Prikladnaâ matematika i mehanika

The article gives a visual illustration of the dynamics of a heavy rigid body around a fixed point in the case of S.V. Kovalevskaya, which arises in the framework of applied problems of space flight. The motion of a rigid body in the case of S.V. Kovalevskaya (the Kovalevskaya top) is equivalent to the dynamics of the attitude of a magnetized satellite around its centre of mass during orbital motion along equatorial circular orbits. The perturbed motion of the magnetized satellite is considered at small deviations from the conditions of the Kovalevskay top, including a small dynamic asymmetry of the satellite, as well as small variations in the magnitude of the external magnetic moment due to weak ellipticity or non-equatoriality of the orbits.

The attitude motion of a rigid body during its contactless transportation by an ion beam is considered. A mathematical model describing the 3D motion of the body and the spacecraft that creates the ion beam is developed. For the case of small asymmetry, when the control system of the spacecraft maintains its constant position relative to the transported body, a simplified mathematical model of the attitude motion of the body is obtained. A generalized energy integral and stationary motions of the system are found for the case of a symmetric body in a circular orbit under the action of gravitational and ion beam torques.

A spatial restricted problem of three bodies attracted by Newton's law is studied. The orbits of the main attracting bodies are assumed to be ellipses of small eccentricity. An approximate mathematical model describing nonlinear oscillations of a passively gravitating body near a Lagrangian triangular libration point is obtained using the normal form method. A detailed study of these oscillations is given in a particular case of third-order resonance. This research was carried out within the framework of the state assignment (registration No. 124012500443-0) at the Ishlinsky Institute for Problems in Mechanics RAS.

The paper studies the problem of motion of a rigid body about a fixed point under the action of gravity and gyroscopic forces in the Hess integrability case. It is shown, that the solution of the problem is reduced to the integration of the second – order linear differential equation with rational coefficients. Using the Kovacic algorithm, we obtain the conditions on the parameters of the problem under which we can find the general solution of the corresponding second order linear differential equation in explicit form. It is also shown that in the case when the rigid body with a fixed point moves under the action of only gyroscopic forces, the general solution of the corresponding linear differential equation can be found in explicit form for any values of parameters of the problem.

The paper considers the problem of propagation of linear internal gravity waves in a layer of viscous stratified medium of finite depth with horizontal background shear currents under critical wave generation conditions. In a flat formulation, new model physical formulations of problems in which critical conditions may arise are discussed, in particular, wave generation by periodic oscillations of the bottom. For arbitrary distributions of shear currents and buoyancy frequency satisfying the Miles–Howard conditions and natural regularity conditions, a model equation describing the main features of solutions near the critical level was proposed. For real parameters of stratified media, using the asymptotics of the model equation, estimates of the spatial scales on which it is necessary to take into account the viscosity of the medium were obtained.

The paper presents formulations and solutions of periodic contact problems for an elastic half-plane and an elastic half-space taking into account the adhesive interaction of contacting bodies’ surfaces. To describe the adhesive forces in the gap between the surfaces, an approximation of the adhesive potential in the form of a piecewise constant function (the Maugis–Dugdale approximation) is used. The dependences of the real contact area, as well as the approach of bodies on the nominal pressure, the parameters of adhesive potential, and the surface relief parameters of the indenting body are investigated. The obtained solutions are compared with the results following from the Johnson, Kendall, Roberts (JKR) model based on the use of a simplified form of the adhesive potential. An analysis of energy dissipation in the surfaces approach–retraction cycle is carried out, and the influence of the parameters of surface microrelief on this contact interaction characteristic is estimated.