We derive criteria for the existence of trapped orbits (orbits which are scattering in the past and bounded in the future). Such orbits exist if the boundary of Hill's region is non-empty and not homeomorphic to a sphere.
For non-trapping energies we introduce a topological degree which can be non-trivial for low energies, and for Coulombic and other singular potentials. A sum of non-trapping potentials of disjoint support is trapping iff at least two of them have non-trivial degree.
For $d \geqslant 2$ dimensions the potential vanishes if for any energy above the non-trapping threshold the classical differential cross section is a continuous function of the asymptotic directions.

1.Classificaton of the algebra of $n$ vortices on a plane
2.Solvable problems of vortex dynamics
3.Algebraization and reduction in a three-body problem
The work [13] introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie–Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works [14,15]. In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem.

An explicit contruction of a nonuniformly hyperbolic invariant set of positive Lebesgue measure in the phase space of an area-preserving map is suggested. The construction is based on the study of the web created by the stable and unstable manifolds of fixed hyperbolic points.

For periodic in time systems, close to the two-dimensional Hamiltonian ones, the problem of the topology of the neighbourhood of degenerate resonance levels is considered. The "truncated" system determining the topology of neighbourhood of degenerate level close to resonance level is conclusion. The behavior of the solutions of this system in dependence on the detuning is investigated and the bifurcations related to the transition from typical nonlinear resonance to degenerate resonance are determined (both in the case of impassable resonances and in the case of partly passable ones).

In the paper integrability of a perturbed billiard in a potential field is considered. Both conditional and nonconditional integrals are discussed. The cases when additional integrals of the first, second and third degrees in momenta exist have been found.

We consider two-body problem and restricted three-body problem in spaces $S^2$ and $L^2$. For two-body problem we have showed the absence of exponential instability of partiбular solutions relevant to roundabout motion on the plane. New libration points are found, and the dependence of their positions on parameters of a system is explored. The regions of existence of libration points in space of parameters were constructed. Basing on a examination of the Hill's regions we found the qualitative estimation of stability of libration points was produced.

We investigate the separatrices splitting of the double mathematical pendulum. The numerical method to find periodic hyperbolic trajectories, homoclinic transversal intersections of its separatreces is discussed. This method is realized for some values of the system parameters and it is found out that homoclinic invariants corresponding to these parameters are not equal to zero.