By A.V. Bolsinov

Integrable Hamiltonian structures were of becoming curiosity during the last 30 years and characterize the most fascinating and mysterious periods of dynamical structures. This ebook explores the topology of integrable structures and the final concept underlying their qualitative homes, singularities, and topological invariants.The authors, either one of whom have contributed considerably to the sector, improve the class conception for integrable platforms with levels of freedom. This conception permits one to differentiate such platforms as much as average equivalence kin: the equivalence of the linked foliation into Liouville tori and the standard orbital equaivalence. The authors exhibit that during either situations, you will discover whole units of invariants that provide the answer of the category challenge. the 1st a part of the publication systematically offers the final development of those invariants, together with many examples and functions. within the moment half, the authors observe the final tools of the category thought to the classical integrable difficulties in inflexible physique dynamics and describe their topological pics, bifurcations of Liouville tori, and native and worldwide topological invariants. They convey how the category conception is helping locate hidden isomorphisms among integrable platforms and current as an instance their evidence that well-known systems--the Euler case in inflexible physique dynamics and the Jacobi challenge of geodesics at the ellipsoid--are orbitally equivalent.Integrable Hamiltonian structures: Geometry, Topology, type deals a distinct chance to discover very important, formerly unpublished effects and procure typically acceptable suggestions and instruments that enable you paintings with a extensive category of integrable platforms.

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And the functions H and f be real analytic. Then in a neighborhood of a non-degenerate singular point x 2 K0 , there exist canonical coordinates (p1 q1 p2 q2 ) in which H and f are simultaneously reduced to one of the following forms. 1) Center{center case : H = H (p21 + q12 p22 + q22 ) f = f (p21 + q12 p22 + q22 ) : 2) Center{saddle case : H = H (p1 q1 p22 + q22 ) f = f (p1 q1 p22 + q22 ) : 3) Saddle{saddle case : H = H (p1 q1 p2 q2 ) f = f (p1 q1 p2 q2 ) : 4) Focus{focus case : H = H (p1 q1 + p2 q2 p1 q2 ; p2 q1 ) f = f (p1 q1 + p2 q2 p1 q2 ; p2 q1 ) : Copyright 2004 by CRC Press LL Remark.

Also note that the subspace L0 , to which we restrict all the Hessians, has dimension 2n ; i . That is why the polynomial P ( ) has 2n ; i roots. However, i of these roots are always equal to zero, since the i -dimensional subspace L L0 lies in the kernel of all Hessians. The second requirement is that all the other roots are di erent. Denote by Ki the set of non-degenerate critical points of rank i . It is possible to prove that this set is a smooth symplectic submanifold in M 2n of dimension 2i .

Some curve (or surface) of may split at the singular point into two tangent curves (or two tangent surfaces). Let us note that the tangency that appears at the moment of splitting has to have in nite order. In the analytic case, such splitting, of course, cannot appear. The smooth splitting is connected to the fact that, in the saddle case, the level line fpq = "g is not connected, but consists of two components. Each of these components can be mapped independently without \knowing anything" about the other one.