Ivan Mamaev

    Ivan Mamaev
    1, Universitetskaya str., Izhevsk 426034, Russia
    Institute of Computer Science, Russia

    Doctor of Physics and Mathematics, Associate Professor

    Professor of the Russian Academy of Sciences

    Head of the Scientific Educational Laboratory of Mobile Systems, Kalashnikov Izhevsk State Technical University (ISTU)

    Professor at Department of Theoretical Physics at Udmurt State University

    Director of the Institute of Computer Science

    Senior Researcher of the Department of Mathematical Methods of Nonlinear Dynamics at Institute of Mathematics and Mechanics UB RAS

    Born: August 25, 1971


    In 1992 graduated from Udmurt State University (UdSU).

    2000: Thesis of Ph.D. (candidate of science). Thesis title: «Numerical and analytical methods in dynamical systems analysis on Lie algebras», Lomonosov Moscow State University

    2005: Doctor in physics and mathematics. Thesis title: «Numerical and analytical methods in nonholonomic mechanics», St Petersburg University.

    Work experience

    2001–2009: Head of the Laboratory of Dynamical Chaos and Nonlinearity, Udmurt State University (UdSU).

    2002–2009: Deputy director of Institute of Computer Science

    since 2004: Senior Researcher at the Department of Mathematical Methods of Nonlinear Dynamics, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences.

    2010–2013: Head of the Sector of Nonholonomic Mechanics of the Laboratory of Nonlinear Analysis and Design of New Types of Vehicles, established under the grant No 11.G34.31.0039 of the Government of the Russian Federation for State Support for Scientific Research Conducted under the Supervision of Leading Scientists at Russian Institutions of Higher Professional Education.

    since 2013: Head of the Scientific Educational Laboratory of Mobile Systems, Kalashnikov Izhevsk State Technical University, Izhevsk. Professor at the Division of Mechatronic Systems

    2015–2019: Leading Researcher at the Department of Mechanics of the Steklov Mathematical Institute, Russian Academy of Sciences.

    2017–2019: Leading Researcher of the Laboratory of Mechatronics and Robotics of the Moscow Institute of Physics and Technology (National Research University).

    since 2017: Professor of the Department of Theoretical Physics at UdSU.

    Scientific prizes and awards

    In 2012, I.S. Mamaev was awarded the Sofya Kovalevskaya Prize for a series of monographs devoted to integrable systems of Hamiltonian mechanics.

    In 2018, the academic rank of Professor of the Russian Academy of Sciences was conferred on him.

    Membership in the editorial boards and scientific organizations

    Deputy Editor-in-Chief of the international scientific journal Regular and Chaotic Dynamics, Editor-in-Chief of the journal Russian Journal of Nonlinear Dynamics. Deputy Editor-in-Chief of the journal Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, member of the editorial board of the journals Computer Research and Modeling, Bulletin of Kalashnikov ISTU.


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    Klekovkin A. V.,  Karavaev Y. L.,  Mamaev I. S.
    This paper presents the design of an aquatic robot actuated by one internal rotor. The robot body has a cylindrical form with a base in the form of a symmetric airfoil with a sharp edge. For this object, equations of motion are presented in the form of Kirchhoff equations for rigid body motion in an ideal fluid, which are supplemented with viscous resistance terms. A prototype of the aquatic robot with an internal rotor is developed. Using this prototype, experimental investigations of motion in a fluid are carried out.
    Keywords: mobile robot, aquatic robot, motion simulation
    Citation: Klekovkin A. V.,  Karavaev Y. L.,  Mamaev I. S., The Control of an Aquatic Robot by a Periodic Rotation of the Internal Flywheel, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp. 265-279
    Mamaev I. S.,  Kilin A. A.,  Karavaev Y. L.,  Shestakov V. A.
    In this paper we present a study of the dynamics of a mobile robot with omnidirectional wheels taking into account the reaction forces acting from the plane. The dynamical equations are obtained in the form of Newton – Euler equations. In the course of the study, we formulate structural restrictions on the position and orientation of the omnidirectional wheels and their rollers taking into account the possibility of implementing the omnidirectional motion. We obtain the dependence of reaction forces acting on the wheel from the supporting surface on the parameters defining the trajectory of motion: linear and angular velocities and accelerations, and the curvature of the trajectory of motion. A striking feature of the system considered is that the results obtained can be formulated in terms of elementary geometry.
    Keywords: omnidirectional mobile robot, reaction force, simulation, nonholonomic model
    Citation: Mamaev I. S.,  Kilin A. A.,  Karavaev Y. L.,  Shestakov V. A., Criteria of Motion Without Slipping for an Omnidirectional Mobile Robot, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 4, pp. 527-546
    Mamaev I. S.,  Tenenev V. A.,  Vetchanin E. V.
    This paper addresses the problem of plane-parallel motion of the Zhukovskii foil in a viscous fluid. Various motion regimes of the foil are simulated on the basis of a joint numerical solution of the equations of body motion and the Navier – Stokes equations. According to the results of simulation of longitudinal, transverse and rotational motions, the average drag coefficients and added masses are calculated. The values of added masses agree with the results published previously and obtained within the framework of the model of an ideal fluid. It is shown that between the value of circulation determined from numerical experiments, and that determined according to the model of and ideal fluid, there is a correlation with the coefficient $\mathcal{R} = 0.722$. Approximations for the lift force and the moment of the lift force are constructed depending on the translational and angular velocity of motion of the foil. The equations of motion of the Zhukovskii foil in a viscous fluid are written taking into account the found approximations and the drag coefficients. The calculation results based on the proposed mathematical model are in qualitative agreement with the results of joint numerical solution of the equations of body motion and the Navier – Stokes equations.
    Keywords: Zhukovskii foil, Navier – Stokes equations, joint solution of equations, finitedimensional model, viscous fluid, circulation, sharp edge
    Citation: Mamaev I. S.,  Tenenev V. A.,  Vetchanin E. V., Dynamics of a Body with a Sharp Edge in a Viscous Fluid, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp. 473-494
    Borisov A. V.,  Mamaev I. S.
    In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics.
    Keywords: Chaplygin sleigh, inhomogeneous nonholonomic constraints, conservation laws, qualitative analysis, resonance
    Citation: Borisov A. V.,  Mamaev I. S., An inhomogeneous Chaplygin sleigh, Rus. J. Nonlin. Dyn., 2017, Vol. 13, no. 4, pp. 625–639
    Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
    This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
    Keywords: invariant submanifold, rotation number, Cantor ladder, limit cycles
    Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The Hess–Appelrot case and quantization of the rotation number, Rus. J. Nonlin. Dyn., 2017, Vol. 13, no. 3, pp. 433-452
    Bizyaev I. A.,  Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
    This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
    Keywords: sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
    Citation: Bizyaev I. A.,  Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups, Rus. J. Nonlin. Dyn., 2017, Vol. 13, no. 1, pp. 129-146
    Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
    This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.
    Keywords: Chaplygin sleigh, invariant measure, nonholonomic mechanics
    Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., Dynamics of the Chaplygin sleigh on a cylinder, Rus. J. Nonlin. Dyn., 2016, Vol. 12, no. 4, pp. 675–687
    Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A.
    In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
    Keywords: nonholonomic mechanics, nonholonomic constraint, d’Alembert–Lagrange principle, permutation relations
    Citation: Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A., Historical and critical review of the development of nonholonomic mechanics: the classical period, Rus. J. Nonlin. Dyn., 2016, Vol. 12, no. 3, pp. 385-411
    Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
    In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.
    Keywords: nonholonomic constraint, wheeled vehicle, reduction, equations of motion
    Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., On the Hadamard–Hamel problem and the dynamics of wheeled vehicles, Rus. J. Nonlin. Dyn., 2016, Vol. 12, no. 1, pp. 145-163
    Borisov A. V.,  Mamaev I. S.
    This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.
    Keywords: reduction, symmetry, tensor invariant, first integral, symmetry group, symmetry field, nonholonomic constraint, Noether theorem
    Citation: Borisov A. V.,  Mamaev I. S., Symmetries and Reduction in Nonholonomic Mechanics, Rus. J. Nonlin. Dyn., 2015, Vol. 11, no. 4, pp. 763–823
    Bizyaev I. A.,  Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S.
    This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie –Poisson bracket of rank 2. This Lie – Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.
    Keywords: nonholonomic hinge, topology, bifurcation diagram, tensor invariants, Poisson bracket, stability
    Citation: Bizyaev I. A.,  Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S., Topology and Bifurcations in Nonholonomic Mechanics, Rus. J. Nonlin. Dyn., 2015, Vol. 11, no. 4, pp. 735–762
    Borisov A. V.,  Karavaev Y. L.,  Mamaev I. S.,  Erdakova N. N.,  Ivanova T. B.,  Tarasov V. V.
    In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
    Keywords: dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
    Citation: Borisov A. V.,  Karavaev Y. L.,  Mamaev I. S.,  Erdakova N. N.,  Ivanova T. B.,  Tarasov V. V., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Rus. J. Nonlin. Dyn., 2015, Vol. 11, no. 3, pp. 547-577
    Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A.
    In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
    Keywords: nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem
    Citation: Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A., The Jacobi Integral in NonholonomicMechanics, Rus. J. Nonlin. Dyn., 2015, Vol. 11, no. 2, pp. 377-396
    Borisov A. V.,  Erdakova N. N.,  Ivanova T. B.,  Mamaev I. S.
    In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system’s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.
    Keywords: dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
    Citation: Borisov A. V.,  Erdakova N. N.,  Ivanova T. B.,  Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Rus. J. Nonlin. Dyn., 2014, Vol. 10, no. 4, pp. 483-495
    Borisov A. V.,  Mamaev I. S.
    This paper discusses new unresolved problems of nonholonomic mechanics. Hypotheses of the possibility of Hamiltonization and the existence of an invariant measure for such systems are advanced.
    Keywords: nonholonomic mechanics, tensor invariant, invariant measure, Poisson structure
    Citation: Borisov A. V.,  Mamaev I. S., Invariant Measure and Hamiltonization of Nonholonomic Systems, Rus. J. Nonlin. Dyn., 2014, Vol. 10, no. 3, pp. 355-359
    Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
    In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
    Keywords: integrability, vortex sources, shape sphere, reduction, homothetic configurations
    Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The dynamics of three vortex sources, Rus. J. Nonlin. Dyn., , Vol. 10, no. 3, pp. 319-327
    Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
    This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figure or its layers uniformly rotate about an axis fixed in space. As is well known, when there is no rotation, only a ball can be a figure of equilibrium.

    It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with inherent constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification.

    We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
    Keywords: self-gravitating fluid, confocal stratification, homothetic stratification, space of constant curvature
    Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Rus. J. Nonlin. Dyn., 2014, Vol. 10, no. 1, pp. 73-100
    Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
    We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of a reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
    Keywords: nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus
    Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., The problem of drift and recurrence for the rolling Chaplygin ball, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 4, pp. 721-754
    Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S.
    This paper develops the theory of the reducing multiplier for a special class of nonholonomic dynamical systems, when the resulting nonlinear Poisson structure is reduced to the Lie–Poisson bracket of the algebra $e(3)$. As an illustration, the Chaplygin ball rolling problem and the Veselova system are considered. In addition, an integrable gyrostatic generalization of the Veselova system is obtained.
    Keywords: nonholonomic dynamical system, Poisson bracket, Poisson structure, reducing multiplier, Hamiltonization, conformally Hamiltonian system, Chaplygin ball
    Citation: Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S., Geometrization of the Chaplygin reducing-multiplier theorem, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 4, pp. 627-640
    Mamaev I. S.,  Ivanova T. B.
    In this paper we consider the dynamics of rigid body whose sharp edge is in contact with a rough plane. The body can move so that its contact point does not move or slips or loses touch with the support. In this paper, the dynamics of the system is considered within three mechanical models that describe different modes of motion. The boundaries of definition range of each model are given, the possibility of transitions from one mode to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces is discussed.
    Keywords: rod, Painlevé paradox, dry friction, separation, frictional impact
    Citation: Mamaev I. S.,  Ivanova T. B., The dynamics of rigid body whose sharp edge is in contact with a inclined surface with dry friction, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 3, pp. 567-594
    Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
    In this paper we investigate two systems consisting of a spherical shell rolling on a plane without slipping and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is fixed at the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of the nonholonomic hinge. The equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics.
    Keywords: nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge
    Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 3, pp. 547-566
    Erdakova N. N.,  Mamaev I. S.
    In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model.

    For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
    Keywords: dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
    Citation: Erdakova N. N.,  Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 3, pp. 521-545
    Borisov A. V.,  Mamaev I. S.,  Karavaev Y. L.
    The paper presents experimental investigation of a homogeneous circular disk rolling on a horizontal plane. In this paper two methods of experimental determination of the loss of contact between the rolling disk and the horizontal surface before the abrupt halt are proposed. Experimental results for disks of different masses and different materials are presented. The reasons for “micro losses” of contact with surface revealed during the rolling are discussed.
    Keywords: Euler disk, loss of contact, experiment
    Citation: Borisov A. V.,  Mamaev I. S.,  Karavaev Y. L., On the loss of contact of the Euler disk, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 3, pp. 499-506
    Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A.
    In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.
    Keywords: nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution
    Citation: Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A., The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 2, pp. 141-202
    Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
    In our earlier paper [2] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.
    Keywords: non-holonomic constraint, control, dry friction, viscous friction, stability, periodic solutions
    Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., How to control the Chaplygin ball using rotors. II, Rus. J. Nonlin. Dyn., 2013, Vol. 9, no. 1, pp. 59-76
    Borisov A. V.,  Mamaev I. S.
    A new integrable system describing the rolling of a rigid body with a spherical cavity over a spherical base is considered. Previously the authors found the separation of variables for this system at the zero level of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.
    Keywords: integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical periodic solution
    Citation: Borisov A. V.,  Mamaev I. S., Topological analysis of one integrable system related to the rolling of a ball over a sphere, Rus. J. Nonlin. Dyn., 2012, Vol. 8, no. 5, pp. 957-975
    Vetchanin E. V.,  Mamaev I. S.,  Tenenev V. A.
    An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier–Stokes equations and equations of motion. A non-stationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid, which is caused by the motion of internal material points, in a gravitational field is explored. The possibility of motion of a body in an arbitrary given direction is shown.
    Keywords: finite-volume numerical method, Navier-Stokes equations, variable internal mass distribution, motion control
    Citation: Vetchanin E. V.,  Mamaev I. S.,  Tenenev V. A., The motion of a body with variable mass geometry in a viscous fluid, Rus. J. Nonlin. Dyn., 2012, Vol. 8, no. 4, pp. 815-836
    Borisov A. V.,  Mamaev I. S.,  Treschev D. V.
    In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.
    Keywords: rolling without slipping, nonholonomic constraint, Chaplygin system, conformally Hamiltonian system
    Citation: Borisov A. V.,  Mamaev I. S.,  Treschev D. V., Rolling of a rigid body without slipping and spinning: kinematics and dynamics, Rus. J. Nonlin. Dyn., 2012, Vol. 8, no. 4, pp. 783-797
    Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S.
    In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
    Keywords: non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability
    Citation: Bolsinov A. V.,  Borisov A. V.,  Mamaev I. S., Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals, Rus. J. Nonlin. Dyn., 2012, Vol. 8, no. 3, pp. 605-616
    Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
    In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic contrability is shown and the control inputs providing motion of the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.
    Keywords: non-holonomic constraint, non-holonomic distribution, control, Chow–Rashevsky theorem, drift
    Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., How to control the Chaplygin sphere using rotors, Rus. J. Nonlin. Dyn., 2012, Vol. 8, no. 2, pp. 289-307

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