Integrable Hamiltonian systems: geometry, topology, by A.V. Bolsinov

By A.V. Bolsinov

Integrable Hamiltonian structures were of becoming curiosity over the last 30 years and characterize essentially the most fascinating and mysterious periods of dynamical platforms. This e-book explores the topology of integrable platforms and the final idea underlying their qualitative houses, singularities, and topological invariants.The authors, either one of whom have contributed considerably to the sphere, advance the type concept for integrable platforms with levels of freedom. This concept permits one to differentiate such structures as much as normal equivalence relatives: the equivalence of the linked foliation into Liouville tori and the standard orbital equaivalence. The authors express that during either instances, you can find whole units of invariants that supply the answer of the type challenge. the 1st a part of the ebook systematically offers the overall development of those invariants, together with many examples and purposes. within the moment half, the authors practice the final equipment of the class idea to the classical integrable difficulties in inflexible physique dynamics and describe their topological pix, bifurcations of Liouville tori, and native and worldwide topological invariants. They express how the category conception is helping locate hidden isomorphisms among integrable platforms and current to illustrate their facts 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Hamiltonian platforms: Geometry, Topology, category bargains a different chance to discover vital, formerly unpublished effects and procure quite often acceptable options and instruments that assist you to paintings with a large category of integrable structures.

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In fact, K (x F ) is completely determined by the Poisson action of the group Rn in a neighborhood of x . Note that the above described procedure is equivalent to the local reduction with respect to the Poisson action of the group Ri generated by the functions gn;i+1 : : : gn independent at the point x : we rst make such a reduction that x becomes xed, and then, for a xed point of rank zero, we repeat the de nition given above for two degrees of freedom. Copyright 2004 by CRC Press LL How to verify that K (x F ) is a Cartan subalgebra in sp(2(n ; i) R)?

This rank, in turn, is equal to the rank of the system fdfi g . Copyright 2004 by CRC Press LL Let us ask ourselves which points of K1 and K0 can naturally be called nondegenerate. We begin with K1 . Consider a point x 2 M 4 such that rank dF (x) = 1. The orbit of this point is one-dimensional and di eomorphic to either the real line or the circle. Suppose, for de niteness, that dH (x) 6= 0. Then, according to the Darboux theorem, there exists a canonical coordinate system (p1 q1 p2 q2 ) in a neighborhood of x such that H = p1 .

Is exact on L C , and we can send the construction onto L C by setting again I s (f ) = 21 f where is a 1-form on L C such that d = F ! , and f is a cycle on L ff g . Finally, we list some general properties of the actions si (f ). First, the value of the action on the Liouville torus does not depend on the choice of i in its homotopy class. jT = 0. Second, the action si is de ned up to an additive constant. This is related to the ambiguity of the choice of . Indeed, let d = d 0 = ! This means that 0 = + , where is a certain closed form.

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