Anatoly Markeev
pr. Vernadskogo 101, str. 1, Moscow, 119526, Russia
Ishlinsky Institute for Problems in Mechanics, RAS
Publications:
Markeev A. P.
On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System
2020, vol. 25, no. 1, pp. 111120
Abstract
This paper is concerned with a onedegreeoffreedom system close to an integrable
system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments,
its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate.
The existence of periodic motions with a period divisible by the period of perturbation is shown
by the Poincaré methods. An algorithm is presented for constructing them in the form of series
(fractional degrees of a small parameter), which is implemented using classical perturbation
theory based on the theory of canonical transformations of Hamiltonian systems. The problem
of the stability of periodic motions is solved using the Lyapunov methods and KAM theory. The
results obtained are applied to the problem of subharmonic oscillations of a pendulum placed
on a moving platform in a homogeneous gravitational field. The platform rotates with constant
angular velocity about a vertical passing through the suspension point of the pendulum, and
simultaneously executes harmonic smallamplitude oscillations along the vertical. Families of
subharmonic oscillations of the pendulum are shown and the problem of their Lyapunov stability
is solved.

Markeev A. P.
On the Stability of the Regular Precession of an Asymmetric Gyroscope at a Secondorder Resonance
2019, vol. 24, no. 5, pp. 502510
Abstract
The motion of a rigid body about a fixed point in a homogeneous gravitational
field is investigated. The body is not dynamically symmetric and its center of gravity lies on
the perpendicular, raised from the fixed point, to one of the circular sections of an ellipsoid
of inertia. A body with such mass geometry may precess regularly about a nonvertical axis
(Grioli’s precession). The problem of the orbital stability of this precession is solved for critical
cases of secondorder resonance, when terms higher than degree four in the series expansion of
the Hamiltonian of the perturbed motion should be taken into account.

Markeev A. P.
On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourthorder Resonance
2017, vol. 22, no. 7, pp. 773781
Abstract
The problem of orbital stability of a periodic motion of an autonomous twodegreeoffreedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure
of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourthorder resonance. It is believed that this case is the critical one for the resonance,
when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion.
Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive. 
Markeev A. P.
On the Birkhoff Transformation in the Case of Complete Degeneracy of the Quadratic Part of the Hamiltonian
2015, vol. 20, no. 3, pp. 309316
Abstract
A timeperiodic onedegreeoffreedom system is investigated. The system is assumed to have an equilibrium point in the neighborhood of which the Hamiltonian is represented as a convergent series. This series does not contain any seconddegree terms, while the terms up to some finite degree $l$ do not depend explicitly on time. An algorithm for constructing a canonical transformation is proposed that simplifies the structure of the Hamiltonian to terms of degree $l$ inclusive.
As an application, a special case is considered when the expansion of the Hamiltonian begins with thirddegree terms. 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.
On the Dynamics of a Rigid Body Carrying a Material Point
2012, vol. 17, no. 34, pp. 234242
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. 17, no. 2, pp. 142149
Abstract
A material system consisting of an outer rigid body (a shell) and an inner body (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 linear stability is studied.

Markeev A. P.
The Dynamics of a Rigid Body Colliding with a Rigid Surface
2008, vol. 13, no. 2, pp. 96129
Abstract
Basic investigation techniques, algorithms, and results are presented for nonlinear oscillations and stability of steady rotations and periodic motions of a rigid body, colliding with a rigid surface, in a uniform gravity field.

Markeev A. P.
On the Steklov case in rigid body dynamics
2005, vol. 10, no. 1, pp. 8193
Abstract
We study the motion of a heavy rigid body with a fixed point. The center of mass is located on mean or minor axis of the ellipsoid of inertia, with the moments of inertia satisfying the conditions $B>A>2C$ or $2B>A>B>C$, $A>2C$ as well as the usual triangle inequalities. Under these circumstances the Euler–Poisson equations have the particular periodic solutions mentioned by V. A. Steklov. We examine the problem of the orbital stability of the periodic motions of a rigid body, which correspond to the Steklov solutions.

Markeev A. P.
On stability of regular precessions of a nonsymmetric gyroscope
2003, vol. 8, no. 3, pp. 297304
Abstract
We consider the motion of a rigid body around a fixed point in homogeneous gravity field. The body is not dynamically symmetric and the center of gravity is situated on the straight line passing through the fixed point perpendicular to circular crosssections of inertia ellipsoid. In 1947, G.Grioli proved that the body with such geometry of mass can be in a state of regular precession around of a nonvertical axis. In this paper we study the stability of this precession.

Markeev A. P.
Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid
2002, vol. 7, no. 2, pp. 149151
Abstract
In this paper we investigate the problem of rolling of a sphere over a fixed horizontal plane; it is assumed that the sphere has a multiply connected cavity with an ideal fluid in vortexfree motion. We show that the solution of the problem can be reduced to quadrature.

Markeev A. P.
On the Dynamics of a Solid on an Absolutely Rough Plane
2002, vol. 7, no. 2, pp. 153160
Abstract
An attempt is made to find a theoretical basis for some dynamic effects discovered experimentally in one problem of solid body dynamics on a plane, namely, the problem of the motion of the "celtic stone" [14]. The main attention is given to oscillations of a solid close to the equilibrium position or steady rotation. The motion is assumed to occur without friction and the supporting plane is fixed. Small oscillations of the body are briefly considered in the neighborhood of its steady rotation about the vertical. An approximate system of equations is obtained which describes nonlinear oscillations of the body in the vicinity of its equilibrium position on a plane and a complete analysis is given. The results of the investigation agree with experimental observations [1,3] of the changes in the direction of rotation the celtic stone about the vertical without any external action, and the origin of rotation in any direction due to oscillations about the horizontal axis.
