Noncommutative Integration Method

1 Maximal Linear Commutative Suhalgebras in the Algebra of Functions on Sym-pletic Manifolds 

we got acquainted with the Liouville theorem which makes it possible to describe the behaviour of integral trajectories of systems possessing a complete set of integrals in involution. In this chapter, we deal with modern methods of integration, a particular case of which is the method of integration by means of the Liouville theorem. We develop here, in particular, the fundamental investigations of E. Cartan, Marsden, Weinstein, Moser, Bernat, Conze, Duflo, and Vergne. 

A linear (infinite-dimensional) space of all smooth functions on a sympletic manifold will be denoted by (7 (A/). As we already know from Ch. 1, this space naturally transforms into an infinite-dimensional Lie algebra with respect to the Poisson bracket /, g, where /, y 6 Different subalgebras (both 

Of course, such commutative subalgebra G(H) may not exist. Then the system V = sgrad H is not integrable in the sense of Liouville. As shown in Ch. 5, most of the systems are not integrable in this sense. That is why one rarely finds a sufficiently large commutative subspace generated by the function H, 

Definition 3.1.1 We will say that on a sympletic manifold a maximal linear commutative subalgebra of functions Go is given (in the Lie algebra C (M) with respect to the Poisson bracket) if dimGo = n and if in Go one can choose an additive basis consisting of n functions /i. /n functionally independent on (almost everywhere). Such an algebra of functions will be sometimes called a complete involutive (commutative) set of functions. 

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The noncommutative integration method described in 1 was proposed by Fomenko and Mishchenko and then developed by Brailov and extended to the case of a larger collection of subalgebras in the Lie algebra C (M). In this subsection, we briefly present the result obtained by Brailov. 

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