We consider the motion of the elastic pendulum in the field of gravity. Periodic solutions of this system are found with the help of numerical methods and their stability is studied. Bifurcation diagrams of stability of periodic solutions are constructed.

We consider generalizations of pairing relations for Kovalevskaya exponents in quasihomogeneous systems with quasihomogeneous tensor invariants. The case of presence of a Poisson structure in the system is investigated in more detail. We give some examples which illustrate general theorems.

The motion on the sphere in a potential $V \simeq (x_1x_2x_3)^{-2/3}$ is considered. The physical origin, the Lax representation and the linearization procedure for this two-dimensional integrable system are considered.

Bifurcations in the multidimensional integrable Hamiltonian systems are presented as a semidirect product of the simplest. We analyze action of finite groups on the two-dimensional components of this representation. Algebraic classification and calculation of the topological genus are suggested for the regular class of these singularities.

Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore is a Hamiltonian vector field. As examples we treat the Goryachev–Chaplygin top, the Toda lattice and the Calogero–Moser system, and construct their global Hamiltonians. We show that the simpliest choice of Hamiltonian leads to the Toda lattice.

There is a well-known example of an integrable conservative system on $S^2$, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. The aim of this paper is to construct new families of examples of conservative systems on $S^2$ possessing an integral of fourth degree in momenta.

Relatively recently in works [3], [4] the topological classification of smooth Hamiltonian systems with one degree of freedom was obtained. When we study the stability of obtained topological invariants, the following natural question arised: is the space of all Morse functions with fixed number of minima and maxima on a closed surface connected? The present paper discusses this question and gives an algorithm of reduction of any Morse function on a closed orientable surface to the so-called canonical form.

The problem on description of dynamics of mechanical systems performing motions in spaces of a constant curvature is known in mechanics. Its investigation can be followed since the publication of Zhukovsky [1], devoted to the problem on motion of a plate on a surface of a pseudo-sphere which was proposed even by Lobachevsky. In recent publications [2, 3] the studies of Zhukovsky were developed. In particular, in the problem on motion of a massive point at a sphere and at a pseudo-sphere in a field of an attracting center there were found the analogs of Kepler's laws, there was studied Bertrand's problem concerning a description of all central force fields, for which all trajectories are closed. There was also integrated a problem on two attracting centers. These studies were continued in [4], where the questions on integrability of the problem on two Newtonian centers in three-dimensional spaces of negative and positive constant curvature as well as on existence of steady motions of two bodies under mutual attraction in these space were considered.
In this paper the more general problem on motion of axisymmetric rigid bodies on the surface of a three-dimensional sphere is considered. Under appropriate assumptions these bodies can be treated as "spherical" planets. The comparison of dynamics of axisymmetric rotating planet with dynamics of analoguous system in a flat space is carried out.

The paper suggests an equation to model a wave process in non-linear deformable systems with moment stresses. The backlund transformation, which helped to obtain an exact kink-antikink and soliton as solutions, was found for this equation.

The integrable problem of a joint motion of a point vortex and the Kirchhoff one is considered. The reduction of the system is carried out in algebraic variables. The phase portrait of the system is constructed, the points of relative equilibrium are investigated. The solutions relevant to elliptic fixed points are found in absolute coordinates. Nonintegrability of interaction of a vortex spot with two point vortices is proved for the restricted problem.