By T.G. Vozmischeva

Introd uction the matter of integrability or nonintegrability of dynamical structures is without doubt one of the important difficulties of arithmetic and mechanics. Integrable instances are of substantial curiosity, when you consider that, through interpreting them, you could learn common legislation of habit for the recommendations of those structures. The classical method of learning dynamical structures assumes a look for particular formulation for the ideas of movement equations after which their research. This procedure influenced the improvement of recent parts in arithmetic, similar to the al gebraic integration and the idea of elliptic and theta features. nonetheless, the qualitative equipment of learning dynamical structures are a lot genuine. It used to be Poincare who based the qualitative idea of differential equa tions. Poincare, figuring out qualitative equipment, studied the issues of celestial mechanics and cosmology during which it's specifically vital to appreciate the habit of trajectories of movement, i.e., the strategies of differential equations at countless time. particularly, starting from Poincare structures of equations (in reference to the learn of the issues of ce lestial mechanics), the right-hand elements of which do not rely explicitly at the autonomous variable of time, i.e., dynamical platforms, are studied.

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**Extra resources for Integrable Problems of Celestial Mechanics in Spaces of Constant Curvature**

**Sample text**

The boundaries of the atoms are glued together by means one-parameter families of Liouville tori that do not contain singular leaves. The scheme for this gluing can be described in terms of a directed graph. , to a certain bifurcation of Liouville tori, and each its edge corresponds to a one-parameter family of tori. Such a graph is called a molecule of the given isoenergy surface. The molecule does not depend on the choice of an integral f. A molecule describes the structure of the Liouville foliation up to the socalled rough Liouville equivalence.

Fn plays the basic role, that is, the linear space G of functions stretched on iI, ... ,In is a commutative Lie algebra of dimension n. In some situations a Hamiltonian system has a set of integrals iI, ... , In which don't make up a commutative Lie algebra, that is, they are not in involution. The theorem of Mischenko and Fomenko generalizes the Liouville theorem in this sense. Let M be a symplectic manifold of dimension 2n, and 11' ... ' Ik: M -+ ~ be smooth independent functions. The linear hull G over the field ~ of functions iI, ...

21) in involution, i. , {h,h} = O. If the functions h, ... : h(p, q, t) = i = 1, ... 21) are integrated by quadratures. ei, 3. Integrability 27 The theorem formulated in [28] generalizes the above cited theorem. Theorem 11 Let lR2n be the phase space of a Hamiltonian system with the standard symplectic structure and the Hamiltonian H(p, q, t). Let the system have n integrals of motion h, 12, .. · ,fn such that {/i, fj} = I: cfjfk , k Cij = const . k If the functions h, 12, ... , In are independent on the set M~ = {(p, q, t) E lR2n X lR: h(P, q, t) = ei, i = 1, ...