The steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability

The positions of relative equilibrium of a satellite carrying a two-degree-of-freedom powered gyroscope, in the axes of the framework of which only dissipative forces can act, are investigated within the limits of a restricted circular problem. For the case when the “satellite - gyroscope” system possesses the property of a gyrostat and the axis of the gyroscope frame is directed parallel to one of the principal central axes of inertia of the satellite, all the equilibrium positions are found as a function of the magnitude of the angular momentum of the rotor. It is established that the minimum number of equilibrium positions is equal to 32 and, in certain ranges of values of the system parameters, it can reach 80. All the positions satisfying the sufficient conditions for stability are also determined. The number of them is either equal to 4 or 8 depending on the values of the system parameters.     

The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope

For a dynamically symmetrical satellite carrying a two-degree-of-freedom powered gyroscope, all the relative equilibria in a circular orbit are found as a function of the angular momentum of the rotor and the angle between the precession axis of the gyroscope and the plane of dynamical symmetry. The case with no spring on the axis of the gyroscope frame and the case with a spring whose stiffness satisfies definite conditions are considered. The secular stability of the equilibria is investigated. For a system with dissipation in the axis of the gyroscope frame, the Barbashin–Krasovskii theorem is used to perform a detailed analysis, which enables the character of the Lyapunov stability of all the equilibria to be determined, with the exception of a few points. The results of a numerical solution of the problem of the optimal values of the system parameters, for which asymptotically stable equilibria are obtained with maximum speed, are presented.     

Analysis of the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axis of the frame

A method that enables a conclusion to be drawn regarding the nature of the Lyapunov stability of the equilibria positions of autonomous systems with partial dissipation from the nature of the secular stability and truncated equations of the linear approximation is developed based on the Barbashin–Krasovskii theorem. The method is used to investigate the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axis of the frame in a circular orbit. For the case when the axis of the gyroscope frame is oriented parallel to one of the principal axes of inertia of the satellite, the nature of the Lyapunov stability of all the equilibria, with the exception of a few branching points, is determined. It is established that gyroscopic stabilization, which is possible when there is no dissipation in the axis of the frame for certain equilibria that are unstable in the secular sense, breaks down when dissipation occurs.     

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.     

Motion with a constant velocity modulus in a central gravitational field

The problem of the motion of a particle (point mass) with a constant velocity modulus in a Newtonian central gravitational field is investigated by two methods: using Lagrange's equations with a multiplier, and using the equations of dynamics proposed earlier [1] for systems with non-holonomic constraints that are non-linear with respect to velocities. A phase diagram of the motion is constructed. The structure of the trajectories as a function of the initial conditions is investigated. Formulae in the form of quadratures are obtained for calculating the time of motion along the trajectory and the angular distance of flight. A qualitative analysis of the properties of improper integrals expressing the angular distance is presented. These properties are illustrated by the results of a numerical investigation. The possibility of carrying out elementary manoeuvres in the vicinity of an attracting centre are analysed.     

The stability boundaries of the steady motion of a satellite with a gyroscope

Parts of the asymptotic stability boundaries of the uniform motion of the centre of mass of a system of bodies consisting of an asymmetrical satellite with a three-axis gyroscope in a circular orbit are investigated by the second Lyapunov method. Terms of the Lyapunov function that are higher than the second order are enlisted for the investigation. The sign-definiteness criterion of inhomogeneous forms is employed for the corresponding function. Parts of the stability boundaries in which the steady motion investigated is asymptotically stable are established using the Lyapunov asymptotic stability theorem. Application of the Barbashin and Krasovskii theorems reveals parts of the stability boundaries in which the steady motion is unstable. It is established that the asymptotic stability of the steady motion investigated is solved by expanding the Lyapunov function to sixth-order terms.     

The loss of stability of symmetrical equilibrium positions

Systems of differential equations possessing a finite (or compact) symmetry group and depending on one parameter are considered. The nature of the loss of stability of equilibrium positions is investigated in cases when, owing to symmetry, the linearized problem has multiple eigenvalues. Conditions are presented that determine whether the loss of stability when the parameter is varied is soft or hard, for double eigenvalues λ - zero or pure imaginary. Cases of triple zero eigenvalues λ corresponding to tetrahedral (or cubic) symmetry, are considered.     

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.     

The properties of the steady motions of a body carrying a system of two-degree-of-freedom powered gyroscopes

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.     

The stress-energy tensor of matter as a gravitational field source

This article discusses the hypothesis that the universally conserved stress-energy tensor of matter is the source of the gravitational field. From this hypothesis, it immediately follows that space-time must be Riemannian. In contrast to the general theory of relativity, in the gravitational theory based on this hypothesis, the concept of an inertial coordinate system, acceleration relative to space, and the laws of conservation of the energy and angular momenta are retained. In the framework of this theory, the gravitational field is a physical field. The theory explains all observable facts of the solar system, predicts the existence of a large hidden mass of matter in a homogeneous and isotropic universe, and assumes that such a universe can only be “flat.” The theory changes the established idea of the collapse of large massive bodies. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 5–24, January, 1997.     

The gravitational field of the Schwarzschild spacetime

The purpose of this paper is to study a gravitational field of the Schwarzschild spacetime. The type of the gravitational field is determined by the characteristic of λ-tensor.     

The equilibrium positions of an ellipsoid with a cavity containing a liquid on a horizontal plane

The equilibrium positions of an ellipsoid with an ellipsoidal cavity, partially filled with an ideal incompressible liquid, on a horizontal plane in a uniform gravitational field are considered. All trivial and non-trivial equilibrium positions are found and the conditions for their stability are obtained. The results are presented in the form of bifurcation diagrams.     

Searching for equilibrium positions in a game of political competition with restrictions

This paper considers a problem of political economy in which a Nash equilibrium study is performed in a proposed game with restrictions where the two major parties in a country vary their position within a politically flexible framework to increase their number of voters. The model as presented fits the reality of many countries. Moreover, it avoids the uniqueness of equilibrium positions. The problem is stated and solved from a geometric point of view.     

The stability of the equilibrium positions of non-linear non-autonomous mechanical systems

The problem of the stability of the equilibrium positions for a certain class of non-linear mechanical systems under the action of time-dependent quasipotential and dissipative-accelerating forces is considered. A method is proposed for constructing Lyapunov functions for these systems. Sufficient conditions for the stability of an equilibrium position both with respect to all of the variables as well as with respect to some of the variables are determined using the direct Lyapunov method.     

The gravitational field of a laser beam and the Generalized Uncertainty Principle

Light carries energy and therefore is source of a gravitational field. In the paper it is proven, in particular, that a laser beam is source of non-linear gravitational waves corresponding, from a quantum point of view, to spin-1 massless particles. This fact suggests both a possible solution to the old problem on the lack of gravitational attraction between two laser beams moving parallel and a new derivation of the Generalized Uncertainty Principle.