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Within the framework of the rigid body hypothesis, the influence of external torques acting on a rotating water lens in a stratified ocean is examined and a hypothesis about the angular motion of objects of such kind is constructed. The structure of the external torques acting on the lens is investigated and their magnitudes and influence on the overall picture of the motion of the lens about its centre of mass are estimated. It is shown that the hydrostatic buoyancy torque is the most important of such torques, it being orders of magnitude greater than the Coriolis torque and the torque due to virtual masses, and also the gravitational torque and other torques. The friction torque can promote stabilization of the angular motion and lead to the appearance of a steady regime. The results obtained are in agreement with the observed motion of oceanic formations.

Problems on the stabilization of the preset positions and steady motions of holonomic mechanical systems, measuring only the generalized coordinates, are investigated. The novelty of the results lies in the construction of control actions that solve the above problems in a fairly general formulation. Problems concerning the stabilization of the preset positions and steady motions of a heavy rigid body with a fixed point and the preset positions of a two-link manipulator with three degrees of freedom are solved as specific applications of the theoretical results obtained.

Perturbed rotational motions of a rigid body, close to the Lagrange case, under the action of a torque that is slowly varying in time are investigated. Conditions for the possibility of averaging the equations of motion with respect to the nutation phase angle are presented and an averaged system of equations is obtained. An example, corresponding to the motion of a body in a medium with linear dissipation, is considered.

The existence conditions for invariant relations and particular solutions of Grioli's equations for the problem of the motion of a rigid body with a fixed point are considered. New classes of solutions of the equations of motion of a rigid body with a fixed point are obtained.

A three-dimensional problem on the contact interaction between a thin elastic layer and an absolutely rigid body sliding over it is considered where wear of both bodies is assumed. The solution is represented by a Fourier series, the coefficients of which are found in an explicit form. The asymptotics of the contact pressure for large times are investigated. The importance of taking the factor of the mutual wear of the contacting bodies into account is demonstrated.

The formulation of the contact problem is considered for the bulk application of intermolecular interaction forces between the contacting bodies as applied to the contact between an absolutely rigid body (an indentor) and an elastic half space. The subsurface stresses and, in particular, the shear stress intensity associated with the fracture of the material are analysed.

The resonance motion modes of a Lagrange top with small mass asymmetry upon passage through low-order resonances are analysed. The motion of a rigid body with a prolate ellipsoid of inertia is considered in the vicinity of a lower, statically stable equilibrium position at small nutation angles. The original system of equations is reduced to the equation of motion of a pendulum with small perturbations using an averaging method. The conditions for plunging the phase trajectory of the system into oscillation regions of the pendulum, i.e., the possibility of capture into resonance under the action of the perturbations considered, are analysed. Cases in which captures into resonance are possible and in which they are not are established. Numerical calculations that illustrate the analytical estimates obtained are presented.

The existence of a soil contact layer when there is shear interaction between a rigid body and the soil has been demonstrated by an analysis of the numerical solution of a one-dimensional unsteady problem on the interaction of a rigid strip with a non-linearly deformable soil medium. The occurrence of a peak value of the shear stress and subsequent structural breakdown in the soil contact layer has been shown, which agrees with experimental results. It is established that, when non-linear soil deformation laws are employed, there is no need to use interaction (friction) laws and conditions of complete adhesion at the boundary of contact between the rigid body and the soil are sufficient in problems on the shear interaction between structures and a soil medium.

The connection between the different methods of interpreting the motion of a body with a fixed point is investigated. A new form of the equations of a stationary angular velocity hodograph is obtained that allows the Poinsot method to be applied using Rodrigues–Hamilton parameters. The properties of the kinematic parameters are studied for the precessing and isoconic motions of the body. The precessing motions in Dokshevich's solution and the isoconic motions in Steklov's solution are considered.

The problem of the optimal control of the spatial orientation of a rotating rigid body with an axis of symmetry is considered. New geometric properties of the extremals of this variational problem are established in the non-degenerate case. The property of a “collapse” of the extremal field, a ranking alternative and the connection with the family of trigonometric extremals constructed earlier in a similar problem are described in detail. The results obtained are based on an analysis of the system of equations obtained as the result of using the formalism of Pontryagin's maximum principle. The intrinsic non-linearity of the equations of motion in problems of optimizing the control of the reorientation of a rotating body 1 , 2 , 3 , 4 , 5 has led to the development of new approaches assuming additional geometric optimal turning properties. For a non-zero initial rotational velocity, the initial problem reduces to the optimization of successive manoeuvres, that is, the body is first completely decelerated and then reorientated from a position of rest to a position of rest. A new property of the extremals has been described 6 in the problem of the optimal control of the reorientation of a rotating spherically-symmetric body. This is associated with the possibility of an abrupt change in the dimension of a certain linear subspace generated by the extremals. It is shown below that similar properties can also be established in the case of the more general problem of the optimal reorientation of an axisymmetric rigid body.

The two-dimensional motion of a system consisting of a hollow rigid body, resting on a rough plane, and two internal mobile point masses, capable of moving parallel to the longitudinal axis of symmetry of the body is considered. Friction in the contact area is modelled by a local Amonton–Coulomb law. A dynamically matched linear model is used to describe the normal stress distribution. The possibility of achieving rotation of the system by a certain relative motion of internal masses is investigated. Two control laws for the movable masses are considered: piecewise-linear and harmonic, for which the equations of motion are integrated numerically for different values of the parameters of the control laws. The features of the motion of the system for the chosen laws are analysed. It is established that, for a two-phase piecewise-linear control law, rotation of the system reaches a steady state. The parameters of the piecewise-linear control law, which deliver a maximum of the mean angular velocity of the body in the steady mode of motion are obtained.

The two-dimensional problem of the control of a rigid body in an ideal fluid with the help of two mobile point masses, is considered. The equations of motion allow of a complete set of Noether integrals, linear in the velocities. For fixed values of these integrals, the equations of motion are reduced to a non-autonomous system of first-order differential equations. The complete controllability of the system for almost all values of the physical parameters is proved. Particular solutions, which enable the control problem to be solved explicitly (transferring the body from the initial position to a position specified in advance), are indicated. The problem of the speed of response is solved numerically using a genetic algorithm.

An area-preserving mapping is considered. It is assumed that the mapping has a fixed point and is analytic in a small neighbourhood near it. A constructive algorithm for obtaining a representation of the mapping in the form of a composite of two area-preserving mappings, one of which is a nearly identity mapping, while the other corresponds to the real normal form of a linearized mapping, is described. The algorithm is used in the problem of the stability of the translational motion of a rigid body in a uniform gravitational field when it undergoes collisions with a fixed horizontal plane and in the problem of the stability of one type of resonant in-plane rotations of a satellite, i.e., a rigid body, in an elliptic orbit.

Geometric mechanics, which develops traditional geometric methods of mechanics, enables one to construct theories of complex coupled systems on the basis of Newtonian axiomatics without recourse to the methods of Lagrangian analytical mechanics, Euler's methods for the dynamics of a rigid body and other theories and principles. Geometric methods simplify the general theory of complex mechanical systems and bring it closer to computerized computational technologies and to engineering practice.

The problem of stabilizing the relative equilibrium of an orbital station in an Earth–Moon system by imparting a small constant-modulus acceleration with constant orientation of its vector with respect to the body of the station, which is assumed to be a rigid body of variable mass, is considered. It is shown that, in the case of a small displacement of the centre of mass of the station (by means of a small reactive acceleration) with respect to the collinear libration point beyond the Moon, its relative equilibrium position can become stable by virtue of the equations of the first approximation.

A kinematic interpretation of the motion of a rigid body with a fixed point is proposed using the rolling of a mobile hodograph, which describes, on the ellipsoid of inertia, a vector collinear with the vector of the angular velocity of the body, with respect to a fixed vector. On the basis of this, an interpretation of the motion of the body in the Steklov, Grioli, Dokshevich and Bobylev – Steklov solutions is obtained. A new formula is derived indicating the connection between the angle of precession and the polar angle of the equations of the fixed hodograph, indicated by Kharlamov.

The problem of the optimal reorientation of an axisymmetric rigid body with a non-zero angular velocity is considered. An integral-quadratic functional, matched with the inertial symmetry of the body and characterizing the total energy consumption, is chosen as the criterion. The net torque of the applied external forces is the control. The control problem is investigated assuming that jet engines are used and the manœuvre leading to the least consumption of the working medium is therefore of interest. An explicit description of the family of analytical extremals, represented by trigonometric functions of time, is obtained. The construction of these extremals is based on a study of the space–time deformations of the solutions of the Euler–Poinsot equations for the free rotation of a body with similar symmetry characteristics.

A dynamically symmetrical rigid body suspended on a string is considered. The suspension point performs periodic oscillations. The loss of stability of the system when it performs high-frequency rotations around the axis of dynamic symmetry is investigated. The sufficient conditions for loss of stability are obtained.

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.

A new family of kinematic parameters for the orientation of a rigid body (global and local) is presented and described. All the kinematic parameters are obtained by mapping the variables onto a corresponding orientated subspace (hyperplane). In particular, a method of stereographically projecting a point belonging to a five-dimensional sphere S5 ⊂ R6 onto an orientated hyperplane R5 is demonstrated in the case of the classical direction cosines of the angles specifying the orientation of two systems of coordinates. A family of global kinematic parameters is described, obtained by mapping the Hopf five-dimensional kinematic parameters defined in the space R5 onto a four-dimensional orientated subspace R4. A correspondence between the five-dimensional and four-dimensional kinematic parameters defined in the corresponding spaces is established on the basis of a theorem on the homeomorphism of two topological spaces (a four-dimensional sphere S4 ⊂ R5 with one deleted point and an orientated hyperplane in R4). It is also shown to which global four-dimensional orientation parameters–quaternions defined in the space R4 the classical local parameters, that is, the three-dimensional Rodrigues and Gibbs finite rotation vectors, correspond. The kinematic differential rotational equations corresponding to the five-dimensional and four-dimensional orientation parameters are obtained by the projection method. All the rigid body kinematic orientation parameters enable one, using the kinematic equations corresponding to them, to solve the classical Darboux problem, that is, to determine the actual angular position of a body in a three-dimensional space using the known (measured) angular velocity of rotation of the object and its specified initial position.

A general theorem on the behaviour of the angular variables of integrable dynamical systems as functions of time is established. Problems on the motion of the nodal line of a Kovalevskaya top and of a three- dimensional rigid body in a fluid are considered in integrable cases as examples. This range of topics is discussed for systems of a more general form obtained from completely integrable systems after changing the time.

The dynamics of a compound system, consisting of a rigid body and a point mass, which moves in a specified way along a curve, rigidly attached to the body is investigated. The system performs free motion in a uniform gravity field. Differential equations are derived which describe the rotation of the body about its centre of mass. In two special cases, which allow of the introduction of a small parameter, an approximate system of equations of motion is obtained using asymptotic methods. The accuracy with which the solutions of the approximate system approach the solutions of the exact equations of motion is indicated. In one case, it is assumed that the point mass has a mass that is small compared with the mass of the body, and performs rapid motion with respect to the rigid body. It is shown that in this case the approximate system is integrable. A number of special motions of the body, described by the approximate system, are indicated, and their stability is investigated. In the second case, no limitations are imposed on the mass of the point mass, but it is assumed that the relative motion of the point is rapid and occurs near a specified point of the body. It is shown that, in the approximate system, the motion of the rigid body about its centre of mass is Euler–Poinsot motion.

The formulation and method of solving of a plane time-varying contact problem with moving boundaries is developed for an absolutely rigid impactor and a half-space filled with a Cosserat medium. A new function, the surface influence function for a Cosserat medium, is constructed and investigated. An original algorithm for constructing it, based on the small-parameter method in conjunction with a simultaneous Fourier–Laplace integral transformation, is developed and implemented. The quantitative and qualitative differences between the influence function constructed and the well-known solution of the plane Lamb problem for an elastic half-space are indicated.

The problem of the optimal control of the rotation of an absolutely rigid body about the centre of mass is investigated. The main purpose of the control is to vary the angular velocity vector from its initial value to the required terminal value in a finite time so that the manoeuvre would require the smallest power consumption, which is characterized by an integral quadratic functional. The principal torque produced by the external forces applied to the body serves as the control. The change in orientation is not taken into account, i.e., the problem of the overspeed–braking control of the body, is studied. A new class of analytic extremals based on the use of space-time deformations of the solutions of the dynamical Euler equations for the free rotation of a rigid body is described. Sufficient conditions for the existence of such extremals for all types of symmetries are presented.

The non-linear oscillations of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its stable equilibrium position are considered. It is assumed that the Hamilton function is sign-definite in the neighbourhood of the equilibrium position and that the values of the frequencies of its linear oscillations are equal or close to one another (1:1 resonance). The investigation is carried out using the example of the problem of the motion of a dynamically symmetrical rigid body (satellite) about its centre of mass in a circular orbit in a central Newtonian gravitational field. In this problem there is relative equilibrium of the rigid body in the orbital system of coordinates, for which its axis of dynamic symmetry is directed along the velocity vector of the centre of mass. Resonance occurs when the ratio of the polar and equatorial principal central moments of inertia is equal to 4/3 or is close to it. The problem of the existence, bifurcation and orbital stability of the periodic motions of a rigid body generated from its relative equilibrium is solved. Some aspects of the existence of quasiperiodic motions are also considered.

The problem of the motion of a heavy rigid body, supported on a rough horizontal plane at three of its points, is considered. The contacts at the support points are assumed to be unilateral and subject to the law of dry (Coulomb) friction. The dynamics of possible motions of such a body under the action of gravity forces and dry friction is investigated. In the case of a plane body, it is possible to obtain particular integrals of the equations of motion.

The motion of a rigid body in a uniform gravity field is investigated. One of the points of the body (the suspension point) performs specified small-amplitude high-frequency periodic or conditionally periodic oscillations (vibrations). The geometry of the body mass is arbitrary. An approximate system of differential equations is obtained, which does not contain the time explicitly and describes the rotational motion of the rigid body with respect to a system of coordinates moving translationally together with the suspension point. The error with which the solutions of the approximate system approximate to the solution of the exact system of equations of motion is indicated. The problem of the stability with respect to the equilibrium of the rigid body, when the suspension point performs vibrations along the vertical, is considered as an application.

The steady motions of a rigid body carrying several two-degree-of-freedom powered gyroscopes in a uniform external field are investigated. It is shown that when the installation scheme of the gyroscopes in the carrying body is collinear, the problem of determining the steady motions of the system and analysing their secular stability reduces for the most part to the previously solved, similar problem for a system with one gyroscope. It is established that when there is dissipation in the axes of the gyroscope frames, the system tends asymptotically to a state of rest if the absolute value of the total angular momentum of the system lies in the segment of possible absolute values of the angular momentum of the gyroscope rotors. The results of an analysis of the steady motions of a system carrying two gyroscopes with a non-collinear installation scheme are presented.

A mechanical system, consisting of an invariable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its components with respect to the carrier is specified), is considered. The system moves in a uniform gravitational field around a fixed point of the carrier. The general form of the quadratic integral is obtained when there is no dynamic symmetry, and the necessary and sufficient conditions for it to exist are found. The conditions when the integral can be split into two independent integrals and the equations of motion are reduced to autonomous form, are obtained.

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.

A comparative analysis of the solutions of the three-dimensional problem of the oblique penetration of a rigid body into soft soil is carried out arsing interaction models based on one-dimensional solutions of the problem of the spherical cavity expansion. Both the well-known self-similar analytical solutions for an incompressible medium as well as the generalized solution for a compressible elastoplastic medium with separation of the shock wave which arises are considered. Use of the incompressible medium hypothesis, disregarding flow separation, in estimating the maximum values of the resistive forces leads to large errors. Taking account of compressibility enables the resistive forces to be refined appreciably and enables a satisfactory estimate of the deviation of the trajectories of bodies from the initial direction of motion to the obtained. In the proposed method of solving oblique penetration problems, a three-dimensional problem is reduced, on the basis of the plane sections hypothesis and disregarding peripheral mass and momentum flows, to the combined solution of a number of axisymmetric problems for each meridional section. It is shown that, with well-known local interaction models, this approach enables the reliability of the calculation of both the force and the kinematic characteristics of the penetration process to be increased considerably due to the fact that the dynamics of the free surface and cavitation effects of the covitating flow are taken into account.

The control problem of moving a rigid body onto the surface of another rigid body in such a way that no collision occurs in investigated. For this, phase constraints must be satisfied in the sense that the surface of one body must not overlap the other, and here an important condition must be met: contact of the bodies is ensured by bounded controls and in a finite time. No impact of the rigid bodies occurs at the instant of contact because a strict rule is observed: if any point of one body lies on the surface of the other, then the velocity of the point is directed along this surface (or away from it). To observe this rule, special controls are used. These controls realize assigned accurate values of the coordinates and velocities of the mechanical system in a finite time.

An orbital tether system, including a satellite (a rigid body), an elastic ponderable tether and a terminal load, is investigated. A mathematical model is obtained using Lagrange's equation of the second kind, which enables the plane translational motion of the centres of mass of the elements of the system and the rotational motion of the satellite and the tether to be investigated. It is shown that the equations of motion for the new independent variable, that is, the true anomaly angle, obtained on the assumption that the motion of the centre of mass of the system is independent of the relative motion of its elements, are an extension of the known mathematical models. The effect of the elasticity of the tether on the angular oscillations of the tether and the satellite is investigated. The model constructed can be used both to analyse of the deployment of a tether system as well as to investigate of the combined behaviour of a satellite and a tether about the natural centres of mass.

The use of Pontryagin's maximum principle to solve spacecraft motion control problems is demonstrated. The problem of the optimal control of the spatial reorientation of a spacecraft (as a rigid body) from an arbitrary initial angular position to an assigned final angular position in the minimum rotation time is investigated in detail. The case in which velocity parameters of the motion are constrained is considered. An analytical solution of the problem is obtained in closed form using the method of quaternions, and mathematical expressions for synthesizing the optimal control programme are given. The kinematic problem of spacecraft reorientation is solved completely. A design scheme for solving the maximum principle boundary-value problem for arbitrary turning conditions and inertial characteristics of the spacecraft is given. A solution of the problem of the optimal control of spatial reorientation for a dynamically symmetrical spacecraft is presented in analytical form (to expressions in elementary functions). The results of mathematical modelling of the motion of a spacecraft under optimal control, which confirm the practical feasibility of the control algorithm developed, are given. Estimates have shown that the turn time of modern spacecraft with a constrained magnitude of the angular momentum can be reduced by 15–25% compared with conventional reorientation methods. The greatest effect is achieved for turns through large angles (90° or more) when the final rotation vector is equidistant from the longitudinal axis and the transverse plane of the spacecraft.

The problems of the motion of two rigid bodies, namely, a uniform thin rod with a fixed end in a gravitational field and a uniform flat circular disk with a fixed centre, are considered. The behaviour of these bodies as a function of the method used to implement the constraint, namely, using an ideal ball joint or different versions of gimbals, are compared. The features of the motion specified by the inertial properties of the parts of the gimbals are found ©2009.

The equations of motion of an ellipsoid on a smooth horizontal plane are similar to the equations of motion of a heavy rigid body with a fixed point. In general, one integral is also lacking in order to integrate them. For a triaxial ellipsoid, the centre of mass of which coincides with the geometrical centre, it is proved that an additional integral is lacking (in the generic case).

The problem of the conditions for the static equilibrium of a body, resting on a rough plane at one, two or three points, is considered. It is assumed that an arbitrary system of active forces is applied to the body, while the friction on the rough supporting plane is anisotropic. This model generalizes the well-known isotropic model of Coulomb dry friction. Explicit analytic formulae, which express the necessary and sufficient conditions for static equilibrium, are obtained. The investigation procedure uses the idea of an anisotropic force of static friction, which enables analytical results for the equilibrium conditions to be obtained more easily.

The problem of determining dry friction forces in the case of the motion of a rigid body with a plane base over a rough surface is discussed. In view of the dependence of the friction forces on the normal load, the solution of this problem involves constructing a model of the contact stresses. The contact conditions impose three independent constraints on the kinematic characteristics, and the model must therefore include three free parameters, which are determined from these conditions at each instant. When the body is supported at three points, these parameters (for which the normal stresses can be taken) completely determine the model, while indeterminacy arises in the case of a larger number of contact points and, in order to remove this, certain physical hypotheses have to be accepted. It is shown that contact models consistent with the dynamics possess certain new qualitative properties compared with the traditional quasi-static models in which the type of motion of the body is not taken into account. In particular, a dependence of the principal vector and principal moment of the friction forces on the direction of sliding or pivoting of the body, as well as on the magnitude of the angular velocity, is possible.

Problems of the existence, stability, and branching of the permanent rotations of a heavy, dynamically symmetrical rigid body suspended on a rod and which has an axisymmetric ellipsoidal cavity filled with a fluid are discussed. The phenomenological model of the friction of the fluid against the cavity wall proposed by Samsonov is used. All the trivial permanent rotations of the system and the non-trivial rotations that branch off from the trivial ones are found. Their stability and branching are investigated using a modified Routh's theory. The results obtained are presented in the form of an atlas of bifurcation diagrams.

The problem of the existence of a solution in the problem of the optimal control of the rotation of an axisymmetric rigid body for the arbitrary case of angular velocity boundary conditions is studied. A square integrable functional, which is consistent with the symmetry of the rotating body and characterizes the power consumption, is chosen as the criterion. The principal moment of the applied external forces serves as the control and the time of termination of a manoeuvre can be both specified as well as free. In the case of a specified termination time, it is shown that the solution (control) belongs to the class of infinitely-differentiable functions of time. The reasoning is based on the use of the singularities of the structure of the differential equations and the possibility of reducing the initial problem to two successive variational problems. The existence of a solution of the first of these problems in the class of square integrable functions is proved using the Cauchy–Bunyakovskii inequality. The second problem reduces to a search for the minimum of a functional which is weakly lower semi-continuous on a weakly compact set and the existence of its solution in the same class of functions follows from the Weierstrass theorem. The required conclusion concerning the smoothness of the solution of the optimal control problem is obtained from the necessary conditions of Pontryagin's maximum principle. In the case of a free termination time, one of the minimizing sequence can be constructed and it can be shown that, in the general case, there is no solution in the class of measurable controls.

The problem of the optimal control of a rigid body moving along a rough horizontal plane due to motion of two internal masses is solved. One of the masses moves horizontally parallel to the line of motion of the main body, while the other mass moves in the vertical direction. Such a mechanical system models a vibration-driven robot–a mobile device able to move in a resistive medium without special propellers (e.g., wheels, legs or caterpillars). Periodic motions are constructed for the internal masses to ensure velocity-periodic motion of the main body with maximum average velocity, provided that the period is fixed and the magnitudes of the accelerations of the internal masses relative to the main body do not exceed prescribed limits. Based on the optimal solution obtained for a fixed period without any constraints imposed on the amplitudes of vibration of the internal masses, a suboptimal solution that takes such constraints into account is constructed.

Planar and spatial non-linear models of the action of a medium on a rigid body which take into account the dependence of the arm of the force on the reduced angular velocity of the body are constructed, when the moment of the force is also a function of the angle of attack. New cases of complete integrability in terms of elementary functions are found, enabling of qualitative similarities to be discovered between the motions of free bodies in a resistive medium and the oscillations of bodies that are partially fixed and are immersed in a flow of the medium. It is shown that if the additional damping action of the medium on the body that appears in the system is significant, the attainment of stability of the rectilinear translational deceleration of the body is possible when it moves with finite angles of attack. The question of the roughness of the description of this phenomenon is of current interest: a finer property of relative roughness is discovered when reduced dynamical systems are investigated. ©2008.

The Cauchy problem for the motion of a dynamically symmetrical rigid body with a cavity, filled with an ideal liquid, which is perturbed from uniform rotation, is considered in a linear formulation. The problem of the simultanious solution of the equations of hydrodynamics and the mechanics of a rigid body is reduced to the solution of an eigenvalue problem which depends solely on the geometry of the cavity and the subsequent integration of a system of differential equations.

The problem of the motion of a rigid body possessing a plane of symmetry over the surface of a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation forces is considered. Approaches to introducing spherical analogues of the concepts of centre of mass and centre of gravity are discussed. The spherical analogue of “satellite approach” in the problem of the motion of a rigid body in a central field, which arises on the assumption that the dimensions of the body are small compared with the distance to the gravitating centre, is studied. Within the framework of satellite approach, assuming plane motion of the body, the question of the existence and stability of steady motions is investigated. A spherical analogue of the equation of the plane oscillations of a body in an elliptic orbit is derived.

The motion of a satellite, i.e., a rigid body, about to the centre of mass under the action of the gravitational moments of a central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is investigated. It is assumed that the satellite is almost dynamically symmetrical. Plane periodic motions for which the ratio of the average value of the absolute angular velocity of the satellite to the average motion of its centre of mass is equal to 3/2 (Mercury-type resonance) are examined. An analytic solution of the non-linear problem of the existence of such motions and their stability to plane perturbations is given. In the special case in which the central ellipsoid of inertia of the satellite is almost spherical, the stability to spatial perturbations is also examined, but only in a linear approximation. ©2008.

New approaches and methods for studying non-linear problems are applied to the classical problem of the motion of a heavy rigid body about a fixed point, i.e., to the system of Euler–Poisson equations. All the asymptotic expansions of the solutions of the Kowalewski equations, to which the Euler–Poisson equations reduce when certain constraints are imposed on the parameters, are found using power geometry. They form 24 families. Then all the exact solutions of the Kowalewski equations of a specific class (which includes almost all the known exact solutions) are found on the basis of these expansions. Five new families of such solutions are found. Instead of the conventional technique of studying the global integrability of the Euler–Poisson equations, studying their local integrability near stationary and periodic solutions is proposed. Normal forms are used for this purpose. Sets of real stationary solutions, in the vicinity of which these equations are locally integrable, are discovered using them. Other real stationary solutions, in the vicinity of which the Euler–Poisson equations are locally non-integrable, are also found. This is established using the theory of resonant normal forms developed and computer calculations of the coefficients of a normal form.

Mechanical systems with non-ideal geometrical constraints are considered. The possible lack of uniqueness of the solution of the problem of determining the generalized accelerations and reactions with respect to specified coordinates and velocities is taken into account in solving the problem of the stability of an equilibrium state. A number of necessary and sufficient conditions of stability are obtained. It is shown that the results are also applicable in the case of unilateral constraints subject to the condition that a specific hypothesis concerning the character of the impacts on the constraints is adopted. A problem on the stability of a rigid body on a rough plane in the two-dimensional case is solved as an example.

The motion about a centre of mass of a rigid body with a tethered system, designed to launch a re-entry capsule from a circular orbit is considered. In the deployment of the tethered system the direction and value of the tensile strength of the tether vary and, if the point of application of the tensile strength does not coincide with the centre of mass of the body, a moment occurs which leads to oscillations of the body with variable amplitude and frequency. A non-linear equation of the perturbed motion of the body about the centre of mass under the action of the tensile force of the tether and the gravitational moment is derived. Assuming that the change in the value and direction of the tensile force is slow and also that the gravitational moment is small, approximate and exact solutions of the non-linear differential equation of the unperturbed motion are obtained in terms of elementary functions and elliptic Jacobi functions. For perturbed motion, the action integral is expressed in terms of complete elliptic integrals of the first and second kind.

The problem of the optimal control of the rotation of an axisymmetric rigid body is investigated. An integral functional, characterizing the power consumption to carry out a manoeuvre is chosen as the criterion, and the boundary conditions for the angular velocity vector are arbitrary. The principal moment of the applied external forces serves as the control. The necessary conditions of the maximum principle are used to solve the problem in the case of a fixed completion time. New non-trivial first integrals are established for the canonical system of direct and conjugate differential equations obtained, which enable the set of all extremals to be parametrized. Hence, the optimal-control problem is reduced to a problem of non-linear mathematical programming. It is shown that there cannot be more than two different solutions in the latter, and a family of boundary conditions is established when the optimum rotation is determined in a uniquely explicit form.

The existence, stability and branching of invariant sets in the problem of the motion of a heavy rigid body with a fixed point, which satisfies the Goryachev–Chaplygin conditions, are discussed. Both trivial invariant sets, in which the pendulum-like motions of a Goryachev–Chaplygin spinning top lie, as well as non-trivial invariant sets, in which the motion of the top is described by elliptic functions of time, are investigated.