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
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 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.
Profiles:
Publications:
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Bizyaev I. A., Mamaev I. S.
Roller Racer with Varying Gyrostatic Momentum: Acceleration Criterion and Strange Attractors
2023, vol. 28, no. 1, pp. 107-130
Abstract
In this paper we investigate a nonholonomic system with parametric excitation,
a Roller Racer with variable gyrostatic momentum. We examine in detail the problem of the
existence of regimes with unbounded growth of energy (nonconservative Fermi acceleration).
We find a criterion for the existence of trajectories for which one of the velocity components
increases withound bound and has asymptotics $t^{1/3}$. In addition, we show that the problem
under consideration reduces to analysis of a three-dimensional Poincaré map. This map exhibits
both regular attractors (a fixed point, a limit cycle and a torus) and strange attractors.
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Bizyaev I. A., Mamaev I. S.
Permanent Rotations in Nonholonomic Mechanics. Omnirotational Ellipsoid
2022, vol. 27, no. 6, pp. 587-612
Abstract
This paper is concerned with the study of permanent rotations of a rigid body
rolling without slipping on a horizontal plane (i. e., the velocity of the point of contact of the
ellipsoid with the plane is zero). By permanent rotations we will mean motions of a rigid body on
a horizontal plane such that the angular velocity of the body remains constant and the point of
contact does not change its position. A more detailed analysis is made of permanent rotations of
an omnirotational ellipsoid whose characteristic feature is the possibility of permanent rotations
about any point of its surface.
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Bizyaev I. A., Mamaev I. S.
Qualitative Analysis of the Dynamics of a Balanced Circular Foil and a Vortex
2021, vol. 26, no. 6, pp. 658-674
Abstract
This paper investigates the dynamics of a point vortex and a balanced circular foil
in an ideal fluid. An explicit reduction to quadratures is performed. A bifurcation diagram is
constructed and a classification of the types of integral manifolds is carried out. The stability
of critical solutions is studied in which the foil and the vortex move in a circle or in a straight
line.
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Artemova E. M., Karavaev Y. L., Mamaev I. S., Vetchanin E. V.
Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass
2020, vol. 25, no. 6, pp. 689-706
Abstract
The motion of a spherical robot with periodically changing moments of inertia,
internal rotors and a displaced center of mass is considered. It is shown that, under some
restrictions on the displacement of the center of mass, the system of interest features chaotic
dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium
point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic
rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the
case of fixed rotors and periodically changing moments of inertia.
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Mamaev I. S., Vetchanin E. V.
Dynamics of Rubber Chaplygin Sphere under Periodic Control
2020, vol. 25, no. 2, pp. 215-236
Abstract
This paper examines the motion of a balanced spherical robot under the action of
periodically changing moments of inertia and gyrostatic momentum. The system of equations
of motion is constructed using the model of the rolling of a rubber body (without slipping and
twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the
equations of motion admit three invariant submanifolds corresponding to plane-parallel motion
of the sphere with rotation about the minor, middle and major axes of inertia. The abovementioned
motions are quasi-periodic, and for the numerical estimate of their stability charts
of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and
amplitude of the moments of inertia. It is shown that rotations about the minor and major axes
of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In
this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle
unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments
of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It
is shown that the nonconservativeness of the system manifests itself in the occurrence of limit
cycles, attracting tori and strange attractors in phase space. Numerical calculations show that
strange attractors may arise through a cascade of period-doubling bifurcations or after a finite
number of torus-doubling bifurcations.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem
2019, vol. 24, no. 5, pp. 560-582
Abstract
This paper addresses the problem of the rolling of a spherical shell with a frame
rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire
system is at the geometric center of the shell.
For the rubber rolling model and the classical rolling model it is shown that, if the angular
velocities of rotation of the frame and the rotors are constant, then there exists a noninertial
coordinate system (attached to the frame) in which the equations of motion do not depend
explicitly on time. The resulting equations of motion preserve an analog of the angular
momentum vector and are similar in form to the equations for the Chaplygin ball. Thus, the
problem reduces to investigating a two-dimensional Poincaré map.
The case of the rubber rolling model is analyzed in detail. Numerical investigation of its
Poincaré map shows the existence of chaotic trajectories, including those associated with a
strange attractor. In addition, an analysis is made of the case of motion from rest, in which the
problem reduces to investigating the vector field on the sphere $S^2$.
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Borisov A. V., Kilin A. A., Mamaev I. S.
A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness
2019, vol. 24, no. 3, pp. 329-352
Abstract
This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.
We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion
in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.
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Borisov A. V., Mamaev I. S., Vetchanin E. V.
Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation
2018, vol. 23, nos. 7-8, pp. 850-874
Abstract
This paper addresses the problem of the self-propulsion of a smooth body in a fluid by periodic oscillations of the internal rotor and circulation. In the case of zero dissipation and constant circulation, it is shown using methods of KAM theory that the kinetic energy of the system is a bounded function of time. In the case of constant nonzero circulation, the trajectories of the center of mass of the system lie in a bounded region of the plane. The method of expansion by a small parameter is used to approximately construct a solution corresponding to directed motion of a circular foil in the presence of dissipation and variable circulation.
Analysis of this approximate solution has shown that a speed-up is possible in the system in the presence of variable circulation and in the absence of resistance to translational motion. It is shown that, in the case of an elliptic foil, directed motion is also possible. To explore the dynamics of the system in the general case, bifurcation diagrams, a chart of dynamical regimes
and a chart of the largest Lyapunov exponent are plotted. It is shown that the transition to chaos occurs through a cascade of period-doubling bifurcations.
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Mamaev I. S., Vetchanin E. V.
The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor
2018, vol. 23, nos. 7-8, pp. 875-886
Abstract
This paper addresses the problem of controlled motion of the Zhukovskii foil in a viscous fluid due to a periodically oscillating rotor. Equations of motion including the added mass effect, viscous friction and lift force due to circulation are derived. It is shown that only limit cycles corresponding to the direct motion or motion near a circle appear in the system at the standard parameter values. The chart of dynamical regimes, the chart of the largest Lyapunov exponent and a one-parameter bifurcation diagram are calculated. It is shown that strange attractors appear in the system due to a cascade of period-doubling bifurcations.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control
2018, vol. 23, nos. 7-8, pp. 983-994
Abstract
In this paper we consider the problem of the motion of the Roller Racer.We assume that the angle $\varphi (t)$ between the platforms is a prescribed function of time. We prove that in this case the acceleration of the Roller Racer is unbounded.
In this case, as the Roller Racer accelerates, the increase in the constraint reaction forces is also unbounded. Physically this means that, from a certain instant onward, the conditions of the rolling motion of the wheels without slipping are violated. Thus, we consider a model in which, in addition to the nonholonomic constraints, viscous friction force acts at the points of contact of the wheels. For this case we prove that there is no constant acceleration and all trajectories of the reduced system asymptotically tend to a periodic solution.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane
2018, vol. 23, no. 6, pp. 665-684
Abstract
This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.
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Borisov A. V., Mamaev I. S., Bizyaev I. A.
Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability
2018, vol. 23, no. 5, pp. 613-636
Abstract
This paper is concerned with the problem of three vortices on a sphere $S^2$ and the
Lobachevsky plane $L^2$. After reduction, the problem reduces in both cases to investigating a
Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to
study it using the methods of Poisson geometry. This paper presents a topological classification
of types of symplectic leaves depending on the values of Casimir functions and system
parameters.
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Borisov A. V., Mamaev I. S., Vetchanin E. V.
Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation
2018, vol. 23, no. 4, pp. 480-502
Abstract
This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.
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Borisov A. V., Kilin A. A., Mamaev I. S.
A Nonholonomic Model of the Paul Trap
2018, vol. 23, no. 3, pp. 339-354
Abstract
In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the
hyperbolic paraboloid is made. A three-dimensional Poincar´e map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.
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Vetchanin E. V., Mamaev I. S.
Dynamics of Two Point Vortices in an External Compressible Shear Flow
2017, vol. 22, no. 8, pp. 893–908
Abstract
This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincar´e map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration
2017, vol. 22, no. 8, pp. 955–975
Abstract
This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.
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Borisov A. V., Mamaev I. S.
An Inhomogeneous Chaplygin Sleigh
2017, vol. 22, no. 4, pp. 435-447
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.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Hess–Appelrot Case and Quantization of the Rotation Number
2017, vol. 22, no. 2, pp. 180-196
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.
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Vetchanin E. V., Kilin A. A., Mamaev I. S.
Control of the Motion of a Helical Body in a Fluid Using Rotors
2016, vol. 21, nos. 7-8, pp. 874-884
Abstract
This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on $SE(3)$ are obtained.
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Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
2016, vol. 21, no. 6, pp. 759-774
Abstract
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.
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Borisov A. V., Mamaev I. S., Bizyaev I. A.
The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
2016, vol. 21, no. 5, pp. 556-580
Abstract
In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.
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Borisov A. V., Mamaev I. S., Bizyaev I. A.
Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period
2016, vol. 21, no. 4, pp. 455-476
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.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Vortex Sources in a Deformation Flow
2016, vol. 21, no. 3, pp. 367-376
Abstract
This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse.
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Borisov A. V., Mamaev I. S.
Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics
2016, vol. 21, no. 2, pp. 232-248
Abstract
The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré – Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
Dynamics of the Chaplygin Sleigh on a Cylinder
2016, vol. 21, no. 1, pp. 136-146
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 (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. |
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Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A.
Qualitative Analysis of the Dynamics of a Wheeled Vehicle
2015, vol. 20, no. 6, pp. 739-751
Abstract
This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined.
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Borisov A. V., Kilin A. A., Mamaev I. S.
On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles
2015, vol. 20, no. 6, pp. 752-766
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 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.
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Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.
Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane
2015, vol. 20, no. 5, pp. 518-541
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 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.
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Borisov A. V., Mamaev I. S.
Symmetries and Reduction in Nonholonomic Mechanics
2015, vol. 20, no. 5, pp. 553-604
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.
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Borisov A. V., Mamaev I. S., Bizyaev I. A.
The Jacobi Integral in Nonholonomic Mechanics
2015, vol. 20, no. 3, pp. 383-400
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.
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Borisov A. V., Kilin A. A., Mamaev I. S.
Dynamics and Control of an Omniwheel Vehicle
2015, vol. 20, no. 2, pp. 153-172
Abstract
A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained.
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Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane
2014, vol. 19, no. 6, pp. 607-634
Abstract
In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough 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.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Three Vortex Sources
2014, vol. 19, no. 6, pp. 694-701
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.
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Bizyaev I. A., Borisov A. V., Mamaev I. S.
Superintegrable Generalizations of the Kepler and Hook Problems
2014, vol. 19, no. 3, pp. 415-434
Abstract
In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.
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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
2014, vol. 19, no. 2, pp. 198-213
Abstract
In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to 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 a nonholonomic hinge. 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. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.
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Mamaev I. S., Ivanova T. B.
The Dynamics of a Rigid Body with a Sharp Edge in Contact with an Inclined Surface in the Presence of Dry Friction
2014, vol. 19, no. 1, pp. 116-139
Abstract
In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.
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Borisov A. V., Kilin A. A., Mamaev I. S.
The Problem of Drift and Recurrence for the Rolling Chaplygin Ball
2013, vol. 18, no. 6, pp. 832-859
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 the 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.
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Borisov A. V., Mamaev I. S.
The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity
2013, vol. 18, no. 5, pp. 490-496
Abstract
We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.
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Borisov A. V., Mamaev I. S.
Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere
2013, vol. 18, no. 4, pp. 356-371
Abstract
A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set 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.
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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
2013, vol. 18, no. 3, pp. 277-328
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 quasi-Hamiltonian behavior.
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Borisov A. V., Kilin A. A., Mamaev I. S.
The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
2013, vol. 18, nos. 1-2, pp. 33-62
Abstract
We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
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Vetchanin E. V., Mamaev I. S., Tenenev V. A.
The Self-propulsion of a Body with Moving Internal Masses in a Viscous Fluid
2013, vol. 18, nos. 1-2, pp. 100-117
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 for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.
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Borisov A. V., Kilin A. A., Mamaev I. S.
How to Control the Chaplygin Ball Using Rotors. II
2013, vol. 18, nos. 1-2, pp. 144-158
Abstract
In our earlier paper [3] 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.
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Bolsinov A. V., Borisov A. V., Mamaev I. S.
Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals
2012, vol. 17, no. 6, pp. 571-579
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 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.
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Bolsinov A. V., Borisov A. V., Mamaev I. S.
The Bifurcation Analysis and the Conley Index in Mechanics
2012, vol. 17, no. 5, pp. 457-478
Abstract
The paper is devoted to the bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We discuss the phenomenon of appearance (disappearance) of equilibrium points under the change of the Morse index of a critical point of a Hamiltonian. As an application of these techniques we find new relative equilibria in the problem of the motion of three point vortices of equal intensity in a circular domain.
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Borisov A. V., Kilin A. A., Mamaev I. S.
How to Control Chaplygin’s Sphere Using Rotors
2012, vol. 17, nos. 3-4, pp. 258-272
Abstract
In the paper we study the control of a balanced dynamically non-symmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.
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Borisov A. V., Kilin A. A., Mamaev I. S.
Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support
2012, vol. 17, no. 2, pp. 170-190
Abstract
We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu.N.Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support – the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.
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Borisov A. V., Mamaev I. S.
Two Non-holonomic Integrable Problems Tracing Back to Chaplygin
2012, vol. 17, no. 2, pp. 191-198
Abstract
The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.
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Borisov A. V., Kilin A. A., Mamaev I. S.
On the Model of Non-holonomic Billiard
2011, vol. 16, no. 6, pp. 653-662
Abstract
In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.
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Bolsinov A. V., Borisov A. V., Mamaev I. S.
Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds
2011, vol. 16, no. 5, pp. 443-464
Abstract
The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics.
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Borisov A. V., Kilin A. A., Mamaev I. S.
Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere
2011, vol. 16, no. 5, pp. 465-483
Abstract
We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.
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Borisov A. V., Kilin A. A., Mamaev I. S.
Hamiltonicity and integrability of the Suslov problem
2011, vol. 16, nos. 1-2, pp. 104-116
Abstract
The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed.
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Borisov A. V., Mamaev I. S., Ramodanov S. M.
Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface
2010, vol. 15, nos. 4-5, pp. 440-461
Abstract
The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.
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Borisov A. V., Kilin A. A., Mamaev I. S.
Superintegrable system on a sphere with the integral of higher degree
2009, vol. 14, no. 6, pp. 615-620
Abstract
We consider the motion of a material point on the surface of a sphere in the field of $2n + 1$ identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional $N$-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed.
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Borisov A. V., Mamaev I. S.
Isomorphisms of geodesic flows on quadrics
2009, vol. 14, nos. 4-5, pp. 455-465
Abstract
We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.
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Borisov A. V., Mamaev I. S., Kilin A. A.
The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids
2009, vol. 14, no. 2, pp. 179-217
Abstract
The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).
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Borisov A. V., Kilin A. A., Mamaev I. S.
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
2009, vol. 14, no. 1, pp. 18-41
Abstract
Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particleinteraction potential homogeneous of degree $\alpha = –2$ are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems.
Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle interaction potential homogeneous of degree $\alpha = –2$ are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane. |
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Borisov A. V., Fedorov Y. N., Mamaev I. S.
Chaplygin ball over a fixed sphere: an explicit integration
2008, vol. 13, no. 6, pp. 557-571
Abstract
We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel–Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time. |
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Borisov A. V., Mamaev I. S.
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
2008, vol. 13, no. 5, pp. 443-490
Abstract
This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.
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Borisov A. V., Kilin A. A., Mamaev I. S.
Stability of Steady Rotations in the Nonholonomic Routh Problem
2008, vol. 13, no. 4, pp. 239-249
Abstract
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.
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Borisov A. V., Kilin A. A., Mamaev I. S.
Absolute and Relative Choreographies in Rigid Body Dynamics
2008, vol. 13, no. 3, pp. 204-220
Abstract
For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev–Chaplygin case, and the Steklov solution. The "genealogy" of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
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Borisov A. V., Kilin A. A., Mamaev I. S.
Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point
2008, vol. 13, no. 3, pp. 221-233
Abstract
In this paper, we consider the transition to chaos in the phase portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotization are indicated: (1) the growth of the homoclinic structure and (2) the development of cascades of period doubling bifurcations. On the zero level of the area integral, an adiabatic behavior of the system (as the energy tends to zero) is noted. Meander tori induced by the break of the torsion property of the mapping are found.
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Borisov A. V., Kozlov V. V., Mamaev I. S.
Asymptotic stability and associated problems of dynamics of falling rigid body
2007, vol. 12, no. 5, pp. 531-565
Abstract
We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
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Borisov A. V., Mamaev I. S.
Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting
2007, vol. 12, no. 2, pp. 153-159
Abstract
Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.
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Borisov A. V., Mamaev I. S.
Interaction between Kirchhoff vortices and point vortices in an ideal fluid
2007, vol. 12, no. 1, pp. 68-80
Abstract
We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found by the variable separation method . A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.
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Borisov A. V., Mamaev I. S.
On the problem of motion of vortex sources on a plane
2006, vol. 11, no. 4, pp. 455-466
Abstract
Equations of motion of vortex sources (examined earlier by Fridman and Polubarinova) are studied, and the problems of their being Hamiltonian and integrable are discussed. A system of two vortex sources and three sources-sinks was examined. Their behavior was found to be regular. Qualitative analysis of this system was made, and the class of Liouville integrable systems is considered. Particular solutions analogous to the homothetic configurations in celestial mechanics are given.
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Borisov A. V., Mamaev I. S.
Superintegrable systems on a sphere
2005, vol. 10, no. 3, pp. 257-266
Abstract
We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, (a compact space of constant curvature). In particular, these generalizations include addition of a spherical analogue of the magnetic monopole (the Poincaré–Appell system) and addition of a more complicated field which is a generalization of the MICZ-system. The mentioned systems are integrable superintegrable, and there exists the vector integral which is analogous to the Laplace–Runge–Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space $L^3$.
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Borisov A. V., Mamaev I. S., Kilin A. A.
Two-body problem on a sphere. Reduction, stochasticity, periodic orbits
2004, vol. 9, no. 3, pp. 265-279
Abstract
We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.
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Borisov A. V., Mamaev I. S., Kilin A. A.
Absolute and relative choreographies in the problem of point vortices moving on a plane
2004, vol. 9, no. 2, pp. 101-111
Abstract
We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the $n$-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.
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Borisov A. V., Mamaev I. S., Ramodanov S. M.
Motion of a circular cylinder and $n$ point vortices in a perfect fluid
2003, vol. 8, no. 4, pp. 449-462
Abstract
The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.
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Mamaev I. S.
New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface
2003, vol. 8, no. 3, pp. 331-335
Abstract
Some new cases when the invariant measure and an additional first integral exist in the problem of a rigid body rolling on a sphere and on an ellipsoid are discussed in the paper. These cases generalize the results obtained previously by V.A.Yaroshchuk and A.V.Borisov, I.S.Mamaev, A.A.Kilin.
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Borisov A. V., Mamaev I. S.
An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid
2003, vol. 8, no. 2, pp. 163-166
Abstract
In this paper we present the nonlinear Poisson structure and two first integrals in the problem on plane motion of circular cylinder and $N$ point vortices in the ideal fluid. A priori this problem is not Hamiltonian. The particular case $N = 1$, i.e. the problem on interaction of cylinder and vortex, is integrable.
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Borisov A. V., Mamaev I. S., Kilin A. A.
Dynamics of rolling disk
2003, vol. 8, no. 2, pp. 201-212
Abstract
In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed.
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Borisov A. V., Mamaev I. S.
The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics
2002, vol. 7, no. 2, pp. 177-200
Abstract
In this paper we study the cases of existence of an invariant measure, additional first integrals, and a Poisson structure in the problem of rigid body's rolling without sliding on a plane and a sphere. The problem of rigid body's motion on a plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of the dynamically symmetrical body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of a body of revolution and the motion of round disk with sharp edge on a sphere. In both cases the systems are Euler–Jacobi integrable and have additional integrals and invariant measure. It can be shown that by an appropriate change of time (determined by reducing multiplier), the reduced system is a Hamiltonian one. Here we consider some particular cases when the integrals and the invariant measure can be presented as finite algebraic expressions. We also consider a generalized problem of rolling of a dynamically nonsymmetric Chaplygin ball. The results of investigations are summarized in tables to illustrate the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure.
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Borisov A. V., Mamaev I. S., Kilin A. A.
The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics
2002, vol. 7, no. 2, pp. 201-219
Abstract
The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.
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Borisov A. V., Mamaev I. S.
Generalization of the Goryachev–Chaplygin Case
2002, vol. 7, no. 1, pp. 21-30
Abstract
In this paper we present a generalization of the Goryachev–Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.
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Borisov A. V., Mamaev I. S.
On the History of the Development of the Nonholonomic Dynamics
2002, vol. 7, no. 1, pp. 43-47
Abstract
The main directions in the development of the nonholonomic dynamics are briefly considered in this paper. The first direction is connected with the general formalizm of the equations of dynamics that differs from the Lagrangian and Hamiltonian methods of the equations of motion's construction. The second direction, substantially more important for dynamics, includes investigations concerning the analysis of the specific nonholonomic problems. We also point out rather promising direction in development of nonholonomic systems that is connected with intensive use of the modern computer-aided methods.
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Borisov A. V., Mamaev I. S.
Euler–Poisson Equations and Integrable Cases
2001, vol. 6, no. 3, pp. 253-276
Abstract
In this paper we propose a new approach to the study of integrable cases based on intensive computer methods' application. We make a new investigation of Kovalevskaya and Goryachev–Chaplygin cases of Euler–Poisson equations and obtain many new results in rigid body dynamics in absolute space. Also we present the visualization of some special particular solutions.
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Borisov A. V., Mamaev I. S., Kholmskaya A. G.
Kovalevskaya top and generalizations of integrable systems
2001, vol. 6, no. 1, pp. 1-16
Abstract
Generalizations of the Kovalevskaya, Chaplygin, Goryachev–Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and action-angle variables is described. Another integration method for the Kovalevskaya top on the bundle is found. This method uses a coordinate transformation that reduces the Kovalevskaya system to the Neumann system. The Kolosov analogy is considered. A generalization of a recent Gaffet system to the bundle of Poisson brackets is obtained at the end of the paper.
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Borisov A. V., Mamaev I. S.
Some comments to the paper by A.M.Perelomov "A note on geodesics on ellipsoid" RCD 2000 5(1) 89-91
2000, vol. 5, no. 1, pp. 92-94
Abstract
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Chernoivan V. A., Mamaev I. S.
The restricted two-body problem and the kepler problem in the constant curvature spaces
1999, vol. 4, no. 2, pp. 112-124
Abstract
In this work we carry out the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$, and construct the analogues of Delaunau variables. We consider the problem of motion of a mass point in the field of moving Newtonian center on $S^2$ and $L^2$. The perihelion deviation is derived by the method of perturbation theory under the small curvature, and a numerical investigation is made, using anology of this problem with rigid body dynamics.
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Bolsinov A. V., Borisov A. V., Mamaev I. S.
Lie algebras in vortex dynamics and celestial mechanics — IV
1999, vol. 4, no. 1, pp. 23-50
Abstract
1.Classificaton of the algebra of $n$ vortices on a plane
2.Solvable problems of vortex dynamics 3.Algebraization and reduction in a three-body problem The work [13] introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie–Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works [14,15]. In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem. |
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Borisov A. V., Mamaev I. S.
Non-linear Poisson brackets and isomorphisms in dynamics
1997, vol. 2, nos. 3-4, pp. 72-89
Abstract
In the paper the equations of motion of a rigid body in the Hamiltonian form on the subalgebra of algebra $e(4)$ are written. With the help of the algebraic methods a number of new isomorphisms in dynamics is established. We consider the lowering of the order as the process of decreasing rank of the Poisson structure with the algebraic point of view and indicate the possibility of arising the nonlinear Poisson brackets at this reduction as well.
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Borisov A. V., Mamaev I. S.
Adiabatic Chaos in Rigid Body Dynamics
1997, vol. 2, no. 2, pp. 65-78
Abstract
We consider arising of adiabatic chaos in rigid body dynamics. The comparison of analytical diffusion coefficient describing probable effects in the chaos zone with numerical experiment is carried out. The analysis of split of asymptotic surfaces is carried out the curves of indfenition in the Poincare-Zhukovsky problem.
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