Bott integral

We assume a Hamiltonian system to posses a second smooth integral which will be called a Bott integral. This is an integral whose critical points may be degenerate but are of necessity organized as nondegenerate smooth (critical) submanifolds. 

Definition 2.1.1 We will call a smooth integral / a Bott integral on a level surface Q if the critical points of the function / and Q form nondegenerate critical smooth submanifolds. 

Definition 2.1.3 We say that a Bott integral / on a surface Q is orientable if all of its critical submanifolds are orientable. If at least one of its critical submanifolds is nonorientable, we say that the integral / is nonorientable. 

An isoenergy surface on which there exists a Bott integral will be sometimes referred to as an integrable surface , for short. 

FVom this it follows that if / is a nonorientable integral on Q, then (Q) 0, and the group xi(Q) contains a subgroup of index two. If, for instance, Q is homeomorphic to the sphere (a particular case in mechanics), then any Bott integral / on 5 is always orientable. 

Proposition 2.1.1. Let a Hamiltonian system v = sgradH be integrable by means of a Bott integral f on some separate constant-energy surface Q homeomorphic to one of the following three-dimensional manifolds S, RP, S x S, Lp q,... 

Consider a geodesic flow of a flat two-dimensional torus, that is, a torus with a locally Euclidean metric. This flow is integrable in the class of Bott integrals and obviously has no closed stable trajectories. By virtue of Proposition 2.1.2, we must have rank i(Q) 2. Indeed, the nonsingular surfaces Q are diffeomorphic here to a three-dimensional torus T, for which Hi T, Z) = Z 0 Z 0 Z. 

V is nonintegrable in the class of smooth Bott integrals on a given surface Q,... 

An application of this assertion is seen, for instance, from CoroUary 2.1.1 on nonintegrability of geodesic flows of general position on a two-dimensional sphere. In the case of a geodesic flow of a flat torus T, we have Q = T, Hi T, 2) = Z, R = S (that is, the conditions of Corollary 2.1.2 are not fulfilled), and although the flow has no closed stable trajectories on Q, it is nonetheless integrable in the class of Bott integrals. 

Corollary 2.1.4. Not nearly each three-dimensional smooth compact closed orientable manifold may play the role of a constant-energy surface of a Hamiltonian system integrated by means of a smooth Bott integral. 

Let be a smooth symplectic manifold (compact or noncompact) and let V = sgradfT be a Hamiltonian system that is Liouville-integrable on a certain nonsingular compact three-dimensional constant-energy surface Q by means of a Bott integral /. Let m by the number of such periodic solutions of the system v on the surface Q on which the integral f attains a strictly local minimum or maximum (then the solutions are stable). Next, let p be the number of two-dimensional... 

Theorem 2.1.3. Let Q be a, compact nonsingular constant-energy surface of a Hamiltonian system v = sgrad on Q integrated by a Bott integral f. Then Q admits the following representation ... 

The former will be called a Hamiltonian decomposition of boenergy surface, and the latter a topological decomposition. It b clear that the Hamiltonian decomposition b more "detailed. It "remembers the structure of the critical submanifolds of the Bott integral. The topological decomposition b rougher. Its elementary blocks have already partially "forgotten the original Hamiltonian picture. In the sequel, we will use one or the other decompositions subject to the problem of interest. 

Class (H) consists of constant-energy (isoenergy) surfaces of integrable Hamiltonian systems (integrated by a Bott integral /). 

Claim 2.1.4 (Fomenko, Zieschang [289], [298]).. Let Q e H), that iSf is a constant-energy surface of an integrable system (by a Bott integral f). Let m be the number of stable periodic solutions of the system, s the number of unstable periodic solutions with nonorientable separatrix diagram, and r the number of critical Klein bottles. Then we always have the following inequalities ... 

Lemma 2.1.1. A smooth Bott integral f cannot have isolated critical points on a nonsingular compact constant-energy surface Q. 

Lemma 2.1.2. The critical points of a smooth Bott integral f on a compact nonsingular surface Q 611 either isolated smooth critical circles or smooth two-dimensional tori, or Klein bottles. 

Classification of Nonorientahle Critical Submanifolds of Bott Integrals... 

We have discovered nonorientable manifolds Ifp, which are minima or maxima of a Bott integral f on a surface In the present subsection we give a complete... 

Theorem 2.3.1. Let v = sgrad H be a Hamiltonian system on Liouville-integrable on one constant- energy surface by means of a Bott integral /. Then each singular saddle level surface of the integral f is obtained by gluing... 

Theorem 6.1.2 (FOMENKO). Let Q be a compact nonsingular isoenergy surface of a system v, with a Hamiltonian H (not necessarily nonresonance), integrable by means of a certain Bott integral /. Then the manifolds U(fc) entering in the decomposition Q = 12c (/c) the following representations depending on... 

X in the algebra so(4) fibre into three-dimensional isoenergy surfaces = = const. These surfaces are such that for all except for a Bnite set, the function Bott integral. Depending on the values of pi and p2,... 

Since each Bott integral is apparently tame, a trivial inclusion (H ) 3 (H) takes place. In other words, extending the class of considered integrals, we have also extended the class of three-dimensional manifolds which are isoenergy surfaces. From the point of view of three-dimensional topology and its applications to Hamiltonian mechanics, the question is of interest whether or not the classes H ) and [H) coincide. To say it differently, to what extent the assumption about the Bott character is essential in many theorems of the Morse-type theory developed in Chapter 2. 

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