0
2013
Impact Factor

Anatoly Markeev

pr. Vernadskogo 101, str. 1, Moscow, 119526, Russia
Ishlinsky Institute for Problems in Mechanics, RAS

Publications:

 Markeev A. P. On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance 2017, vol. 22, no. 7, pp.  773-781 Abstract The problem of orbital stability of a periodic motion of an autonomous two-degreeof-freedom 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 fourth-order 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. Keywords: Hamilton’s equations, stability, canonical transformations Citation: Markeev A. P.,  On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance, Regular and Chaotic Dynamics, 2017, vol. 22, no. 7, pp. 773-781 DOI:10.1134/S1560354717070012
 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.  309-316 Abstract A time-periodic one-degree-of-freedom 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 second-degree 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 third-degree terms. For this case, sufficient conditions for instability of the equilibrium are obtained depending on the forms of the fourth and fifth degrees. Keywords: Hamiltonian system, canonical transformations, stability Citation: Markeev A. P.,  On the Birkhoff Transformation in the Case of Complete Degeneracy of the Quadratic Part of the Hamiltonian, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 309-316 DOI:10.1134/S1560354715030077
 Markeev A. P. On the Dynamics of a Rigid Body Carrying a Material Point 2012, vol. 17, no. 3-4, pp.  234-242 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. Keywords: rigid body dynamics, collision, periodic motion, stability Citation: Markeev A. P.,  On the Dynamics of a Rigid Body Carrying a Material Point, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 234-242 DOI:10.1134/S1560354712030021
 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.  142-149 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 straight-line 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. Keywords: rigid body dynamics, collision, periodic motion, stability Citation: Markeev A. P.,  On a Periodic Motion of a Rigid Body Carrying a Material Point in the Presence of Impacts with a Horizontal Plane, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 142-149 DOI:10.1134/S1560354712020049
 Markeev A. P. The Dynamics of a Rigid Body Colliding with a Rigid Surface 2008, vol. 13, no. 2, pp.  96-129 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. Keywords: rigid body, constraints, collision, stability Citation: Markeev A. P.,  The Dynamics of a Rigid Body Colliding with a Rigid Surface, Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 96-129 DOI:10.1134/S1560354708020044
 Markeev A. P. On the Steklov case in rigid body dynamics 2005, vol. 10, no. 1, pp.  81-93 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. Keywords: rigid body dynamics, Euler–Poisson equations, Steklov solutions, orbital stability of the periodic motions Citation: Markeev A. P.,  On the Steklov case in rigid body dynamics , Regular and Chaotic Dynamics, 2005, vol. 10, no. 1, pp. 81-93 DOI:On the Steklov case in rigid body dynamics
 Markeev A. P. On stability of regular precessions of a non-symmetric gyroscope 2003, vol. 8, no. 3, pp.  297-304 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 cross-sections 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. Citation: Markeev A. P.,  On stability of regular precessions of a non-symmetric gyroscope, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 297-304 DOI:10.1070/RD2003v008n03ABEH000245
 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.  149-151 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 vortex-free motion. We show that the solution of the problem can be reduced to quadrature. Citation: Markeev A. P.,  Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 149-151 DOI:10.1070/RD2002v007n02ABEH000201
 Markeev A. P. On the Dynamics of a Solid on an Absolutely Rough Plane 2002, vol. 7, no. 2, pp.  153-160 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" [1-4]. 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 non-linear 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. Citation: Markeev A. P.,  On the Dynamics of a Solid on an Absolutely Rough Plane, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 153-160 DOI:10.1070/RD2002v007n02ABEH000202