February 26, 2017

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 symbolize some of the most interesting and mysterious periods of dynamical platforms. This ebook explores the topology of integrable platforms and the final concept underlying their qualitative houses, singularities, and topological invariants.The authors, either one of whom have contributed considerably to the sphere, advance the class idea for integrable structures with levels of freedom. This thought permits one to differentiate such platforms as much as typical equivalence kin: the equivalence of the linked foliation into Liouville tori and the standard orbital equaivalence. The authors express that during either instances, you'll find entire units of invariants that provide the answer of the category challenge. the 1st a part of the booklet systematically provides the overall development of those invariants, together with many examples and purposes. within the moment half, the authors practice the overall equipment of the class concept to the classical integrable difficulties in inflexible physique dynamics and describe their topological photos, bifurcations of Liouville tori, and native and international topological invariants. They exhibit how the type idea 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Hamiltonian platforms: Geometry, Topology, category bargains a distinct chance to discover vital, formerly unpublished effects and obtain as a rule appropriate strategies and instruments that allow you to paintings with a large type of integrable structures.

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Fig. 6). 6 We now can give a visual explanation of the condition that the integral f is a Bott function on the isoenergy surface Q = fH = constg . Take the line H = const in the two-dimensional plane R2 (H f ). Suppose that this line does not pass through singular points of and intersects the smooth pieces of transversally. Then f will be a Bott function on Q (notice that Q is the preimage of this line in M 4 ). , among them there will be neither tori nor Klein bottles. , x is a xed point of the Poisson action of R2 .

If H (x) = h , then we denote the corresponding isoenergy surface by Qh . As we already know, it is always invariant with respect to the Hamiltonian eld v . Consider an isoenergy three-dimensional surface Q = Q3 . In what follows, we shall assume it to be a smooth compact submanifold in M 4 . In particular, we shall consider only those 3-surfaces on which dH 6= 0. In the case of two degrees of freedom, for integrability of the system v it is su cient to have just one additional integral f functionally independent of the energy integral H .

17. , the set = F (K ) Rn , is called the bifurcation diagram. Thus, the bifurcation diagram is the set of all critical values of F . According to the Sard theorem, the set has zero measure in Rn . In most examples of integrable systems appearing in physics and mechanics, the bifurcation diagram is a manifold with singularities. In other words, it consists of several strata (pieces) i each of which is a smooth i -dimensional surface in Rn . 4 as = 0 + 1 + : : : + n;1 , where di erent strata do not intersect with each other and the union of them gives the whole of .

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