Ivan Mamaev
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
Education
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 EditorinChief of the international scientific journal Regular and Chaotic Dynamics, EditorinChief of the journal Russian Journal of Nonlinear Dynamics. Deputy EditorinChief 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.
Profiles:
Publications:
Klekovkin A. V., Karavaev Y. L., Mamaev I. S.
Abstract
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.

Mamaev I. S., Kilin A. A., Karavaev Y. L., Shestakov V. A.
Abstract
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.

Mamaev I. S., Tenenev V. A., Vetchanin E. V.
Abstract
This paper addresses the problem of planeparallel 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.

Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
This paper is concerned with two systems from subRiemannian 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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
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 (twodimensional) 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.

Borisov A. V., Mamaev I. S., Bizyaev I. A.
Abstract
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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
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 quasivelocities. 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 onewheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S.
Abstract
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.

Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.
Abstract
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 highspeed 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 wellknown experimental and theoretical results in this area.

Borisov A. V., Mamaev I. S., Bizyaev I. A.
Abstract
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.

Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
Abstract
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. Computeraided 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.

Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
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.

Bizyaev I. A., Borisov A. V., Mamaev I. S.
Abstract
This paper is concerned with the figures of equilibrium of a selfgravitating ideal fluid with density stratification and a steadystate 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. 
Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
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.

Bolsinov A. V., Borisov A. V., Mamaev I. S.
Abstract
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.

Mamaev I. S., Ivanova T. B.
Abstract
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.

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
Abstract
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.

Erdakova N. N., Mamaev I. S.
Abstract
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 wellknown experimental and theoretical results in this area. 
Borisov A. V., Mamaev I. S., Karavaev Y. L.
Abstract
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.

Borisov A. V., Mamaev I. S., Bizyaev I. A.
Abstract
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 quasiHamiltonian behavior.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In our earlier paper [2] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with noslip 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 dissipationfree periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.

Borisov A. V., Mamaev I. S.
Abstract
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.

Vetchanin E. V., Mamaev I. S., Tenenev V. A.
Abstract
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 nonstationary threedimensional solution to the problem is found. The motion of a sphere and a dropshaped 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.

Borisov A. V., Mamaev I. S., Treschev D. V.
Abstract
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 rubberrolling (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.

Bolsinov A. V., Borisov A. V., Mamaev I. S.
Abstract
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 2tori; 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The noslip 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.
