Anatoly Markeev
pr. Vernadskogo 101, str. 1, Moscow, 119526, Russia
Ishlinsky Institute for Problems in Mechanics, RAS
Publications:
Markeev A. P.
On the Dynamics of a Gravitational Dipole
2021, Vol. 17, no. 3, pp. 247261
Abstract
An orbital gravitational dipole is a rectilinear inextensible rod of negligibly small mass which
moves in a Newtonian gravitational field and to whose ends two point loads are fastened. The
gravitational dipole is mainly designed to produce artificial gravity in a neighborhood of one of
the loads. In the nominal operational mode on a circular orbit the gravitational dipole is located
along the radius vector of its center of mass relative to the Newtonian center of attraction. The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a neighborhood of its nominal mode. The orbit of the center of mass is assumed to be circular or elliptic with small eccentricity. Consideration is given both to planar and arbitrary spatial deviations of the gravitational dipole from its position corresponding to the nominal mode. The analysis is based on the classical Lyapunov and Poincaré methods and the methods of Kolmogorov – Arnold – Moser (KAM) theory. The necessary calculations are performed using computer algorithms. An analytic representation is given for conditionally periodic oscillations. Special attention is paid to the problem of the existence of periodic motions of the gravitational dipole and their Lyapunov stability, formal stability (stability in an arbitrarily high, but finite, nonlinear approximation) and stability for most (in the sense of Lebesgue measure) initial conditions. 
Markeev A. P., Chekhovskaya T. N.
On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance
2020, Vol. 16, no. 4, pp. 607623
Abstract
The points of suspension of two identical pendulums moving in a homogeneous gravitational
field are located on a horizontal beam performing harmonic oscillations of small amplitude along
a fixed horizontal straight line passing through the points of suspension of the pendulums. The
pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial
frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal
oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the
frequency of external periodic action on the system is equal simultaneously to two its frequencies
of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and
stability of periodic motions of pendulums with a period equal to the period of oscillations of the
beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov,
Arnold and Moser) theory, and algorithms of computer algebra. The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation). 
Markeev A. P.
On Nonlinear Resonant Oscillations of a Rigid Body Generated by Its Conical Precession
2018, Vol. 14, no. 4, pp. 503518
Abstract
The motion of a dynamically symmetric rigid body relative to its center of mass in the central Newtonian gravitational field in a circular orbit is investigated. This problem involves
motion (called conical precession) where the dynamical symmetry axis of the body is located all
the time in the plane perpendicular to the velocity vector of the center of mass of the body and makes a constant angle with the direction of the radius vector of the center of mass relative to the
attracting center. This paper deals with a special case in which this angle is $\pi/4$ and the ratio
between the polar and the equatorial principal central moments of inertia of the body is equal to
the number $2/3$ or is close to it. In this case, the conical precession is stable with respect to the
angles that define the position of the symmetry axis in an orbital coordinate system and with
respect to the time derivatives of these angles, and the frequencies of small (linear) oscillations
of the symmetry axis are equal or close to each other (that is, the 1:1 resonance takes place).
Using classical perturbation theory and modern numerical and analytical methods of nonlinear
dynamics, a solution is presented to the problem of the existence, bifurcations and stability
of periodic motions of the symmetry axis of a body which are generated from its relative (in the
orbital coordinate system) equilibrium corresponding to conical precession. The problem of the
existence of conditionally periodic motions is also considered.

Markeev A. P.
On stability of motion of the Maxwell pendulum
2017, Vol. 13, No. 2, pp. 207226
Abstract
We investigate the stability of motion of the Maxwell pendulum in a uniform gravity field [1, 2]. The threads on which the axis and the disk of the pendulum have been suspended are assumed to be weightless and inextensible, and the characteristic linear size of the disk is assumed to be small compared to the lengths of threads. In the unperturbed motion the angle the threads make with the vertical is zero, and the disk moves along the vertical and rotates around its horizontal axis. The nonlinear problem of stability of this motion is solved with respect to small deviations of the threads from the vertical. By means of canonical transformations and the Poincar´e section surface method, the problem is reduced to the study of stability of the fixed point of the areapreserving mapping of the plane into itself. In the space of dimensionless parameters of the problem, regions of stability and instability are found. 
Markeev A. P.
On the stability of the twolink trajectory of the parabolic Birkhoff billiards
2016, Vol. 12, No. 1, pp. 7590
Abstract
We study the inertial motion of a material point in a planar domain bounded by two coaxial parabolas. Inside the domain the point moves along a straight line, the collisions with the boundary curves are assumed to be perfectly elastic. There is a twolink periodic trajectory, for which the point alternately collides with the boundary parabolas at their vertices, and in the intervals between collisions it moves along the common axis of the parabolas. We study the nonlinear problem of stability of the twolink trajectory of the point.

Markeev A. P.
On stability of permanent rotation of a disk that collides with an horizontal plane
2015, Vol. 11, No. 4, pp. 685–707
Abstract
Stability of the motion of a thin homogeneous disk in a uniform gravitational field above a fixed horizontal plane is investigated. Collisions between the disk and the plane are assumed to be absolutely elastic, and friction is negligible. In unperturbed motion, the disk rotates at a constant angular velocity about its vertical diameter, and its center of gravity makes periodic oscillations along a fixed vertical as a result of collisions. The stability problem depends on two dimensionless parameters characterizing the magnitude of the angular velocity of the disk and the height of his jump above the plane in the unperturbed motion. An exact solution of the problem of stability is obtained for all physically admissible values of these parameters.

Markeev A. P.
On the fixed points stability for the areapreserving maps
2015, Vol. 11, No. 3, pp. 503545
Abstract
We study areapreserving maps. The map is assumed to have a fixed point and be analytic in its small neighborhood. The main result is a developed constructive algorithm for studying the stability of the fixed point in critical cases when members of the first degrees (up to the third degree inclusive) in a series specifying the map do not solve the issue of stability. As an application, the stability problem is solved for a vertical periodic motion of a ball in the presence of impacts with an ellipsoidal absolutely smooth cylindrical surface with a horizontal generatrix. Study of areapreserving maps originates in the Poincaré section surfaces method [1]. The classical works by Birkhoff [2–4], LeviCivita [5], Siegel [6, 7], Moser [7–9] are devoted to fundamental aspects of this problem. Further consideration of the objectives is contained in the works by Russman [10], Sternberg [11], Bruno [12, 13], Belitsky [14] and other authors. 
Markeev A. P.
G. Birkhoff’s transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian
2015, Vol. 11, No. 2, pp. 343352
Abstract
A timeperiodic system with one degree of freedom is investigated. The system is assumed to have an equilibrium position, in the vicinity of which the Hamiltonian is represented as a convergent series.This series does not contain members of the second degree, whereas the members to some finite degree $\ell$ do not depend explicitly on time. The algorithm for constructing a canonical transformation is proposed that simplifies the structure of the Hamiltonian in members to degree $\ell$, inclusive. As an application, a special case is considered when the expansion of the Hamiltonian begins with members of the third degree. For this case, sufficient conditions for instability of the equilibrium are obtained depending on the forms of the fourth and fifth degrees.

Markeev A. P.
Simplifying the structure of the third and fourth degree forms in the expansion of the Hamiltonian with a linear transformation
2014, Vol. 10, No. 4, pp. 447464
Abstract
We consider the canonical differential equations describing the motion of a system with one degree of freedom. The origin of the phase space is assumed to be an equilibrium position of the system. It is supposed that in a sufficiently small neighborhood of the equilibrium Hamiltonian function can be represented by a convergent series. This series does not include terms of the second degree, and the terms of the third and fourth degrees are independent of time. Linear real canonical transformations leading the terms of the third and fourth degrees to the simplest forms are found. Classification of the systems in question being obtained on the basis of these forms is used in the discussion of the stability of the equilibrium position.

Markeev A. P.
A motion of connected pendulums
2013, Vol. 9, No. 1, pp. 2738
Abstract
A motion of two identical pendulums connected by a linear elastic spring with an arbitrary stiffness is investigated. The system moves in an homogeneous gravitational field in a fixed vertical plane. The paper mainly studies the linear orbital stability of a periodic motion for which the pendulums accomplish identical oscillations with an arbitrary amplitude. This is one of two types of nonlinear normal oscillations. Perturbational equations depend on two parameters, the first one specifies the spring stiffness, and the second one defines the oscillation amplitude. Domains of stability and instability in a plane of these parameters are obtained. Previously [1, 2] the problem of arbitrary linear and nonlinear oscillations of a small amplitude in a case of a small spring stiffness was investigated. 
Markeev A. P.
On the dynamics of a rigid body carrying amaterial point
2012, Vol. 8, No. 2, pp. 219229
Abstract
In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases. In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler’s case. It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas. 
Markeev A. P.
On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane
2012, Vol. 8, No. 1, pp. 7181
Abstract
A material system consisting of a «carrying» rigid body (a shell) and a body «being carried» (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straightline segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its stability in the linear approximation is studied.

Markeev A. P.
On nonlinear Meissner’s equation
2011, Vol. 7, No. 3, pp. 531547
Abstract
A nonlinear equation of motion for a 0pendulumtype system is investigated. It differs from the classical equation of a mathematical pendulum in the presence of a parametric disturbance. The potential energy of the «pendulum» is a twostage periodic step function of time. The equation depends on two parameters that characterize the timeaveraged value of a parametric disturbance and the depth of its «ripple». These parameters can take on arbitrary values. There exist two equilibrium configurations corresponding to the hanging and inverse «pendulum». The problem of stability of these equilibria is considered. In the first approximation it necessitates an analysis of the wellknown linear Meissner equation. A detailed investigation of this equation is carried out supplementing and specifying the known results. The nonlinear problem of stability of equilibria is solved.

Markeev A. P.
Nonlinear oscillations of sympathetic pendulums
2010, Vol. 6, No. 3, pp. 605621
Abstract
Nonlinear problem of motion of two identical pendulums connected by an elastic spring in the neighborhood of their stable vertical equilibrium is investigated. Stiffness of the spring is supposed small, i. e. the case close to resonance 1:1 is considered. The problem of existence and orbital stability of periodical motions of the pendulums arising from the equilibrium is solved. It is indicated existence of motions asymptotic to one of the periodical motions. An analysis of quasiperiodical motions of an approximate system s given in which members up to the forth order inclusively in the normalizing Hamiltonian of the problem are taken into account. Using KAMtheory the question is considered of preservation of these motions in the complete nonlinear system in which members of all orders in the series expansion of Hamiltonian in the sufficiently small neighborhood of the equilibrium are taken account.

Markeev A. P.
To the theory of Resonant Rotation of the Mercury
2009, Vol. 5, No. 1, pp. 8798
Abstract
The motion of a rigid body about its center of mass under action of gravitational moments of the central Newtonian force field is investigated. The orbit of the center of mass is proposed to be an elliptical one, the eccentricity of the orbit is equal to the one of the Mercury. The central ellipsoid of inertia of the body is arbitrary. The problem of existence of planar periodical rotations under resonance 3:2 of the Mercurian type is considered and their stability (in Liapunov) is investigated. In a case of planar perturbations the nonlinear problem of stability is solved. In a case of arbitrary perturbations of periodical rotations the stability in first (linear) approximation is investigated.

Markeev A. P.
Dynamics of a rigid body that collides with a rigid surface
2008, Vol. 4, No. 1, pp. 138
Abstract
The paper deals with nonlinear oscillations and analysis of stability of stationary rotations and periodic motions of a rigid body that collides with a rigid surface in a uniformgravity field. Along with new results an overview of the fundamental methods and algorithms engaged is given.
