We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.

A chaotic invariant set is constructed for the planar three-body problem. The orbits in the invariant set exhibit many close approaches to triple collision and also excursions near infinity. The existence proof is based on finding appropriate «windows» in the phase space which are stretched across one another by flow-defined Poincare’ maps.

The Kowalevski gyrostat in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in three-dimensional space of the first integrals constants.

Nonlinear Schrodinger equation (NSE) has many applications in mathematical physics (nonlinear optics, wave theory and so on). Gagnon and Winternitz have constructed symmetry algebra $L_{12}$ and optimal system of subalgebras for NSE (1989). It’s an extension of Galilei algebra $L_{11}$ admitted gas dynamics equations. Its three-dimensional symmetry subalgebras generate 27 different submodels. List of all solutions corresponding to these algebras has been received in this paper. Most of this solutions have not investigate previously.