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.     

The motion of a body with a cavity partly filled with a viscous liquid

Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered.     

Instability in the critical case of pair of pure imaginary roots for a class of systems with aftereffect

The stability of a system described by Volterra integrodifferential equations is investigated in the critical case when the characteristic equation has a pair of pure imaginery roots. Conditions for instability, analogous to the well-known conditions from the theory of differential equations [1], are derived. (A similar result was established previously in [2] for integrodifferential equations of simpler structure with integral kernels of exponential-polynomial type). For the proof, several manipulations are used to simplify the original equation and, in particular, to reduce the linearized equation to the form of a differential equation with constant diagonal matrix. (An analogous approach was used to analyse instability for Volterra integrodifferential equations in the critical case of zero root in [3, 4]). As an example, the sign of the Lyapunov constant in the problem of the rotational motion of a rigid body with viscoelastic supports is calculated.     

Three-dimensional non-linear oscillations of a rod with hinged supports

The free and forced flexural oscillations of a rod with hinged supports are investigated analytically and numerically. The geometrical non-linearity due to the change in the length of the central line of the rod accompanying its three-dimensional motion is taken into account. The oscillations of a rod with different natural frequencies in two mutually perpendicular directions as a consequence of the variance in the flexural stiffnesses of the rod or the stiffnesses of the supports in the different directions, are considered. It is shown in the case of natural oscillations that, together with two planar forms of motion, a form exists when a certain threshold value is exceeded, which corresponds to the motion of the cross-sections of the rod in a circle. The amplitude-frequency and phase-frequency characteristics of the system are constructed and qualitatively investigated in the neighbourhood of the principal resonance.     

Non-linear oscillations of a hamiltonian system with one degree of freedom and fourth-order resonance

Non-linear oscillations of a 2π-periodic Hamiltonian system with one degree of freedom are considered . It is assumed that the origin of coordinates is an equilibrium position, the linearized system is assumed to be stable, its characteristic exponents ±iv are pure imaginary, and the value of 4v is close to an integer. When the methods of classical perturbation theory are used, the investigation reduces to an analysis of a model system which can be described by the typical Hamiltonian of problems on the motion of Hamiltonian systems with one degree of freedom in the case of fourth-order resonance. The system is analysed in detail. The results for the model system are applied to the total system using Poincaré's theory of periodic motion and the KAM-theory. The existence, number and stability of 8π-periodic motions of the initial system are investigated. Trajectories of motion which start in a fairly small neighbourhood of the origin of coordinates are bounded. An estimate of the size of that neighbourhood is given. The examples considered are of a point mass above a curve in the shape of an ellipse which collides with the curve, and plane non-linear oscillations of a satellite in an elliptical orbit in the case of fourth-order resonance.     

First integrals and families of symmetric periodic motions of a reversible mechanical system

A reversible mechanical system which allows of first integrals is studied. It is established that, for symmetric motions, the constants of the asymmetric integrals are equal to zero. The form of the integrals of a reversible linear periodic system corresponding to zero characteristic exponents and the structure of the corresponding Jordan Boxes are investigated. A theorem on the non-existence of an additional first integral and a theorem on the structural stabilities of having a symmetric periodic motion (SPM) are proved for a system with m symmetric and k asymmetric integrals. The dependence of the period of a SPM on the constants of the integrals is investigated. Results of the oscillations of a quasilinear system in degenerate cases are presented. Degeneracy and the principal resonance: bifurcation with the disappearance of the SPM and the birth of two asymmetric cycles, are investigated. A heavy rigid body with a single fixed point is studied as the application of the results obtained. The Euler-Poisson equations are used. In the general case, the energy integral and the geometric integral are symmetric while the angular momentum integral turns out to be asymmetric. In the special case, when the centre of gravity of the body lies in the principal plane of the ellipsoid of inertia, all three classical integrals become symmetric. It is ascertained here that any SPM of a body contains four zero characteristic exponents, of which two are simple and two form a Jordan Box. In typical situation, the remaining two characteristic exponents are not equal to zero. All of the above enables one to speak of an SPM belonging to a two-parameter family and the absence of an additional first integral. It is established that a body also executes a pendulum motion in the case when the centre of gravity is close to the principal plane of the ellipsoid of inertia.     

Stability and control of transversal oscillations of a tethered satellite system

The tethered satellite system is characterized by weak nonlinearities but it practically works in conditions of internal resonance which produces unstable oscillations. The effect of a longitudinal control force is investigated. Since the displacement component in the orbit plane is always present in the motion due to the nonlinear coupling, the control force is assumed depending only on this component and also when a prevailing out-of-plane oscillation is considered. The harmonic balance method and numerical solutions of amplitude modulated equations are used to obtain stationary and nonstationary oscillations, respectively; the Floquet theory is followed in the stability analysis. The assumed control force is shown to be effective in reducing the primary and secondary instability regions of oscillations perturbed by internally resonant disturbance components.     

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.     

Stochastic Schrodinger Equation for a Quantum Oscillator with Dissipation

In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.     

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 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.     

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 cylindrical body in a stratified fluid under the action of a radiation force

The free motion of a thin cylindrical body is investigated based on a previously derived expression for the radiation force acting on moving point sources in a stratified fluid. The fundamental equations of motion are derived, the limits of applicability of the approximation used are indicated and the results of calculations of typical trajectories of a body which begins to move with a specified velocity from a position of neutral buoyancy at an angle to the horizon are presented. Calculations of the trajectory of motion of a thin cylindrical body in a stratified fluid when the total radiation force is taken into account show that the effect of the lateral component of this force is considerable and leads not only to quantitative corrections but also to qualitative effects (for example, to an increase in the oscillations of the body and a change in its direction of motion). The results obtained pertain both to the motion of solids in fluids and to the translational motion of vortex dipoles in weakly stratified media.     

Evolution of the precessional motion of unbalanced gyrostats of variable structure

The precessional motion of an unbalanced gyrostat of variable structure when acted upon by dissipative and accelerating external and internal moments, which depend on the angular velocities of the bodies (the carrier and the rotor) is considered. A qualitative method of analysing the phase space of non-autonomous dynamical systems is developed, based on the determination of the curvature of the phase trajectory. The motion is analysed and the conditions for obtaining the required modes of nutational-precessional motion of unbalanced gyrostats of variable structure are synthesized using this method. A number of cases of the motion of a gyrostat of variable structure, including free motion, motion when there are constant internal and reactive moments and, also, under the action of the moments of resistance forces, proportional to the angular velocities, is investigated. The possible evolutions in the above-mentioned cases of motion and the causes of these evolutions are determined. The conditions for evolution with a decreasing amplitude of the nutational oscillations are obtained.     

The critical case of a pair of zero roots in a two-degree-of-freedom hamiltonian system

The problem of the motion of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its equilibrium position is considered. It is assumed that the characteristic equation of the linearized system has a pair of pure imaginary roots. The roots of the other pair are assumed to be close to or equal to zero, and in the latter case non-simple elementary dividers correspond to these roots. The problem of the existence, bifurcations and orbital stability of families of periodic motions, generated from the equilibrium position, is solved. Conditionally periodic motions are analysed. The problem of the boundedness of the trajectories of the system in the neighbourhood of the equilibrium position in the case when it is Lyapunov unstable, is considered. Non-linear oscillations of an artificial satellite in the region of its steady rotation around the normal to the orbit plane are investigated as an application.     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

The equations of the perturbed motion of a rocket as a thin-walled structure with a liquid

The perturbed motion of a rocket as an elastic thin-walled structure with compartments partially filled with liquid propellant is considered. It is assumed that the normal modes of the hydroelastic oscillations of the rocket are determined under the condition that the velocity potential on the free surface of the liquid is equal to zero and with standard remaining conditions. Certain features of these modes with zero fundamental frequencies are pointed out and the “loss” of mass effect associated with this is explained. Equations are derived for the perturbed motion of a rocket taking account of the hydroelastic oscillations of its structure and the oscillations of the liquid with deviations of the free surface from the equilibrium position under the action of mass forces. The coefficients of these equations, characterizing the relation between the different type of oscillations, are expressed in terms of known hydrodynamic parameters and the values of the oscillation modes at certain points.     

Resonance interactions of waves in an ice channel

The properties of low-amplitude surface waves propagating in an ice channel are investigated in the shallow-water approximation. The ice cover is modelled either by a rigid cap or by a thin elastic plate floating on a liquid surface. It is shown that an ice channel is a waveguide for surface waves. The dispersive properties of the natural oscillations of the liquid in the channel are investigated. The resonance velocities of the motion of the load on the channel surface, at which the amplitude of the forced oscillations of the liquid increases without limit in time, are determined. The decay instability of the natural oscillations of high harmonics with respect to waves of the first mode is demonstrated. The process is described by the standard equations for non-linear three-wave interaction. The investigations lead to the conclusion that critical modes of motion of a boat are realizable in an ice channel.     

The stability of a class of reversible systems

The problem of the stability of the point of rest of an autonomous system of ordinary differential equations from a class of reversible systems [1] characterized by the critical case of m zero roots and n pairs of pure imaginary roots is considered. When there are no internal resonances [2, 3], the point of rest always has Birkhoff complete stability [2]. Internal resonances may lead to Lyapunov instability. The conditions of stability and instability of the model system when there are third-order resonances may be obtained from a criterion previously developed [4] for the case of pure imaginary roots. The results are used to analyse the stability of the translational-rotational motion of an active artificial satellite in a non-Keplerian circular orbit, including a geostationary satellite in any latitude [4, 5]. The region of stability of relative equilibria and regular precession of the satellite is constructed assuming a central gravitational field and the resonance modes are analysed.     

Finite dimensional characteristic functions of Brownian rough path

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2: Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in ?^d and obtain an explicit formula for the case when d = 2