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2013
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    Anatoly Markeev

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

    Publications:

    Markeev A. P.
    On stability of motion of the Maxwell pendulum
    2017, Vol. 13, No. 2, pp.  207-226
    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 area-preserving mapping of the plane into itself. In the space of dimensionless parameters of the problem, regions of stability and instability are found.
    Keywords: stability, map, canonical transformations
    Citation: Markeev A. P.,  On stability of motion of the Maxwell pendulum, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 2, pp.  207-226
    DOI:10.20537/nd1702005
    Markeev A. P.
    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 two-link 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 two-link trajectory of the point.
    Keywords: map, canonical transformations, Hamilton system, stability
    Citation: Markeev A. P.,  On the stability of the two-link trajectory of the parabolic Birkhoff billiards, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp.  75-90
    DOI:10.20537/nd1601005
    Markeev A. P.
    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.
    Keywords: stability, map, canonical transformations
    Citation: Markeev A. P.,  On stability of permanent rotation of a disk that collides with an horizontal plane, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 4, pp.  685–707
    DOI:10.20537/nd1504005
    Markeev A. P.
    Abstract
    We study area-preserving 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 area-preserving maps originates in the Poincaré section surfaces method [1]. The classical works by Birkhoff [2–4], Levi-Civita [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.
    Keywords: map, canonical transformations, Hamilton system, stability
    Citation: Markeev A. P.,  On the fixed points stability for the area-preserving maps, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 3, pp.  503-545
    DOI:10.20537/nd1503005
    Markeev A. P.
    Abstract
    A time-periodic 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.
    Keywords: Hamiltonian system, canonical transformation, stability
    Citation: Markeev A. P.,  G. Birkhoff’s transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp.  343-352
    DOI:10.20537/nd1502009
    Markeev A. P.
    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.
    Keywords: Hamiltonian system, canonical transformation, stability
    Citation: Markeev A. P.,  Simplifying the structure of the third and fourth degree forms in the expansion of the Hamiltonian with a linear transformation, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp.  447-464
    DOI:10.20537/nd1404005
    Markeev A. P.
    A motion of connected pendulums
    2013, Vol. 9, No. 1, pp.  27-38
    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.
    Keywords: pendulum, nonlinear oscillation, stability
    Citation: Markeev A. P.,  A motion of connected pendulums, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 1, pp.  27-38
    DOI:10.20537/nd1301003
    Markeev A. P.
    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 amaterial point, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp.  219-229
    DOI:10.20537/nd1202002
    Markeev A. P.
    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 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 stability in the linear approximation 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, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp.  71-81
    DOI:10.20537/nd1201005
    Markeev A. P.
    On nonlinear Meissner’s equation
    2011, Vol. 7, No. 3, pp.  531-547
    Abstract
    A nonlinear equation of motion for a 0pendulum-type 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 two-stage periodic step function of time. The equation depends on two parameters that characterize the time-averaged 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 well-known 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.
    Keywords: parametric oscillations, stability, resonance, mapping
    Citation: Markeev A. P.,  On nonlinear Meissner’s equation, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp.  531-547
    DOI:10.20537/nd1103009
    Markeev A. P.
    Nonlinear oscillations of sympathetic pendulums
    2010, Vol. 6, No. 3, pp.  605-621
    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 quasi-periodical 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 KAM-theory 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.
    Keywords: pendulum, nonlinear oscillation, resonance, stability
    Citation: Markeev A. P.,  Nonlinear oscillations of sympathetic pendulums, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp.  605-621
    DOI:10.20537/nd1003009
    Markeev A. P.
    To the theory of Resonant Rotation of the Mercury
    2009, Vol. 5, No. 1, pp.  87-98
    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.
    Keywords: Mercury, resonance, periodic motion, stability
    Citation: Markeev A. P.,  To the theory of Resonant Rotation of the Mercury, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 1, pp.  87-98
    DOI:10.20537/nd0901011
    Markeev A. P.
    Abstract
    The paper deals with non-linear 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.
    Keywords: rigid body, constraint, collision, stability
    Citation: Markeev A. P.,  Dynamics of a rigid body that collides with a rigid surface, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 1, pp.  1-38
    DOI:10.20537/nd0801001

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