ~ ( D 2 X T n - a ) _ ~ p ( T ~ X

D I) + q(N2XT~-I)-~ ~s~(A~),-l-rKp,

i.e., X n+1 is obtained by gluing together the boundary tori

=~0

(by means of certain diffeomorphisms) of the following "elementary bricks": m dissipative solid tori, p cylinders, q toric oriented saddles, s ~ s ~

toric nonoriented saddles, and r

manifolds Kp. The number m is equal to the number of limiting degenerations of the system v on X n+~ on which H achieves a minimum or maximum. Theorem 2.1.3 follows from Theorem 2.2.2 for n = 2. All the results enumerated are also valid for Hamiltonian systems admitting "noncommutative integration." In these cases the Hamiltonian H is included in a noncommutative Lie algebra G of functions M 2n such that rank G + dimG = dimM 2n. The trajectories of the system then move along tori T r, where r = rankG. The following assertions are used in the proof of the results enumerated. LEMMA 2.2.1. Suppose Mq is a stratum of the critical set N on which precisely q integrals of the system v are dependent. Then the intersection of Nq with the singular fiber B c = F-1(c) consists of the disjoint union of tori TJ (if the intersection is compact) where

n - - q + l~]