The relative equilibria of a tethered gyrostat in a central Newtonian field

The motion of an orbital tether system comprising a massive body and a gyrostat of small mass attached to it by a non-extensible weightless tether is examined. The body performs unperturbed motion in a Kepler orbit. There are several different equilibria of the system relative to a uniformly rotating system of coordinates. These equilibria are interpreted geometrically using Mohr circles. Despite being the simplest example of an orbital tether system with a gyrostat, it exhibits a wealth of dynamic properties. There are also more complex orbital tether systems which contain more than one gyrostat [1].     

First integrals of the equations of motion of a symmetric gyrostat on a perfectly rough plane

The problem of the motion of a dynamically symmetric gyrostat without slipping on a fixed horizontal plane is investigated. When the surface of the gyrostat and the distribution of the masses in it satisfy a certain condition, supplementing and developing the results obtained by Mushtari [Mushtari KhM. The rolling of a heavy solid of revolution on a fixed horizontal plane. Mat Sbornik 1932; 39(1, 2):105–26], an explicit form of two first integrals of the equations of motion of the gyrostat, in addition to the energy integral, is presented.     

Extensions of the Appelrot classes for the generalized gyrostat in a double force field

For the integrable system on e(3, 2) found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.     

Free surface of a drop in a symmetric force field

We investigate the problem of the free surface of a drop in a force field. Under the assumption of a certain symmetry and smallness of the force field, this problem reduces to an equation with a contraction operator from which there follows the existence and uniqueness of a nearly spherical solution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 160–174, 1975.In conclusion, the author expresses his deep gratitude to N. N. Ural'tseva for suggesting the problem and for a series of critical remarks as well as to O. A. Ladyzhenskaya for her interest in the paper.     

Nonlinear free oscillations of a rotating solid body in a Newtonian force field

Nonlinear free oscillations of a rotating axisymmetrical solid body are considered with respect to the center of mass and with the body moving in a Newtonian force field. To construct periodic solutions of nonlinear differential equations of the motion, some algorithms, which are based on a modification of the extension method of solution with respect to a parameter, are used. The stability of nonlinear oscillations of the rotating solid body are studied with respect to stationary motions, some amplitude-frequency characteristics and forms of oscillations of the body are formulated for different values of its inertial parameters.Translated from Dinamicheskie Sistemy, No. 8, pp. 3–8, 1989.     

The orbital motion of a tetrahedral gyrostat

The orbital motion of a gyrostat whose mass distribution admits of the symmetry group of a regular tetrahedron is examined. The equations of motion and their first integrals are presented. The order of the equations of motion is reduced using a Routh–Lyapunov approach. The reduced potential and the equations for its critical points are presented. Some solutions of these equations are indicated, and a mechanical interpretation of the steady motions corresponding to them is given. Equations of motion similar to the well known equations of relative motion of a gyrostat in an elliptical orbit in the satellite approximation are derived assuming that the dimensions of the body are small compared with its distance from the attracting centre. A three-dimensional analogue of Beletskii's equation that relies on the use of the true anomaly as the independent variable is presented. Three classes of steady configurations are determined by Routh's method in the case of a circular orbit, and the conditions for their stability are investigated.     

Quasiperiodic solutions of variational problems of motion in a central force field

A method is proposed for computing nearly optimal trajectories of dynamic systems with a small parameter by splitting the original variational problem into two separate problems for "fast" and "slow" variables. The problem for "fast" variables is solved by improving the zeroth approximation — the extremals of the linearized problem — by the Ritz method. The solution of the problem for "slow" variables is constructed by passing from a discrete argument — the number of revolutions around the attracting center— to a continuous argument. The proposed method does not require numerical integration of systems of differential equations and produces a highly accurate approximate solution of the problem.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 113–118, 1989.     

The stability of the steady motion of a gyrostat with a liquid in a cavity

A symmetrical rigid body with a spherical base, carrying a rotor and having a cavity in the shape of an ellipsoid of revolution, completely filled with an ideal incompressible liquid in uniform vortex motion, is moving along an absolutely rough plane. It is shown that this system admits of an energy integral, Jellett's integral, the integral of constant vorticity and a geometric integral. The construction of a Lyapunov function as a linear combination of first integrals [1] yields the sufficient conditions for the rotation of the gyrostat about the vertically positioned axis of symmetry to be stable. The conditions for the gyrostat's rotation to be unstable are found. It is shown that the rotor may prove to have either a stabilizing or destabilizing effect on the system and that the gyrostat admits of motions of the type of regular precession. The sufficient conditions for the stability of these motions are obtained.     

Integrable cases of the problem of the free motion of a gyrostat

The free spatial motion of a gyrostat in the form of a carrier body with a triaxial ellipsoid of inertia and an axisymmetric rotor is considered. The bodies have a common axis of rotation, which coincides with one of the principal axes of inertia of the carrier. In the Andoyer–Deprit variables the equations of motion reduce to a system with one degree of freedom. Stationary solutions of this system are found, and their stability is analysed. Cases in which the longitudinal moment of inertia of the carrier is greater than the largest of the transverse moments of inertia of the system of bodies, is smaller than the smallest or belongs to a range composed of the moments of inertia indicated, are investigated. General analytical solutions that describe the motion on separatrices and in regions corresponding to oscillations and rotation on the phase portrait are obtained for each case. The results can be interpreted as a development of the Euler case of the motion of a rigid body about a fixed point when one degree of freedom, namely, relative rotation of the bodies, is added.     

On the motion of a solid in a magnetic field

The problem of the motion of a magnetic solid in a constant uniform magnetic field, taking gyromagnetic effects into account, is considered. The equations of motion are derived, the Hamiltonian structure is studied, and the cases of integrability indicated. Certain classes of stationary motions are studied and their stability examined.

The gyromagnetic effects arise because the electrons have magnetic and mechanical spin moments /1/. The rotation of the body causes it to become magnetized (the Barnett effect) and when a freely suspended body is magnetized, it begins to rotate (the Einsteinde Haas effect). It is found that gyromagnetic phenomena must be taken into account when analysing the motion of gyroscopic precision systems.     

Periodic solutions of symmetric autonomous Newtonian systems

In this paper we show the existence and bifurcation of T-periodic solutions of a special form for an autonomous Newtonian system with symmetry. If the phase-space R2n is equipped with the structure of an orthogonal representation (W,ρW) and the potential is invariant, then for every such a solution the set of indices of nonvanishing Fourier coefficients is finite and depends on W only. If the potential V depends on the squares of complex coordinates, then for every such a solution T is the minimal period.     

On stability of motion of a symmetric body placed on a vibrating base

We consider the problem of stabilization of a symmetric solid body rotating about a fixed point and show that its unstable states can be stabilized by vertical vibration.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1661–1666, December, 1995.     

The steady motions of a gyrostat suspended on a rod in a central gravitational field

The problem of the existence, stability and bifurcation of the steady motions of two bodies in an orbital tethered system, when one of the bodies is a symmetrical satellite with a rotor on the axis of symmetry, is considered. One-parameter families of steady motions are indicated, and their stability and bifurcations are investigated. The conditions which relate the parameters of the system for which stabilization of the families obtained is possible using a rotating rotor are obtained.     

On a certain two-sided symmetric condition in magnetic field analysis and computations

A special two-sided condition for the incremental magnetic reluctivity is introduced which guarantees the unique existence of both the weak and the approximate solutions of the nonlinear stationary magnetic field distributed on a region composed of different media, as well as a certain estimate of the error between the two solutions. The condition, being discussed from the physical as well as the mathematical point of view, can be easily verified and is fulfilled for various magnetic reluctivity models used in electrotechnical practice.     

Direct simulation of the motion of a settling ellipsoid in Newtonian fluid

In this paper, we discuss the generalization of a Lagrange multiplier-based fictitious domain method to the simulation of the motion of general shape particles in Newtonian fluid. Preliminary numerical results of a settling ellipsoid in a narrow channel filled with a Newtonian fluid are presented. As expected, the ellipsoid turns its broadside to the stream in the simulations.     

Planar motion and stability for a rigid body with a beam in a field of central gravitational force

Planar motion for a rigid body with an elastic beam in a field of central gravitational force was investigated, and both of the orbital motion and attitude motion were under consideration. The equations of motion of the system were derived by the variational principle, and on view point of generalized Hamiltonian dynamics, the sufficient conditions for the stability of one class of relative equilibria were given by the energymomentum method.