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By Victor A. Sadovnichiy, Mikhail Z. Zgurovsky

Focused on contemporary advances, this booklet covers theoretical foundations in addition to numerous purposes. It provides sleek mathematical modeling techniques to the qualitative and numerical research of ideas for complicated engineering difficulties in physics, mechanics, biochemistry, geophysics, biology and climatology. Contributions via a global staff of revered authors bridge the distance among summary mathematical techniques, akin to utilized tools of recent research, algebra, primary and computational mechanics, nonautonomous and stochastic dynamical structures at the one hand, and useful purposes in nonlinear mechanics, optimization, choice making conception and keep an eye on idea at the different. As such, the ebook should be of curiosity to mathematicians and engineers operating on the interface of those fields.

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N , (which is called the action-angle variables), where s1 , . . , sn are coordinates on the disc Dn and ϕ1 , . . , ϕn are standard angle coordinates on the torus, such that • ω = Σdϕi ∧ dsi , are functions of the integrals, • the action variables si are functions of the integrals f1 , . . , fn , • in the action-angle variables s1 , . . , sn , ϕ1 , . . , ϕn , the Hamiltonian flow v is straightened on each of the Liouville tori in the neighborhood U, that is, s˙i = 0, ϕ˙i = qi (s1 , .

339(3), 293–296 (1994) 8. : Orbital classification of the geodesic flows on two- dimensional ellipsoids. The Jacobi problem is orbitally equivalent to the integrable Euler case in rigid body dynamics. Funkts. Analiz i ego Prilozh. 29(3), 1–15 (1995) 9. : Integrable Hamiltonian Systems: Geometry, Topology, Classification, 1, 2. Regulyarnaya i Khaolichcskaya Dinamika, Izhevsk (1999). [in Russian] 10. : Methods of calculation of the Fomenko – Ziesehang invariant. In: Topological Classification of Integrable Systems – Advances in Soviet Mathematics, vol 6, 147–183.

T. Fomenko Thus, as a result of the introduction of generalized billiards we have been able not only fully simulate the Euler case, but also to get a large number of systems, whose Fomenko–Zieschang invariants coincide with those calculated previously for many systems of rigid body dynamics. This has allowed to simulate a wide class of problems of rigid body dynamics, though not completely. References 1. : The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability.

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