Liouville tori

Stability of a trajectory implies that a normal two-dimensional dbk (of small radius) b fibred fully, without gaps, into concentric cbcles. Since we are primarily concerned with integrable systems, the above definition of stability coincides with the traditional notion of strong stability. The fact b that nonsingular level surfaces of a second integral of such a system are two-dimensional Liouville tori, and there-... 

Theorem 2.1.4. (Fomenko. Classification op bifurcations of two-dimensional Liouville tori). 

The general classification theorem for bifurcations of multidimensional Liouville tori is formulated below. For more detailed information, see Ch. 6. 

Topological Surgery on Liouville Tori of an Integrable Hamiltonian System upon... 

Reversing the consideration, we arrive at a modification of two Liouville tori into one. In the nonorientable case, both components and of the separatrix dia gram P of the critical circle 5 are homeomorphic to a Mobius strip (separately). Their boundaries are therefore connected and homeomorphic to a circle. This implies that one torus is modified to exactly one torus and Lemma 2.1.6 follows. 

Lemma 2.1.8. It may always be assumed fin the study of surgeiy on Liouville tori) that on each critical level Ba there exists exactly one critical saddle circle. In other words, it may always be assumed that round handles or thick Mobius strips are glued successively and not simultaneously. 

In both versions of the surgery, it is obvious that a deformed circle (or a doubled circle r ) describe a two-dimensional disk with two holes. Moreover, after one revolution along the circles 5, this disk returns to its initial place, but the two holes of this disk exchange places. We have obtained none other than the manifold fibred over the base with fibre JV. Let be a critical circle of index 2 or 0. Consider its tubular neighbourhood U(S ), Since / is a Bott function, it follows that U S ) fibres into tori which contract onto the circle and are level surfaces of the function /. Since these are Liouville tori, the integral trajectories of the system v lie on them. Then the circle 5 is obviously stable. 

Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams... 

Thus, let dim E = n — 1. Examine a point c on E and investigate the surgery on Liouville tori when a smooth curve 7 (the trace of the motion of the point a) punctures the diagram E at the point c. It suffices to examine only a small neighbourhood U = U(c) of the point c in R. ... 

Definition 2.2.1 We will say that the surgery on Liouville tori which form a nonsingular fibre Ba is the surgery on general position if in the neighbourhood of a modified torus V the surface is a compact and nonsingular submanifold,... 

Proof If T = then T is a common (singular) level surface of all n integrals /i) > /n Level surfaces Rj close to this surface, are nonsingular compact Liouville tori. It is clear that R is the boundary of a tubular neighbourhood of the... 

If a critical torus has the dimension n, then it is either the set of the local minimum or of the local maximum of the energy H. In this case, either two close nonsingular Liouville tori flow into one torus or the torus T splits into two tori r. Let P2 = P2 T ) and = P T ) be, respectively, in and out separatrix diagrams of the critical submanifold... 

A similar Morse-type theory holds for Hamiltonian systems admitting noncom-mutative integration which we described in [188], [l43j. All the assertions proved in the present chapter hold true, with the only exception that n-dimensional Liouville tori are everywhere replaced by r-dimensional tori, where r n. 

Any Composition of Elementary Bifurcations (of Three Types) of Liouville Tori Is Realized for a Certain Integrable System On an Appropriate Symplectic Manifold... 

A similar question arises, of course, in the multidimensional case and is also answered in the affirmative. We have seen that any surgery on Liouville tori of general position splits into a composition of five canonical simple modifications of types I, II, III, rV, V. 

As proved above, each singular fibre containing a saddle critical circle is obtained as follows. One takes two nonsingular Liouville tori, sets a cycle on each of them (that is, draws noncontractible circles), after which glues both tori by these... 

In 2 we have described the surgery on Liouville tori which occurs at the moment when a point a, moving along a smooth segment 7 in R", intersects the bifurcation diagram nonsingular point on a stratum of dimension n — 1. 

In this theorem, we did not imply any concrete Hamiltonian system, but described the properties of the whole class of fields of the form sgrad h generated by the annihilator of the covector of general position. A particular case of Theorem S.1.1 is, of course, the classical Liouville theorem. Indeed, if the maximal linear subalgebra of functions G is commutative then its index r = ind G is equal to its dimension k and, theorefore, the maximality condition becomes A + A = dim Af = 2n, that is A = n. In this case, all the tori 17 from Theorem 3.1.1 are ordinary n-dimensional Liouville tori. 

Definition 6.1.1 Let us caU the Hamiltonian H nonresonance on a given isoenergy surface if in Q everywhere dense are Liouville tori on which integral trar jectories of a system t form a dense irrational winding. 

Digits in Fig. 94 indicate the number if Liouville tori of which consists the complete... 

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