Non-linear oscillations of a satellite at 1:1:1 resonance

The motion of a satellite about its centre of mass in a central Newtonian gravitational field is investigated. The satellite is considered to be a dynamically symmetrical rigid body. It is assumed that the ratio of the polar and equatorial principal central moments of inertia of the satellite is 4/3, or close to this. The orbit of the centre of mass is elliptic, and the orbit eccentricity is assumed to be small. In the limit case, when the orbit of the centre of mass is circular, a steady motion exists (corresponding to relative equilibrium of the satellite in the orbital system of coordinates) in which the axis of dynamic symmetry is directed along the velocity vector of the centre of mass of the satellite; here, the frequencies of the small linear oscillations of the axis of symmetry are equal or close to one another. But in an elliptic orbit of small eccentricity, multiple 1:1:1 resonance occurs in this case, as the oscillation frequencies mentioned are equal or close to the frequency of motion of the centre of mass of the satellite in orbit. The non-linear problem of the existence, bifurcations and stability of periodic motions of the satellite with a period equal to the rotation period of its centre of mass in orbit is investigated.     

Quadratic integrals and the reducibility of the equations of motion of a complex mechanical system in a central field

A mechanical system, consisting of a non-variable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its elements with respect to the carrier is specified), is considered. The system moves in a central force field at a distance from its centre which considerably exceeds the dimensions of the system. The effect of the system motion about the centre of mass on the motion of the centre of mass, which is assumed to be known, is ignored (the analogue of the limited problem [1] for a rigid body). The necessary and sufficient conditions for a quadratic integral of the motion around the centre of mass to exist are obtained in the case when there is no dynamic symmetry. It is shown that, for a quadratic integral to exist, it is necessary that the trajectory of the motion of the centre of mass should be on the surface of a certain circular cone, fixed in inertial space, with its vertex at the centre of the force field. If the trajectory does not lie on the generatrix of the cone, only one non-trivial quadratic integral can exist and the initial system, in the presence of this quadratic integral, reduces to autonomous form. For the motion of the centre of mass along the generatrix or the motion of the system around a fixed centre of mass, the necessary and sufficient conditions for a non-trivial quadratic integral to exist are obtained, which are generalizations of the energy integral, the de Brun integral [2] and the integral of the projection of the kinetic moment. When three non-trivial quadratic integrals exist, the condition for reduction to an autonomous system describing the rotation of the rigid body around the centre of mass and integrable in quadratures are indicated [3, 4].     

The steady motions of gyrostats with equal moments of inertia in a central force field

The problem of the motion of a gyroscope in a central force field is considered. It is assumed that the principal central moments of inertia of the gyrostat are equal to one another, while the centre of mass moves in a circular orbit in a plane passing through the attracting centre. The steady motions of the gyrostat and their stability are investigated. The case when the mass distribution allows of the symmetry group of a tetrahedron is considered as an example.     

Using infinite series to calculate the centre of mass of triangular plates

We present an alternative and pedagogical method to calculate the centre of mass of homogeneous triangular plates by using scaling, symmetry and geometric infinite series. This work also aims to better understand problems that involve concepts of centre of mass of discrete and continuous systems.     

The motion of a body with a plane of symmetry over a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation

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.     

Rapid cylindrically and spherically symmetric strong compression of a perfect gas

The problem of the rapid cylindrically and spherically symmetric strong compression of a perfect (non-viscous and non-heat-conducting) gas is solved. The term “rapid” denotes that the compression time is much less than the run time of a sound wave across the initial cylindrical or spherical volume, while the term “strong” in this case means the simultaneous attainment of as large a density and temperature as desired. By definition, rapid compression must begin in a strong shock wave, which propagates to the axis or centre of symmetry. When the shock wave approaches the centre of symmetry this flow is described by the self-similar Guderley equation with an unbounded rise in temperature, pressure and velocity and a finite increase in the density at the centre of symmetry both behind the arriving and behind the reflected shock waves. To obtain as high an increase in the density as desired one must add on a centred compression wave with focus at the centre of symmetry to the overtaking shock wave at the instant it arrives at the centre of symmetry C⁻-characteristic. Outside a small neighbourhood of the focus one can calculate, by the method of characteristics, the centred wave and the trajectory of the piston which produces it. As for any centred wave, this calculation must be carried out from the centre of symmetry. Since some of the parameters at the focus (certainly the pressure, temperature and velocity of the gas) are unbounded, it is necessary to preface the calculation by the method of characteristics by constructing an analytic solution which holds in a small neighbourhood of the centre of symmetry. Below, after constructing the required solution, the centred waves corresponding to it and the trajectories of the piston producing them are calculated.     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

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.     

Non-linear oscillations of a 1:1 resonance Hamiltonian system

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 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.     

The reflection of a shock wave from a centre or axis of symmetry at adiabatic exponents from 1.2 to 3

The self-similar problem of the reflection of a shock wave from a centre or axis of symmetry for adiabatic exponents from 1.2 to 3 with a maximum step of 0.1 is solved. The distributions of the main parameters behind the reflected shock wave are obtained.     

Approximate equations of rotational motion of a rigid body carrying a movable point mass

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.     

Stabilization of the motions of mechanical systems with variable masses

A holonomic mechanical system with variable masses and cyclic coordinates is considered. Such a system can have generalized steady motions in which the positional coordinates are constant and the cyclic velocities under the action of reactive forces vary according to a given law. Sufficient Routh-Rumyantsev-type conditions for the stability of such motions are determined. The problem of stabilizing a given translational-rotational motion of a symmetric satellite in which its centre of mass moves in a circular orbit and the satellite executes rotational motion about its axis of symmetry is solved.     

The motions of a spheroid on a horizontal plane with viscous friction

The stability conditions of the steady motions of a heavy spheroid on a plane with viscous friction are analysed. A geometrical interpretation of the results is given. The results are compared with the corresponding results in the case of an absolutely smooth and absolutely rough surface. The unsteady motions of the spheroid are investigated numerically.     

Self-similar time-varying flows of an ideal gas with a change in the adiabatic exponent in a “reflected” shock wave

Self-similar one-dimensional time-varying problems are considered under the assumption that there is a change in the adiabatic exponent in a shock wave (SW) running (“reflected”) from a centre or axis of symmetry (later from a centre of symmetry, CS) or from a plane. The medium is an ideal (inviscid and non-heat-conducting) perfect gas with constant heat capacities. In problems with strong SW, the change in the adiabatic exponent in a gas approximately simulates physicochemical processes such as dissociation and ionization and, in the problem of the collapse of a spherical cavity in a liquid, the conversion of liquid into vapour. In both cases, the adiabatic exponent decreases on passing across a reflected SW. Problems of the collapse of a spherical cavity, the reflection of a strong SW from a centre of symmetry and a simpler problem with a self-similarity index of one are examined. When it is assumed that there is an increase in the adiabatic exponent, the self-similar solutions of the first two problems are rejected due to the decrease in entropy from the instant when the SW is reflected. When it is assumed that there is a decrease in the adiabatic exponent, the solutions of these problems only become unsuitable after a finite time has elapsed for the same reason. Up to this time when the decrease in the adiabatic exponent has not reached a certain threshold, the structure of the self-similar solution does not undergo qualitative changes. When the above-mentioned threshold is exceeded, a self-similar solution is possible if a cylindrical or spherical piston expands according to a special law from the instant of SW reflection from the CS. When there is no piston, the flow behind the reflected wave becomes non-self- similar. In the case of the deceleration of a plane flow, conditions are possible with the joining of SW from different sides to a centred rarefaction wave.     

Cylindrically and spherically symmetrical rapid intense compression of an ideal perfect gas with adiabatic exponents from 1.001 to 3

The problem of the rapid intense cylindrically or spherically symmetrical compression of an ideal (non-viscous and non-heat-conducting) perfect gas with different adiabatic exponents is considered. We mean by rapid and intense a compression in a time much less than the time taken for the sound wave to propagate through the uncompressed target up to temperatures and densities as high as desired. It is found that the solution previously obtained with a focused non-self-similar compression wave at the point where the shock wave is reflected from the axis or centre of symmetry (henceforth the centre of symmetry) holds for adiabatic exponents not exceeding 1.9092 and 1.8698 respectively in the cylindrical and spherical cases. It was not possible to construct a complete solution with focusing at the centre of symmetry for gases with higher adiabatic exponents. On the other hand, one can focus the compression waves into a cylinder or sphere of as small, but finite, radius as desired at the instant of arrival on them, for example, of a special characteristic or reflected shock wave of the Guderley problem. It is shown that for high degrees of compression, the time dependences of the coordinates of the pistons which produce such focusing, and of the gas density on them are close to power laws.     

The motion of a particle in the non-stationary field of a logarithmic potential

The plane motion of a material point, driven by a force inversely proportional to the distance from a fixed centre of variable mass, is studied. Consideration is given to the case in which the motion may be integrated by using a specially obtained first integral.     

The oscillations of a satellite about a direction fixed in absolute space

The stability of the plane oscillations of a satellite about the centre of mass in a central Newtonian gravitational field is investigated. The orbit of the centre of mass is circular and the principal central moments of inertia of the satellite are different. In unperturbed motion, one of the axes of inertia is perpendicular to the plane of the orbit, while the satellite performs periodic oscillations about a direction fixed in absolute space. The problem of the stability of these oscillations with respect to plane and spatial perturbations is investigated.     

Spatial chaotic vibrations when there is a periodic change in the position of the centre of mass of a body in the atmosphere

The spatial chaotic motion of a blunt body in the atmosphere when there is a periodic change in the position of the centre of mass is considered. A restoring moment, described by a biharmonic dependence on the spatial angle of attack, a small perturbing moment, due to the periodic change in the position of the centre of mass, and also a small damping moment, acts on the body. The motion when the velocity head remains constant is investigated. When there are no small perturbations, the phase portrait of the system can have points of stable and unstable equilibrium. The behaviour of the system in the neighbourhood of the separatrice is investigated using Mel’nikov's method. An analytic solution of the equation of the body motion along the separatrice is obtained. The criteria for the occurrence of chaos are obtained and the results of numerical modelling, which confirm the correctness of the solutions obtained, are presented.     

Statistical Methods for the Detection of a Centre of Symmetry in a Crystal

It is important in the determination of crystal structures to establish whether a centre of symmetry is present, since this will affect the choice of space group, the equations used for structure‐factor and electron‐density calculations? the size of the asymmetric unit, and whether molecules occupy ‘special’ or ‘general’ positions in the unit cell. As a centre of symmetry does not give rise to a systematic pattern of absent reflections in the X‐ray diffraction data, evidence for its presence or absence must be sought by other methods such as pyroelectric and piezoelectric effects, or statistical tests applied to the intensities of the X‐ray reflections. The theory underlying statistical methods applied to this problem is outlined. Practical tests suggested by several authors are reviewed. These illustrate the application of statistical methods in the field of crystallography and solid state chemistry. A computer program is described which performs the N(z)test and calculates the Wilson ratio; operating instructions and a FORTRAN IV listing are given.