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.     

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

The limiting motions of a heavy gyroscope, simulated by a system of rigid bodies, are considered when there is internal friction. The whole set of limiting motions is determined and the nature of their stability is studied in detail for cases when the carried body of the gyroscope has a) three degrees and b) one degree of freedom with respect to the supporting body. The results of an analysis of case a are extended to the motion of a gyroscope with a fluid filling. For case b, the values of the parameters are determined for which the gyroscope, apart from steady rotations, has unsteady limiting motions that are integrable motions in the special Bobylev-Steklov case.     

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.     

Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

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.     

On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.     

On the orbital stability of pendulum-like motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev-Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.     

Oscillations of a rigid body containing an elastic element with distributed parameters

The motions of a hybrid (discrete-continual) system, consisting of a carrier rigid body and an elastic element with distributed parameters fastened to it are investigated. Two types of fastening are considered: (1) both ends are clamped, and (2) one of the ends is clamped while the other is free. A closed system of integro-differential equations is obtained which describes the state of the system under arbitrary initial conditions and forces applied to the rigid body. The perturbed motion of the rigid body in the case of a quasi-linear restoring force is investigated using asymptotic methods. The motions are studied both when there is internal resonance between the oscillations of the rigid body and the natural oscillations of the element, and when there are no such resonances. Qualitative effects are found.     

The motion of a body supported at three points on a rough plane

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.     

Invariant sets in the Goryachev–Chaplygin problem: existence,stability and branching

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.     

On the fall of a heavy rigid body in an ideal fluid

We consider a problem about the motion of a heavy rigid body in an unbounded volume of an ideal irrotational incompressible fluid. This problem generalizes a classical Kirchhoff problem describing the inertial motion of a rigid body in a fluid. We study different special statements of the problem: the plane motion and the motion of an axially symmetric body. In the general case of motion of a rigid body, we study the stability of partial solutions and point out limiting behaviors of the motion when the time increases infinitely. Using numerical computations on the plane of initial conditions, we construct domains corresponding to different types of the asymptotic behavior. We establish the fractal nature of the boundary separating these domains.     

The motion in a perfect fluid of a body containing a moving point mass

The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.     

On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat

We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.     

The steady motions of a heavy ellipsoid filled with liquid on a plane with friction

The motion of a heavy uniform thin-walled ellipsoid of revolution, completely filled with an ideal incompressible liquid, performing uniform vortex motion is investigated. It is assumed that the ellipsoid is situated on a horizontal plane, from the side of which a normal reaction and a force of viscous sliding friction act on it. The equations of motion of the system, suitable both in the general case and in limiting cases of zero ellipsoid mass or zero liquid mass, are set up. Steady and periodic motions of the ellipsoid with the liquid are obtained. The conditions for uniform rotations of the ellipsoid about a vertically situated axis of symmetry to be stable are obtained.     

Invariant sets and symmetric periodic motions of reversible mechanical systems

A method of constructing and classifying all symmetric periodic motions of a reversible mechanical system is proposed. The principal solution of the above problem is given for the Hill problem, the restricted three-body problem (including the photogravitational problem), the problem of a heavy rigid body with a fixed point, and that of a heavy rigid body on a rough plane. In particular, problems requiring a systematic numerical study are therby formulated.     

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.     

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.     

The stability of the steady motions of a rigid body in a central field

The problem of the translational-rotational motion of a rigid body with a triaxial ellipsoid of inertia in a central gravitational field is considered. The body is modelled by a weightless sphere, at the ends of the three mutually perpendicular diameters of which there are point masses. It is shown that, unlike the cases when the approximate expression for the potential of the gravity forces is used, there are not only “trivial” steady motions of the body, for which the main central axes of inertia of the body coincide with the axes of the orbital system of coordinates, but also other classes of steady motions. In addition, the stability of these “trivial” steady motions is investigated, and the possibility of secular stability of the motions, unstable in the satellite approximation, is pointed out.     

Permanent rotations of a body with a viscous filler suspended on a rod

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 limiting steady rotations of a body on a string with a suspension point on the axis of symmetry

The steady motions of an axisymmetrical rigid body suspended from a fixed base by a weightless undeformable rod or a non-twisting inextensible string are investigated. The case when the rod is fastened to the body at a point situated on its axis of dynamic symmetry is considered. All types of limiting equilibrium configurations which are possible when there is an unlimited increase in the angular velocity of rotation of the system about the vertical are analysed. Domains in which each type of limiting regular precession and permanent rotation can exist are constructed in the space of dimensionless parameters, and the nature of their asymptotic behaviour when the angular velocity increases is determined. The limiting motions which are possible in the case of suspension on a rod and impossible in the case of suspension on a string are investigated.