Narrow Results

Some problems connected with the theory of qualitative investigation of integrable Hamiltonian systems are discussed in this paper. The main topological object under consideration is the Liouville foliation either on the whole phase space of the system or on some part of this space.

This paper contains main results on the topological classification of integrable Hamiltonian systems (in other words, on the classification of corresponding Liouville foliations). All results are cited without proof.

The main aim of this paper is to formulate a number of problems which can be solved with the help of a computer, for example, algorithmic classification of topological objects coding singularities of the system, visualization of various effects which is not completely described yet, calculation of invariants for the specific integrable systems arising in mechanics and mathematical physics.

A symplectic reduction method for symplectic G-spaces is given in this paper without using the existence of momentum mappings. By a method similar to the above one, the arthors give a symplectic reduction method for the Poisson action of Poisson Lie groups on symplectic manifolds, also without using the existence of momentum mappings. The symplectic reduction method for momentum mappings is thus a special case of the above results.