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# Vol. 18, no. 4, 2022 Anatoly Pavlovich Markeev. On the Occasion of his 80th Birthday

 Abstract Citation: Anatoly Pavlovich Markeev. On the Occasion of his 80th Birthday, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 467-472 DOI:10.20537/nd221224
 Markeev A. P. Abstract This paper studies a material system with a finite number of degrees of freedom the motion of which is described by differential Lagrange’s equations of the second kind. A twice continuously differentiable change of generalized coordinates and time is considered. It is well known that the equations of motion are covariant under such transformations. The conventional proof of this covariance property is usually based on the integral variational principle due to Hamilton and Ostrogradskii. This paper gives a proof of covariance that differs from the generally accepted one. In addition, some methodical examples interesting in theory and applications are considered. In some of them (the equilibrium of a polytropic gas sphere between whose particles the forces of gravitational attraction act and the problem of the planar motion of a charged particle in the dipole force field) Lagrange’s equations are not only covariant, but also possess the invariance property. Keywords: analytical mechanics, Lagrange’s equations, transformation methods in mechanics Citation: Markeev A. P., On a Change of Variables in Lagrange’s Equations, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 473-480 DOI:10.20537/nd220701
 Kholostova O. V. Abstract We consider the motions of a near-autonomous Hamiltonian system $2\pi$-periodic in time, with two degrees of freedom, in a neighborhood of a trivial equilibrium. A multiple parametric resonance is assumed to occur for a certain set of system parameters in the autonomous case, for which the frequencies of small linear oscillations are equal to two and one, and the resonant point of the parameter space belongs to the region of sufficient stability conditions. Under certain restrictions on the structure of the Hamiltonian of perturbed motion, nonlinear oscillations of the system in the vicinity of the equilibrium are studied for parameter values from a small neighborhood of the resonant point. Analytical boundaries of parametric resonance regions are obtained, which arise in the presence of secondary resonances in the transformed linear system (the cases of zero frequency and equal frequencies). The general case, for which the parameter values do not belong to the parametric resonance regions and their small neighborhoods, and both cases of secondary resonances are considered. The question of the existence of resonant periodic motions of the system is solved, and their linear stability is studied. Two- and threefrequency conditionally periodic motions are described. As an application, nonlinear resonant oscillations of a dynamically symmetric satellite (rigid body) relative to the center of mass in the vicinity of its cylindrical precession in a weakly elliptical orbit are investigated. Keywords: multiple parametric resonance, normalization, nonlinear oscillations, stability, periodic motions, satellite, cylindrical precession Citation: Kholostova O. V., On Nonlinear Oscillations of a Time-Periodic Hamiltonian System at a 2:1:1 Resonance, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 481-512 DOI:10.20537/nd221101
 Cabral H. E.,  Carvalho A. C. Abstract We study the mechanical system consisting of the following variant of the planar pendulum. The suspension point oscillates harmonically in the vertical direction, with small amplitude $\varepsilon$, about the center of a circumference which is located in the plane of oscillations of the pendulum. The circumference has a uniform distribution of electric charges with total charge $Q$ and the bob of the pendulum, with mass $m$, carries an electric charge $q$. We study the motion of the pendulum as a function of three parameters: $\varepsilon$, the ratio of charges $\mu = \frac qQ$ and a parameter $\alpha$ related to the frequency of oscillations of the suspension point and the length of the pendulum. As the speed of oscillations of the mass $m$ are small magnetic effects are disregarded and the motion is subjected only to the gravity force and the electrostatic force. The electrostatic potential is determined in terms of the Jacobi elliptic functions. We study the parametric resonance of the linearized equations about the stable equilibrium finding the boundary surfaces of stability domains using the Deprit – Hori method. Keywords: planar charged pendulum, Hamiltonian systems, parametric resonance, Deprit – Hori method, Jacobi elliptic integrals Citation: Cabral H. E.,  Carvalho A. C., Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 513-526 DOI:10.20537/nd220703
 Podvigina O. M. Abstract We investigate the temporal evolution of the rotation axis of a planet in a system comprised of the planet (which we call an exo-Earth), a star (an exo-Sun) and a satellite (an exo-Moon). The planet is assumed to be rigid and almost spherical, the difference between the largest and the smallest principal moments of inertia being a small parameter of the problem. The orbit of the planet around the star is a Keplerian ellipse. The orbit of the satellite is a Keplerian ellipse with a constant inclination to the ecliptic, involved in two types of slow precessional motion, nodal and apsidal. Applying time averaging over the fast variables associated with the frequencies of the motion of exo-Earth and exo-Moon, we obtain Hamilton’s equations for the evolution of the angular momentum axis of the exo-Earth. Using a canonical change of variables, we show that the equations are integrable. Assuming that the exo-Earth is axially symmetric and its symmetry and rotation axes coincide, we identify possible types of motions of the vector of angular momentum on the celestial sphere. Also, we calculate the range of the nutation angle as a function of the initial conditions. (By the range of the nutation angle we mean the difference between its maximal and minimal values.) Keywords: nutation angle, exoplanet, averaging, Hamiltonian dynamics Citation: Podvigina O. M., Rotation of a Planet in a Three-body System: a Non-resonant Case, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 527-541 DOI:10.20537/nd221001
 Bardin B. S.,  Avdyushkin A. N. Abstract The stability of the collinear libration point $L_1^{}$ in the photogravitational three-body problem is investigated. This problem is concerned with the motion of a body of infinitely small mass which experiences gravitational forces and repulsive forces of radiation pressure coming from two massive bodies. It is assumed that the massive bodies move in circular orbits and that the body of small mass is located in the plane of their motion. Using methods of normal forms and KAM theory, a rigorous analysis of the Lyapunov stability of the collinear libration point lying on the segment connecting the massive bodies is performed. Conclusions on the stability are drawn both for the nonresonant case and for the case of resonances through order four. Keywords: collinear libration point, photogravitational three-body problem, normal forms, KAM theory, Lyapunov stability, resonances Citation: Bardin B. S.,  Avdyushkin A. N., On Stability of the Collinear Libration Point $L_1^{}$ in the Planar Restricted Circular Photogravitational Three-Body Problem, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 543-562 DOI:10.20537/nd221202
 Sukhov E. A.,  Volkov E. V. Abstract We address the planar restricted four-body problem with a small body of negligible mass moving in the Newtonian gravitational field of three primary bodies with nonnegligible masses. We assume that two of the primaries have equal masses and that all primary bodies move in circular orbits forming a Lagrangian equilateral triangular configuration. This configuration admits relative equilibria for the small body analogous to the libration points in the threebody problem. We consider the equilibrium points located on the perpendicular bisector of the Lagrangian triangle in which case the bodies constitute the so-called central configurations. Using the method of normal forms, we analytically obtain families of periodic motions emanating from the stable relative equilibria in a nonresonant case and continue them numerically to the borders of their existence domains. Using a numerical method, we investigate the orbital stability of the aforementioned periodic motions and represent the conclusions as stability diagrams in the problem’s parameter space. Keywords: Hamiltonian mechanics, four-body problem, periodic motions, orbital stability Citation: Sukhov E. A.,  Volkov E. V., Numerical Orbital Stability Analysis of Nonresonant Periodic Motions in the Planar Restricted Four-Body Problem, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 563-576 DOI:10.20537/nd221201
 Krasil'nikov P. S.,  Ismagilov A. R. Abstract This paper discusses and analyzes the dumb–bell equilibria in a generalized Sitnikov problem. This has been done by assuming that the dumb–bell is oriented along the normal to the plane of motion of two primaries. Assuming the orbits of primaries to be circles, we apply bifurcation theory to investigate the set of equilibria for both symmetrical and asymmetrical dumb–bells. We also investigate the linear stability of the trivial equilibrium of a symmetrical dumb–bell in the elliptic Sitnikov problem. In the case of the dumb–bell length $l\geqslant 0.983819$, an instability of the trivial equilibria for eccentricity $e \in (0,\,1)$ is proved. Keywords: Sitnikov problem, dumb–bell, equilibrium, linear stability Citation: Krasil'nikov P. S.,  Ismagilov A. R., On the Dumb-Bell Equilibria in the Generalized Sitnikov Problem, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 577-588 DOI:10.20537/nd221203
 Bardin B. S.,  Chekina E. A.,  Chekin A. M. Abstract The orbital stability of planar pendulum-like oscillations of a satellite about its center of mass is investigated. The satellite is supposed to be a dynamically symmetrical rigid body whose center of mass moves in a circular orbit. Using the recently developed approach [1], local variables are introduced and equations of perturbed motion are obtained in a Hamiltonian form. On the basis of the method of normal forms and KAM theory, a nonlinear analysis is performed and rigorous conclusions on orbital stability are obtained for almost all parameter values. In particular, the so-called case of degeneracy, when it is necessary to take into account terms of order six in the expansion of the Hamiltonian function, is studied. Keywords: rigid body, satellite, oscillations, orbital stability, Hamiltonian system, local coordinates, normal form Citation: Bardin B. S.,  Chekina E. A.,  Chekin A. M., On the Orbital Stability of Pendulum Oscillations of a Dynamically Symmetric Satellite, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 589-607 DOI:10.20537/nd221211
 Maciejewski A. J.,  Przybylska M. Abstract In this paper, we investigate the gyrostat under influence of an external potential force with the Suslov nonholonomic constraint: the projection of the total angular velocity onto a vector fixed in the body vanishes. We investigate cases of free gyrostat, the heavy gyrostat in the constant gravity field, and we discuss certain properties for general potential forces. In all these cases, the system has two first integrals: the energy and the geometric first integral. For its integrability, either two additional first integrals or one additional first integral and an invariant $n$-form are necessary. For the free gyrostat we identify three cases integrable in the Jacobi sense. In the case of heavy gyrostat three cases with one additional first integral are identified. Among them, one case is integrable and the non-integrability of the remaining cases is proved by means of the differential Galois methods. Moreover, for a distinguished case of the heavy gyrostat a co-dimension one invariant subspace is identified. It was shown that the system restricted to this subspace is super-integrable, and solvable in elliptic functions. For the gyrostat in general potential force field conditions of the existence of an invariant $n$-form defined by a special form of the Jacobi last multiplier are derived. The class of potentials satisfying them is identified, and then the system restricted to the corresponding invariant subspace of co-dimension one appears to be integrable in the Jacobi sense. Keywords: gyrostat, Suslov constraint, integrability, nonholonomic systems Citation: Maciejewski A. J.,  Przybylska M., Gyrostatic Suslov Problem, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 609-627 DOI:10.20537/nd221210
 Gadzhiev M. M.,  Kuleshov A. S. Abstract The problem of the motion, in the free molecular flow of particles, of a rigid body with a fixed point bounded by the surface of an ellipsoid of revolution is considered. This problem is similar in many aspects to the classical problem of the motion of a heavy rigid body about a fixed point. In particular, this problem possesses the integrable cases corresponding to the classical Euler – Poinsot, Lagrange and Hess cases of integrability of the equations of motion of a heavy rigid body with a fixed point. A natural question arises about the existence of analogues of other integrable cases that exist in the problem of motion of a heavy rigid body with a fixed point (Kovalevskaya case, Goryachev – Chaplygin case, etc) for the system considered. Using the standard Euler angles as generalized coordinates, the Hamiltonian function of the system is derived. Equations of motion of the body in the flow of particles are presented in Hamiltonian form. Using the theorem on the Liouville-type nonintegrability of Hamiltonian systems near elliptic equilibrium positions, which has been proved by V. V. Kozlov, necessary conditions for the existence in the problem under consideration of an additional analytic first integral independent of the energy integral are presented. We have proved that the necessary conditions obtained are not fulfilled for a rigid body with a mass distribution corresponding to the classical Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point. Thus, we can conclude that this system does not possess an integrable case similar to the Kovalevskaya integrable case in the problem of the motion of a heavy rigid body with a fixed point. Keywords: rigid body with a fixed point, free molecular flow of particles, Hamiltonian system, nonintegrability Citation: Gadzhiev M. M.,  Kuleshov A. S., Nonintegrability of the Problem of the Motion of an Ellipsoidal Body with a Fixed Point in a Flow of Particles, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 629-637 DOI:10.20537/nd221216
 Burov A. A.,  Kosenko I.,  Nikonov V. I. Abstract The motion of a spacecraft containing a moving massive point in the central field of Newtonian attraction is considered. Within the framework of the so-called “satellite approximation”, the center of mass of the system is assumed to move in an unperturbed elliptical Keplerian orbit. The spacecraft’s dynamics about its center of mass is studied. Conditions under which the spacecraft rotates about a perpendicular to the plane of the orbit uniformly with respect to the true anomaly are found. Such uniform rotations are achieved using a specially selected rule for changing the position of a massive point with respect to the spacecraft. Necessary conditions for these uniform rotations are studied numerically. An analysis of the nonintegrability of a special class of spacecraft’s rotation is carried out using the method of separatrix splitting. Poincaré sections are constructed for certain parameter values. Several linearly stable periodic motions are pointed out and investigated. Keywords: spacecraft attitude dynamics, spacecraft in an elliptic orbit, spacecraft with variable mass distribution, spacecraft’s chaotic oscillations, spacecraft’s periodic motions Citation: Burov A. A.,  Kosenko I.,  Nikonov V. I., Spacecraft with Periodic Mass Redistribution: Regular and Chaotic Behaviour, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 639-649 DOI:10.20537/nd221222
 Shatina A. V.,  Djioeva M. I.,  Osipova L. S. Abstract This paper considers the rotational motion of a satellite equipped with flexible viscoelastic rods in an elliptic orbit. The satellite is modeled as a symmetric rigid body with a pair of flexible viscoelastic rods rigidly attached to it along the axis of symmetry. A planar case is studied, i. e., it is assumed that the satellite’s center of mass moves in a Keplerian elliptic orbit lying in a stationary plane and the satellite’s axis of rotation is orthogonal to this plane. When the rods are not deformed, the satellite’s principal central moments of inertia are equal to each other. The linear bending theory for thin inextensible rods is used to describe the deformations. The functionals of elastic and dissipative forces are introduced according to this model. The asymptotic method of motions separation is used to derive the equations of rotational motion reflecting the influence of the fluctuations, caused by the deformations of the rods. The method of motion separation is based on the assumption that the period of the autonomous oscillations of a point belonging to the rod is much smaller than the characteristic time of these oscillations’ decay, which, in its turn, is much smaller than the characteristic time of the system’s motion as a whole. That is why only the oscillations induced by the external and inertial forces are taken into account when deriving the equations of the rotational motion. The perturbed equations are described by a third-order system of ordinary differential equations in the dimensionless variable equal to the ratio of the satellite’s absolute value of angular velocity to the mean motion of the satellite’s center of mass, the angle between the satellite’s axis of symmetry and a fixed axis and the mean anomaly. The right-hand sides of the equation depend on the mean anomaly implicitly through the true anomaly. A new slow angular variable is introduced in order to perform the averaging for the perturbed system near the 3:2 resonance, and the averaging is performed over the mean anomaly of the satellite’s center of mass orbit. In doing so the wellknown expansions of the true anomaly and its sine and cosine in powers of the mean anomaly are used. The steady-state solutions of the resulting system of equations are found and their stability is studied. It is shown that, if certain conditions are fulfilled, then asymptotically stable solutions exist. Therefore, the 3:2 spin-orbital resonance capture is explained. Keywords: Keplerian elliptical orbit, satellite, spin-orbit resonance, dissipation Citation: Shatina A. V.,  Djioeva M. I.,  Osipova L. S., Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 651-660 DOI:10.20537/nd220803
 Moiseev G. N.,  Zobova A. A. Abstract We consider the dynamics of an omnidirectional vehicle moving on a perfectly rough horizontal plane. The vehicle has three omniwheels controlled by three direct current motors. We study constant voltage dynamics for the symmetric model of the vehicle and get a general analytical solution for arbitrary initial conditions which is shown to be Lyapunov stable. Piecewise combination of the trajectories produces a solution to boundary-value problems for arbitrary initial and terminal mass center coordinates, course angles and their derivatives with one switch point. The proposed control combining translation and rotation of the vehicle is shown to be more energy-efficient than a control splitting these two types of motion. For the nonsymmetrical vehicle configuration, we propose a numerical procedure of solving boundary-value problems that uses parametric continuation of the solution obtained for the symmetric vehicle. It shows that the proposed type of control can be used for an arbitrary vehicle configuration. Keywords: omnidirectional vehicle, omniwheel, universal wheel, dynamics-based control, piecewise control, point-to-point path planning Citation: Moiseev G. N.,  Zobova A. A., Dynamics-Based Piecewise Constant Control of an Omnivehicle, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 661-680 DOI:10.20537/nd221102
 Artemova E. M.,  Kilin A. A. Abstract This paper is concerned with the controlled motion of a three-link wheeled snake robot propelled by changing the angles between the central and lateral links. The limits on the applicability of the nonholonomic model for the problem of interest are revealed. It is shown that the system under consideration is completely controllable according to the Rashevsky – Chow theorem. Possible types of motion of the system under periodic snake-like controls are presented using Fourier expansions. The relation of the form of the trajectory in the space of controls to the type of motion involved is found. It is shown that, if the trajectory in the space of controls is centrally symmetric, the robot moves with nonzero constant average velocity in some direction. Keywords: nonholonomic mechanics, wheeled vehicle, snake robot, controllability, periodic control Citation: Artemova E. M.,  Kilin A. A., A Nonholonomic Model and Complete Controllability of a Three-Link Wheeled Snake Robot, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 681-707 DOI:10.20537/nd221204
 Karavaev Y. L. Abstract This paper describes the existing designs of spherical robots and reviews studies devoted to investigating their dynamics and to developing algorithms for controlling them. An analysis is also made of the key features and the historical aspects of the development of their designs, in particular, taking into account various areas of application. Keywords: spherical robot, rolling, design, modeling Citation: Karavaev Y. L., Spherical Robots: An Up-to-Date Overview of Designs and Features, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 4, pp. 709-750 DOI:10.20537/nd221207

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