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

Kholostova O. V.
Abstract
We consider the motions of a nearautonomous 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.

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.

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 exoEarth), a star (an exoSun) and a satellite (an exoMoon).
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 exoEarth and exoMoon, we obtain Hamilton’s equations for the
evolution of the angular momentum axis of the exoEarth. Using a canonical change of variables,
we show that the equations are integrable. Assuming that the exoEarth 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.)

Bardin B. S., Avdyushkin A. N.
Abstract
The stability of the collinear libration point $L_1^{}$ in the photogravitational threebody 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.

Sukhov E. A., Volkov E. V.
Abstract
We address the planar restricted fourbody 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 socalled 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.

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. 
Bardin B. S., Chekina E. A., Chekin A. M.
Abstract
The orbital stability of planar pendulumlike 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 socalled case of degeneracy, when it is necessary to take into account terms of
order six in the expansion of the Hamiltonian function, is studied.

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 nonintegrability of the remaining cases is proved by means
of the differential Galois methods. Moreover, for a distinguished case of the heavy gyrostat
a codimension one invariant subspace is identified. It was shown that the system restricted to
this subspace is superintegrable, 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 codimension one appears
to be integrable in the Jacobi sense.

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 Liouvilletype 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.

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 socalled “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.

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 thirdorder 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 righthand 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 steadystate 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 spinorbital resonance capture is explained.

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 boundaryvalue 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 energyefficient than a control splitting these two types of motion. For the nonsymmetrical vehicle configuration, we propose a numerical procedure of solving boundaryvalue 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. 
Artemova E. M., Kilin A. A.
Abstract
This paper is concerned with the controlled motion of a threelink 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 snakelike 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.

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.
