By Borisov A.V., Mamaev I.S.

The booklet discusses major types of equations of movement of a inflexible physique, together with the movement in strength fields, in fluid (Kirchhoffs equations), and movement of a inflexible physique with cavities jam-packed with fluid. The booklet includes stipulations of the order aid of those equations, and lifestyles of cyclic variables. It collects just about all integrable circumstances almost immediately identified, and strategies in their specific integration. For the aim of research, the pc strategies, permitting bright illustration of the movie, are well-known. the vast majority of effects provided within the publication belongs to the authors.For scholars and graduate scholars of mechanical, mathematical and actual departments of universities, mathematical physicists, and experts on dynamical structures.

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2. The problem about the separation of variables distinctly stated by K. Jacobi in his “Lectures on Dynamics” (1842–43) [183], is still under serious investigations. J. Liouville and P. Shtekkel found more general forms of Hamiltonians, allowing separation of variables. It turned out that if only configurational space transformations (point transformations) be used, the separation of variables is closely connected with the presence of the complete set of first integrals, quadratic with respect to momenta.

Such a linearization resulted in a remarkable fact that the general solution of a system extends to single-valued holomorphic functions into the complex domain of time, i. e. the solution has only poles as singularities. The expression of the general solution for the majority of integrable problems of rigid body dynamics in single-valued elliptic (in a complex sense) functions of time is conditioned by the fact that the general level of the first integrals, representing the intersection of rather simple algebraic surfaces, like quadric ones, allows extension into the complex domain to the Abelian manifolds (Abelian tori), allowing parameterization by means of theta-functions.

I˙ = − ∂H = 0, ∂ϕ ˙ = ∂H = ω(I), ϕ ∂I Hence, I(t) = I 0 , ω(I) = ω(I 0 ) = (ω1 , . . , ωn ). Variables action I “number” invariant tori T n = Mf in M 2n , and variables angle ϕ are uniformly changing on them with n various frequencies ω 1 , . . , ωn . Such a motion is called quasi-periodic. Variables action-angle are very important in the perturbation theory. In some cases the number of independent variables may exceed n = = 1 dimM 2n . And not all of them are in involution and lead to noncommu2 tative integrability of the system.