Speedup of the Chaplygin Top by Means of Rotors
Doklady Physics, 2019, vol. 64, no. 3, pp. 120-124
Abstract
pdf (486.65 Kb)
In this paper we consider the control of the motion of a dynamically asymmetric unbalanced ball
(Chaplygin top) by means of two perpendicular rotors. We propose a mechanism for control by periodically
changing the gyrostatic momentum of the system, which leads to an unbounded speedup. We then formulate
a general hypothesis of the mechanism for speeding up spherical bodies on a plane by periodically changing
the system parameters.
A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness
Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 329-352
Abstract
pdf (1.48 Mb)
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 Poincar´e sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.
Borisov A. V., Kilin A. A., Mamaev I. S., A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness, Regular and Chaotic Dynamics, 2019, vol. 24, no. 3, pp. 329-352
The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass
Nonlinear Dynamics, 2019, vol. 95, no. 1, pp. 699-714
Abstract
pdf (2.97 Mb)
For a Chaplygin sleigh moving in the presence of weak friction, we present and investigate two mechanisms of arising acceleration due to oscillations of an internal mass. In certain parameter regions, the mechanism induced by small oscillations determines acceleration which is on average one-directional. The role of friction is that the velocity reached in the process of the acceleration is stabilized at a certain level. The second mechanism is due to the effect of the developing oscillatory parametric instability in the motion of the sleigh. It occurs when the internal oscillating particle is comparable in mass with the main platform and the oscillations are of a sufficiently large amplitude. In the nonholonomic model the magnitude of the parametric oscillations and the level of mean energy achieved by the system turn out to be bounded if the line of the oscillations of the moving particle is displaced from the center of mass; the observed sustained motion is in many cases associated with a chaotic attractor. Then, the motion of the sleigh appears to be similar to the process of two-dimensional random walk on the plane.
Bizyaev I. A., Borisov A. V., Kuznetsov S. P., The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass, Nonlinear Dynamics, 2019, vol. 95, no. 1, pp. 699-714
Stabilization of the motion of a spherical robot using feedbacks
Applied Mathematical Modelling, 2019, vol. 69, pp. 583-592
Abstract
pdf (1005.01 Kb)
The paper is concerned with the problem of stabilizing a spherical robot of combined type during its motion. The focus is on the application of feedback for stabilization of the robot which is an example of an underactuated system. The robot is set in motion by an inter- nal wheeled platform with a rotor placed inside the sphere. The results of experimental investigations for a prototype of the spherical robot are presented.
Borisov A. V., Kilin A. A., Karavaev Y. L., Klekovkin A. V., Stabilization of the motion of a spherical robot using feedbacks, Applied Mathematical Modelling, 2019, vol. 69, pp. 583-592
Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard
International Journal of Bifurcation and Chaos, 2019, vol. 29, no. 3, 1930008, 19 pp.
Abstract
pdf (2.79 Mb)
In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two
wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on
which the sportsman executes necessary body movements. From the theoretical point of view,
this system is a direct generalization of the classical nonholonomic system of the Chaplygin
sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical
system. An important feature of the problem is that the common level set of first integrals is
a compact two-dimensional surface of genus 5. We specify conditions under which the reaction
forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming
is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more
complex mechanical models incorporating sliding, elasticity, etc.
Keywords:
Nonholonomic mechanics, snakeboard, qualitative analysis, bifurcations, regularization (blowing up singularities), system on a torus, nonholonomic jamming, bifurcation analysis
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard, International Journal of Bifurcation and Chaos, 2019, vol. 29, no. 3, 1930008, 19 pp.
Reduction and relative equilibria for the two-body problem on spaces of constant curvature
Celestial Mechanics and Dynamical Astronomy, 2018, vol. 130, no. 6, pp. 1-36
Abstract
pdf (1.38 Mb)
We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each $q>0$ we show there are two relative equilibria where the masses are separated by a distance $q$. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except $\pi/2$. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At $\pi/2$, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function.
Borisov A. V., García-Naranjo L., Mamaev I. S., Montaldi J., Reduction and relative equilibria for the two-body problem on spaces of constant curvature, Celestial Mechanics and Dynamical Astronomy, 2018, vol. 130, no. 6, pp. 1-36
Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body
Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 803-820
Abstract
pdf (1014 Kb)
This paper addresses the problem of a rigid body moving on a plane (a platform) whose motion is initiated by oscillations of a point mass relative to the body in the presence of the viscous friction force applied at a fixed point of the platform and having in one direction a small (or even zero) value and a large value in the transverse direction. This problem is analogous to that of a Chaplygin sleigh when the nonholonomic constraint prohibiting motions of the fixed point on the platform across the direction prescribed on it is replaced by viscous friction. We present numerical results which confirm correspondence between the phenomenology of complex dynamics of the model with a nonholonomic constraint and a system with viscous friction —
phase portraits of attractors, bifurcation diagram, and Lyapunov exponents. In particular, we show the possibility of the platform’s motion being accelerated by oscillations of the internal mass, although, in contrast to the nonholonomic model, the effect of acceleration tends to saturation. We also show the possibility of chaotic dynamics related to strange attractors of equations for generalized velocities, which is accompanied by a two-dimensional random walk of the platform in a laboratory reference system. The results obtained may be of interest to applications in the context of the problem of developing robotic mechanisms for motion in a fluid under the action of the motions of internal masses.
Borisov A. V., Kuznetsov S. P., Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 803-820
Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation
Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 850-874
Abstract
pdf (4.03 Mb)
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.
Keywords:
self-propulsion in a fluid, smooth body, viscous fluid, periodic oscillation of circulation, control of a rotor
Citation:
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, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 850-874
Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control
Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 983-994
Abstract
pdf (1.99 Mb)
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.
Keywords:
Roller Racer, speed-up, nonholonomic mechanics, Rayleigh dissipation function, viscous friction, integrability by quadratures, control, constraint reaction force
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 983-994
An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane
Regular and Chaotic Dynamics, 2018, vol. 23, no. 6, pp. 665-684
Abstract
pdf (662.06 Kb)
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.
Keywords:
nonholonomic mechanics, regularization, blowing-up, invariant measure, ergodic theorems, normal hyperbolic submanifold, Poincaré map, first integrals
Citation:
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, Regular and Chaotic Dynamics, 2018, vol. 23, no. 6, pp. 665-684
Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability
Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, pp. 613-636
Abstract
pdf (3.34 Mb)
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.
Keywords:
Poisson geometry, point vortices, reduction, quadratic Poisson bracket, spaces of constant curvature, symplectic leaf, collinear configurations
Citation:
Borisov A. V., Mamaev I. S., Bizyaev I. A., Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability, Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, pp. 613-636
Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation
Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 480-502
Abstract
pdf (2.88 Mb)
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.
Keywords:
self-propulsion in a fluid, motion with speed-up, parametric excitation, viscous dissipation, circulation, period-doubling bifurcation, Neimark – Sacker bifurcation, Poincaré map, chart of dynamical regimes, chart of Lyapunov exponents, strange att
Citation:
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, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 480-502
A Nonholonomic Model of the Paul Trap
Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, pp. 339-354
Abstract
pdf (6.87 Mb)
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.
Keywords:
Paul trap, stability, nonholonomic system, three-dimensional map, gyroscopic stabilization, noninertial coordinate system, Poincaré map, nonholonomic constraint, rolling without slipping, region of linear stability
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., A Nonholonomic Model of the Paul Trap, Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, pp. 339-354
Theoretical and experimental investigations of the rolling of a ball on a rotating plane (turntable)
European Journal of Physics, 2018, vol. 39, no. 6, 065001, 13 pp.
Abstract
pdf (510.14 Kb)
In this work we investigate the motion of a homogeneous ball rolling without slipping on uniformly rotating horizontal and inclined planes under the action of a constant external force supplemented with the moment of rolling friction, which depends linearly on the angular velocity of the ball. We systematise well-known results and supplement them with the stability analysis of partial solutions of the system. We also perform an experimental investigation whose results support the adequacy of the rolling friction model used. Comparison of numerical and experimental results has shown a good qualitative agreement.
Keywords:
rolling, rotating surface, tilted turntable, non-holonomic constraint, rolling ball, rolling friction, qualitative analysis
Citation:
Borisov A. V., Ivanova T. B., Karavaev Y. L., Mamaev I. S., Theoretical and experimental investigations of the rolling of a ball on a rotating plane (turntable), European Journal of Physics, 2018, vol. 39, no. 6, 065001, 13 pp.
Dynamics of the Chaplygin ball on a rotating plane
Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, pp. 423-433
Abstract
pdf (872.8 Kb)
This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincaré map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system’s dynamics is chaotic.
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin ball on a rotating plane, Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, pp. 423-433
Rigid Body Dynamics
Ser. De Gruyter Studies in Mathematical Physics, Vol. 52, Berlin/Boston: Higher Education Press and de Gruyter GmbH, 2018, 526 pp.
Abstract
This book provides an up-to-date overview of results in rigid body dynamics, including material concerned with the analysis of nonintegrability and chaotic behavior in various related problems. The wealth of topics covered makes it a practical reference for researchers and graduate students in mathematics, physics and mechanics.
Citation:
Borisov A. V., Mamaev I. S., Rigid Body Dynamics, Ser. De Gruyter Studies in Mathematical Physics, Vol. 52, Berlin/Boston: Higher Education Press and de Gruyter GmbH, 2018, 526 pp.
Hamilton’s Principle and the Rolling Motion of a Symmetric Ball
Doklady Physics, 2017, vol. 62, no. 6, pp. 314-317
Abstract
pdf (206.42 Kb)
In this paper, we show that the trajectories of a dynamical system with nonholonomic constraints
can satisfy Hamilton’s principle. As the simplest illustration, we consider the problem of a homogeneous ball
rolling without slipping on a plane. However, Hamilton’s principle is formulated either for a reduced system
or for a system defined in an extended phase space. It is shown that the dynamics of a nonholonomic homogeneous
ball can be embedded in a higher-dimensional Hamiltonian phase flow. We give two examples of
such an embedding: embedding in the phase flow of a free system and embedding in the phase flow of the
corresponding vakonomic system.
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Hamilton’s Principle and the Rolling Motion of a Symmetric Ball, Doklady Physics, 2017, vol. 62, no. 6, pp. 314-317
Retrograde motion of a rolling disk
Physics-Uspekhi, 2017, vol. 60, no. 9, pp. 931-934
Abstract
pdf (364.25 Kb)
This paper presents results of theoretical and experi-
mental research explaining the retrograde final-stage rolling of
a disk under certain relations between its mass and geometric
parameters. Modifying the no-slip model of a rolling disk by
including viscous rolling friction provides a qualitative explana-
tion for the disk's retrograde motion. At the same time, the
simple experiments described in the paper completely reject
the aerodynamical drag torque as a key reason for the retro-
grade motion of a disk considered, thus disproving some recent
hypotheses.
Keywords:
retrograde turn, rolling disk, nonholonomic model, rolling friction
Citation:
Borisov A. V., Kilin A. A., Karavaev Y. L., Retrograde motion of a rolling disk, Physics-Uspekhi, 2017, vol. 60, no. 9, pp. 931-934
Control of the Motion of a Triaxial Ellipsoid in a Fluid Using Rotors
Mathematical Notes, 2017, vol. 102, no. 4, pp. 455-464
Abstract
pdf (616.14 Kb)
The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.
Keywords:
ideal fluid, motion of a rigid body, Kirchhoff equations, control by rotors, gate
Citation:
Borisov A. V., Vetchanin E. V., Kilin A. A., Control of the Motion of a Triaxial Ellipsoid in a Fluid Using Rotors, Mathematical Notes, 2017, vol. 102, no. 4, pp. 455-464
Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics
Russian Mathematical Surveys, 2017, vol. 72, no. 5, pp. 783-840
Abstract
pdf (1.09 Mb)
This is a survey of the main forms of equations of dynamical
systems with non-integrable constraints, divided into two large groups.
The first group contains systems arising in vakonomic mechanics and optimal
control theory, with the equations of motion obtained from the variational
principle, and the second contains systems in classical non-holonomic
mechanics, when the constraints are ideal and therefore the D’Alembert–Lagrange principle holds.
Borisov A. V., Mamaev I. S., Bizyaev I. A., Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics, Russian Mathematical Surveys, 2017, vol. 72, no. 5, pp. 783-840
The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration
Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 955–975
Abstract
pdf (1.91 Mb)
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.
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 955–975
An Inhomogeneous Chaplygin Sleigh
Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 435-447
Abstract
pdf (676.55 Kb)
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.
The Hess–Appelrot Case and Quantization of the Rotation Number
Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
Abstract
pdf (991.1 Kb)
This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords:
invariant submanifold, rotation number, Cantor ladder, limit cycles
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot Case and Quantization of the Rotation Number, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
An inhomogeneous Chaplygin sleigh
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 625–639
Abstract
pdf (518.79 Kb)
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.
The Hess–Appelrot case and quantization of the rotation number
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 433-452
Abstract
pdf (556.32 Kb)
This paper is concerned with the Hess case in the Euler –Poisson equations and with its
generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces
to investigating the vector field on a torus and that the graph showing the dependence of the
rotation number on parameters has horizontal segments (limit cycles) only for integer values of
the rotation number. In addition, an example of a Hamiltonian system is given which possesses
an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation
number on parameters is a Cantor ladder.
Keywords:
invariant submanifold, rotation number, Cantor ladder, limit cycles
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot case and quantization of the rotation number, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 433-452
Regular and chaotic dynamics in the rubber model of a Chaplygin top
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 277-297
Abstract
pdf (2.3 Mb)
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Keywords:
Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact
Citation:
Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and chaotic dynamics in the rubber model of a Chaplygin top, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 277-297
Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 1, pp. 129-146
Abstract
pdf (2.82 Mb)
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords:
sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
Citation:
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S., Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 1, pp. 129-146
Chaplygin sleigh with periodically oscillating internal mass
EPL, 2017, vol. 119, no. 6, 60008, 7 pp.
Abstract
pdf (854.88 Kb)
We consider the movement of Chaplygin sleigh on a plane that is a solid body with
imposed nonholonomic constraint, which excludes the possibility of motions transversal to the constraint element (“knife-edge”), and complement the model with an attached mass, periodically oscillating relatively to the main platform of the sleigh. Numerical simulations indicate the occurrence of either unrestricted acceleration of the sleigh, or motions with bounded velocities and
momenta, depending on parameters. We note the presence of phenomena characteristic to nonholonomic systems with complex dynamics; in particular, attractors occur responsible for chaotic
motions. In addition, quasiperiodic regimes take place similar to those observed in conservative nonlinear dynamics.
Citation:
Bizyaev I. A., Borisov A. V., Kuznetsov S. P., Chaplygin sleigh with periodically oscillating internal mass, EPL, 2017, vol. 119, no. 6, 60008, 7 pp.
Generalizations of the Kovalevskaya Case and Quaternions
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 33-44
Abstract
pdf (190.82 Kb)
This paper provides a detailed description of various reduction schemes in rigid
body dynamics. The analysis of one of such nontrivial reductions makes it possible to put the
cases already found in order and to obtain new generalizations of the Kovalevskaya case to $e(3)$.
Note that the indicated reduction allows one to obtain in a natural way some singular additive
terms that were proposed earlier by D.N. Goryachev.
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Generalizations of the Kovalevskaya Case and Quaternions, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 33-44
The Hess–Appelrot system and its nonholonomic analogs
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 294, pp. 268-292
Abstract
pdf (1.23 Mb)
This paper is concerned with the nonholonomic Suslov problem
and its generalization proposed by Chaplygin. The issue of the existence
of an invariant measure with singular density (having singularities at some
points of phase space) is discussed.
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot system and its nonholonomic analogs, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 294, pp. 268-292
Bifurcation Analysis of the Motion of a Cylinder and a Point Vortex in an Ideal Fluid
Mathematical Notes, 2016, vol. 99, no. 6, pp. 834-839
Abstract
pdf (545.15 Kb)
We consider an integrable Hamiltonian system describing the motion of a circular cylinder and a vortex filament in an ideal fluid. We construct bifurcation diagrams and bifurcation complexes for the case in which the integral manifold is compact and for various topological structures of the symplectic leaf. The types of motions corresponding to the bifurcation curves and their stability are discussed.
Borisov A. V., Ryabov P. E., Sokolov S. V., Bifurcation Analysis of the Motion of a Cylinder and a Point Vortex in an Ideal Fluid, Mathematical Notes, 2016, vol. 99, no. 6, pp. 834-839
Describing the Motion of a Body with an Elliptical Cross Section in a Viscous Uncompressible Fluid by Model Equations Reconstructed from Data Processing
Technical Physics Letters, 2016, vol. 42, no. 9, pp. 886-890
Abstract
pdf (508.12 Kb)
From analysis of time series obtained on the numerical solution of a plane problem on the motion
of a body with an elliptic cross section under the action of gravity force in an incompressible viscous fluid, a
system of ordinary differential equations approximately describing the dynamics of the body is reconstructed.
To this end, coefficients responsible for the added mass, the force caused by the circulation of the velocity
field, and the resisting force are found by the least square adjustment. The agreement between the finitedimensional
description and the simulation on the basis of the Navier–Stokes equations is illustrated by
images of attractors in regular and chaotic modes. The coefficients found make it possible to estimate the
actual contribution of different effects to the dynamics of the body.
Citation:
Borisov A. V., Kuznetsov S. P., Mamaev I. S., Tenenev V. A., Describing the Motion of a Body with an Elliptical Cross Section in a Viscous Uncompressible Fluid by Model Equations Reconstructed from Data Processing, Technical Physics Letters, 2016, vol. 42, no. 9, pp. 886-890
Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 792-803
Abstract
pdf (1.65 Mb)
For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.
Borisov A. V., Kuznetsov S. P., Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 792-803
Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901
Abstract
pdf (2.21 Mb)
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of period-doubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Keywords:
Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact
Citation:
Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901
Spiral Chaos in the Nonholonomic Model of a Chaplygin Top
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954
Abstract
pdf (2.21 Mb)
This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn
about the influence of “strangeness” of the attractor on the motion pattern of the top.
Borisov A. V., Kazakov A. O., Sataev I. R., Spiral Chaos in the Nonholonomic Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954
Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
Abstract
pdf (3.48 Mb)
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3, 6, 14)$, the other is defined by two generatrices and growth vector $(2, 3, 5, 8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords:
sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
Citation:
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S., Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580
Abstract
pdf (1.45 Mb)
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.
Keywords:
celestial mechanics, space of constant curvature, reduction, rigid body dynamics, Poincaré section
Citation:
Borisov A. V., Mamaev I. S., Bizyaev I. A., The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580
Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period
Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
Abstract
pdf (1.87 Mb)
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., Mamaev I. S., Bizyaev I. A., Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
The Dynamics of Vortex Sources in a Deformation Flow
Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376
Abstract
pdf (1.37 Mb)
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.
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Vortex Sources in a Deformation Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376
Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics
Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 232-248
Abstract
pdf (941.28 Kb)
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).
Keywords:
adiabatic invariants, Liouville system, transition through resonance, adiabatic chaos
Citation:
Borisov A. V., Mamaev I. S., Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 2, pp. 232-248
Dynamics of the Chaplygin Sleigh on a Cylinder
Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146
Abstract
pdf (268.54 Kb)
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.
Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin Sleigh on a Cylinder, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146
Dynamics of the Chaplygin sleigh on a cylinder
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 675–687
Abstract
pdf (331.42 Kb)
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.
Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin sleigh on a cylinder, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 675–687
Historical and critical review of the development of nonholonomic mechanics: the classical period
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 3, pp. 385-411
Abstract
pdf (1.9 Mb)
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., Mamaev I. S., Bizyaev I. A., Historical and critical review of the development of nonholonomic mechanics: the classical period, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 3, pp. 385-411
Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 263-287
Abstract
pdf (3.54 Mb)
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 263-287
On the Hadamard–Hamel problem and the dynamics of wheeled vehicles
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 145-163
Abstract
pdf (445.79 Kb)
In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.
Keywords:
nonholonomic constraint, wheeled vehicle, reduction, equations of motion
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., On the Hadamard–Hamel problem and the dynamics of wheeled vehicles, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 145-163
The Hojman Construction and Hamiltonization of Nonholonomic Systems
Symmetry, Integrability and Geometry: Methods and Applications, 2016, vol. 12, 012, 19 pp.
Abstract
pdf (571.09 Kb)
In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hojman Construction and Hamiltonization of Nonholonomic Systems, Symmetry, Integrability and Geometry: Methods and Applications, 2016, vol. 12, 012, 19 pp.
Rigid Body Dynamics in Non-Euclidean Spaces
Russian Journal of Mathematical Physics, 2016, vol. 23, no. 4, pp. 431-454
Abstract
pdf (704.17 Kb)
In this paper, we focus on the study of two-dimensional plate dynamics on the Lobachevskii plane $L^2$. First of all, we consider the free motion of such a plate, which is a pseudospherical analog of dynamics of the Euler top, and also present an analog of the Euler–Poisson equations enabling us to study the motion of the body in potential force fields having rotational symmetry. We present a series of integrable cases, having analogs in Euclidean space, for different fields. Moreover, in the paper, a partial qualitative analysis of the dynamics of free motion of a plate under arbitrary initial conditions is made and a number of computer illustrations are presented which show a substantial difference of the motion from the case of Euclidean space. The study undertaken in the present paper leads to interesting physical onsequences, which enable us to detect the influence of curvature on the body dynamics.
Citation:
Borisov A. V., Mamaev I. S., Rigid Body Dynamics in Non-Euclidean Spaces, Russian Journal of Mathematical Physics, 2016, vol. 23, no. 4, pp. 431-454
Dynamics of a body sliding on a rough plane and supported at three points
Theoretical and Applied Mechanics, 2016, vol. 43, no. 2, pp. 169-190
Abstract
pdf (1.81 Mb)
This paper is concerned with the problem of a rigid body (tripod)
moving with three points in contact with a horizontal plane under the action of
dry friction forces. It is shown that the regime of asymptotic motion (final dy-
namics) of the tripod can be pure rotation, pure sliding, or sliding and rotation
can cease simultaneously, which is determined by the position of the tripod’s
supports relative to the radius of inertia. In addition, the dependence of the
trajectory of the center of mass on the system parameters is investigated. A
comparison is made with the well-known theoretical and experimental studies
on the motion of bodies with a flat base
Borisov A. V., Mamaev I. S., Erdakova N. N., Dynamics of a body sliding on a rough plane and supported at three points, Theoretical and Applied Mechanics, 2016, vol. 43, no. 2, pp. 169-190
A New Integrable System of Nonholonomic Mechanics
Doklady Physics, 2015, vol. 60, no. 6, pp. 269-271
Abstract
pdf (255.48 Kb)
A new integrable problem of nonholonomic mechanics is considered and its mechanical realization is proposed. This problem is a generalization of the well-known problem of А. P. Veselov and L. E. Veselova concerning the rolling motion of the Chaplygin ball in a straight line. Particular cases are found in which integration can be reduced to explicit quadratures.
Citation:
Borisov A. V., Mamaev I. S., A New Integrable System of Nonholonomic Mechanics, Doklady Physics, 2015, vol. 60, no. 6, pp. 269-271
Notes on new friction models and nonholonomic mechanics
Physics-Uspekhi, 2015, vol. 58, no. 12, pp. 1220-1222
Abstract
pdf (262.98 Kb)
This is a reply to the comment by V.F. Zhuravlev (see Usp. Fiz. Nauk 185 1337 (2015) [Phys. Usp. 58 (12) (2015)]) on the inadequacy of the nonholonomic model when applied to the rolling of rigid bodies. The model of nonholonomic mechanics is discussed. Using recent results as examples, it is shown that the validity and potential of the nonholonomic model are not inferior to those of other dynamics and friction models.
Keywords:
nonholonomic model, dry friction, rattleback, rolling motion of a rigid body
Citation:
Borisov A. V., Mamaev I. S., Notes on new friction models and nonholonomic mechanics, Physics-Uspekhi, 2015, vol. 58, no. 12, pp. 1220-1222
Figures of equilibrium of an inhomogeneous self-gravitating fluid
Celestial Mechanics and Dynamical Astronomy, 2015, vol. 122, no. 1, pp. 1-26
Abstract
pdf (651.34 Kb)
This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a 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 corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
Keywords:
Self-gravitating fluid, Confocal stratification, Homothetic stratification, Chaplygin problem, Axisymmetric equilibrium figures, Space of constant curvature
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Celestial Mechanics and Dynamical Astronomy, 2015, vol. 122, no. 1, pp. 1-26
On the loss of contact of the Euler disk
Nonlinear Dynamics, 2015, vol. 79, no. 4, pp. 2287-2294
Abstract
pdf (829.12 Kb)
This paper is an experimental investigation of a round uniform disk rolling on a horizontal surface. Two methods for experimentally determining the loss of contact of the rolling disk from the horizontal surface before its stop are proposed. Results of experiments for disks having different masses and manufactured from different materials are presented. Causes of “microlosses of contact” detected in the processes of motion are discussed.
Keywords:
Euler’s disk, Loss of contact, Experiment
Citation:
Borisov A. V., Mamaev I. S., Karavaev Y. L., On the loss of contact of the Euler disk, Nonlinear Dynamics, 2015, vol. 79, no. 4, pp. 2287-2294
Qualitative Analysis of the Dynamics of a Wheeled Vehicle
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751
Abstract
pdf (445.93 Kb)
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.
Keywords:
nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A., Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751
On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 752-766
Abstract
pdf (265.93 Kb)
In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.
Keywords:
nonholonomic constraint, wheeled vehicle, reduction, equations of motion
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 752-766
Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
Abstract
pdf (640.12 Kb)
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
Symmetries and Reduction in Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 553-604
Abstract
pdf (539.38 Kb)
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.
Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 518-541
Abstract
pdf (516.92 Kb)
In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords:
dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation:
Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V., Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 518-541
The Jacobi Integral in Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 383-400
Abstract
pdf (990.04 Kb)
In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
Keywords:
nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem
Citation:
Borisov A. V., Mamaev I. S., Bizyaev I. A., The Jacobi Integral in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 383-400
Dynamics and Control of an Omniwheel Vehicle
Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 153-172
Abstract
pdf (1.11 Mb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., Dynamics and Control of an Omniwheel Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 153-172
Symmetries and Reduction in Nonholonomic Mechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 763–823
Abstract
pdf (909.32 Kb)
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.
Borisov A. V., Mamaev I. S., Symmetries and Reduction in Nonholonomic Mechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 763–823
Topology and Bifurcations in Nonholonomic Mechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 735–762
Abstract
pdf (561.73 Kb)
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.
Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and Bifurcations in Nonholonomic Mechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 735–762
On the dynamics of a body with an axisymmetric base sliding on a rough plane
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 547-577
Abstract
pdf (2.38 Mb)
In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords:
dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation:
Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 547-577
The Jacobi Integral in NonholonomicMechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 377-396
Abstract
pdf (1.9 Mb)
In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
Keywords:
nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem
Citation:
Borisov A. V., Mamaev I. S., Bizyaev I. A., The Jacobi Integral in NonholonomicMechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 377-396
Geometrisation of Chaplygin's reducing multiplier theorem
Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318
Abstract
pdf (156.41 Kb)
We develop the reducing multiplier theory for a special class of nonholonomic
dynamical systems and show that the non-linear Poisson brackets naturally
obtained in the framework of this approach are all isomorphic to the Lie–Poisson $e$(3)-bracket. As two model examples, we consider the Chaplygin ball
problem on the plane and the Veselova system. In particular, we obtain an
integrable gyrostatic generalisation of the Veselova system.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Geometrisation of Chaplygin's reducing multiplier theorem, Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318
Topology and Bifurcations in Nonholonomic Mechanics
International Journal of Bifurcation and Chaos, 2015, vol. 25, no. 10, 1530028, 21 pp.
Abstract
pdf (645.53 Kb)
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 two. 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.
Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and Bifurcations in Nonholonomic Mechanics, International Journal of Bifurcation and Chaos, 2015, vol. 25, no. 10, 1530028, 21 pp.
Hamiltonization of Elementary Nonholonomic Systems
Russian Journal of Mathematical Physics, 2015, vol. 22, no. 4, pp. 444-453
Abstract
pdf (115.49 Kb)
In this paper, we develop the method of Chaplygin’s reducing multiplier; using this method, we obtain a conformally Hamiltonian representation for three nonholonomic systems, namely, for the nonholonomic oscillator, for the Heisenberg system, and for the Chaplygin sleigh. Furthermore, in the case of oscillator and nonholonomic Chaplygin sleigh, we show that the problem reduces to the study of motion of a mass point (in a potential field) on a plane and, in the case of Heisenberg system, on the sphere. Moreover, we consider an example of a nonholonomic system (suggested by Blackall) to which one cannot apply the method of reducing multiplier.
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Hamiltonization of Elementary Nonholonomic Systems, Russian Journal of Mathematical Physics, 2015, vol. 22, no. 4, pp. 444-453
Nonlinear dynamics of the rattleback: a nonholonomic model
Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
Abstract
pdf (750.09 Kb)
For a solid body of convex form moving on a rough horizontal plane that is known as a rattleback, numerical simulations are used to discuss and illustrate dynamical phenomena that are characteristic of the motion due to a nonholonomic nature of the mechanical system; the relevant feature is the nonconservation of the phase volume in the course of the dynamics. In such a system, a local compression of the phase volume can produce behavior features similar to those exhibited by dissipative systems, such as stable equilibrium points corresponding to stationary rotations; limit cycles (rotations with oscillations); and strange attractors. A chart of dynamical regimes is plotted in a plane whose axes are the total mechanical energy and the relative angle between the geometric and dynamic principal axes of the body. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. A number of strange attractors are considered, for which phase portraits, Lyapunov exponents, and Fourier spectra are presented.
Citation:
Borisov A. V., Kazakov A. O., Kuznetsov S. P., Nonlinear dynamics of the rattleback: a nonholonomic model , Physics-Uspekhi, 2014, vol. 184, no. 5, pp. 453-460
On the Nonlinear Poisson Bracket Arising in Nonholonomic Mechanics
Mathematical Notes, 2014, vol. 95, no. 3, pp. 308-315
Abstract
pdf (509.63 Kb)
Nonholonomic systems describing the rolling of a rigid body on a plane and their relationship with various Poisson structures are considered. The notion of generalized conformally Hamiltonian representation of dynamical systems is introduced. In contrast to linear Poisson structures defined by Lie algebras and used in rigid-body dynamics, the Poisson structures of nonholonomic systems turn out to be nonlinear. They are also degenerate and the Casimir functions for them can be expressed in terms of complicated transcendental functions or not appear at all.
Keywords:
Poisson bracket, nonholonomic system, Poisson structure, dynamical system, con- formally Hamiltonian representation, Casimir function, Routh sphere, rolling of a Chaplygin ball
Citation:
Borisov A. V., Mamaev I. S., Tsiganov A. V., On the Nonlinear Poisson Bracket Arising in Nonholonomic Mechanics , Mathematical Notes, 2014, vol. 95, no. 3, pp. 308-315
Non-holonomic dynamics and Poisson geometry
Russian Mathematical Surveys, 2014, vol. 69, no. 3, pp. 481-538
Abstract
pdf (917.58 Kb)
This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie-Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them.
Keywords:
non-holonomic systems, Poisson bracket, Chaplygin ball, Suslov system, Veselova system
Citation:
Borisov A. V., Mamaev I. S., Tsiganov A. V., Non-holonomic dynamics and Poisson geometry , Russian Mathematical Surveys, 2014, vol. 69, no. 3, pp. 481-538
The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
Abstract
pdf (1.29 Mb)
In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.
Keywords:
rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
The Dynamics of Three Vortex Sources
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701
Abstract
pdf (244.27 Kb)
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.
The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 607-634
Abstract
pdf (965.59 Kb)
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.
Keywords:
dry friction, linear pressure distribution, planar motion, Coulomb law
Citation:
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, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 607-634
Superintegrable Generalizations of the Kepler and Hook Problems
Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434
Abstract
pdf (300.95 Kb)
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.
Keywords:
superintegrable systems, Kepler and Hook problems, isomorphism, central projection, reduction, highest degree polynomial superintegrals
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Superintegrable Generalizations of the Kepler and Hook Problems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434
The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside
Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 198-213
Abstract
pdf (241.48 Kb)
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.
Keywords:
nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 198-213
Paul Painlevé and His Contribution to Science
Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 1-19
Abstract
pdf (1.89 Mb)
The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the $N$-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.
On the dynamics of a body with an axisymmetric base sliding on a rough plane
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 483-495
Abstract
pdf (667.35 Kb)
In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system’s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.
Keywords:
dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation:
Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 483-495
Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Abstract
pdf (1.3 Mb)
We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.
Keywords:
Chaplygin’s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Invariant Measure and Hamiltonization of Nonholonomic Systems
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 355-359
Abstract
pdf (283.75 Kb)
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.
Borisov A. V., Mamaev I. S., Invariant Measure and Hamiltonization of Nonholonomic Systems, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 355-359
The dynamics of three vortex sources
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 319-327
Abstract
pdf (413.32 Kb)
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., The dynamics of three vortex sources, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 319-327
Figures of equilibrium of an inhomogeneous self-gravitating fluid
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 73-100
Abstract
pdf (492.78 Kb)
This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figure or its layers uniformly rotate about an axis fixed in space. As is well known, when there is no rotation, only a ball can be a figure of equilibrium.
It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with inherent constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification.
We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
Keywords:
self-gravitating fluid, confocal stratification, homothetic stratification, space of constant curvature
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 73-100
The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid
Fluid Dynamics Research, 2014, vol. 46, no. 3, 031415, 16 pp.
Abstract
pdf (746.69 Kb)
We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible and viscous fluid. Using the numerical simulation of the Navier–Stokes equations, we confirm the existence of leapfrogging of three equal vortex rings and suggest the possibility of detecting it experimentally. We also confirm the existence of leapfrogging of two vortex rings with opposite-signed vorticities in a viscous fluid.
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Tenenev V. A., The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid , Fluid Dynamics Research, 2014, vol. 46, no. 3, 031415, 16 pp.
The Problem of Drift and Recurrence for the Rolling Chaplygin Ball
Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 832-859
Abstract
pdf (702.93 Kb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 832-859
The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity
Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 490-496
Abstract
pdf (405.38 Kb)
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.
Borisov A. V., Mamaev I. S., The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 490-496
Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere
Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 356-371
Abstract
pdf (488.37 Kb)
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.
Borisov A. V., Mamaev I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 356-371
The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere
Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 277-328
Abstract
pdf (2.69 Mb)
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.
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, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 277-328
How to Control the Chaplygin Ball Using Rotors. II
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 144-158
Abstract
pdf (1.73 Mb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., How to Control the Chaplygin Ball Using Rotors. II, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 144-158
The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 33-62
Abstract
pdf (857.35 Kb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 33-62
The problem of drift and recurrence for the rolling Chaplygin ball
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 721-754
Abstract
pdf (875.6 Kb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., The problem of drift and recurrence for the rolling Chaplygin ball, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 721-754
Geometrization of the Chaplygin reducing-multiplier theorem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 627-640
Abstract
pdf (373.67 Kb)
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.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Geometrization of the Chaplygin reducing-multiplier theorem, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 627-640
The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 547-566
Abstract
pdf (441.83 Kb)
In this paper we investigate two systems consisting of a spherical shell rolling on a plane without slipping and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is fixed at the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of the nonholonomic hinge. The equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics.
Keywords:
nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 547-566
On the loss of contact of the Euler disk
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 499-506
Abstract
pdf (362.06 Kb)
The paper presents experimental investigation of a homogeneous circular disk rolling on a horizontal plane. In this paper two methods of experimental determination of the loss of contact between the rolling disk and the horizontal surface before the abrupt halt are proposed. Experimental results for disks of different masses and different materials are presented. The reasons for “micro losses” of contact with surface revealed during the rolling are discussed.
Keywords:
Euler disk, loss of contact, experiment
Citation:
Borisov A. V., Mamaev I. S., Karavaev Y. L., On the loss of contact of the Euler disk, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 499-506
The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 141-202
Abstract
pdf (7.91 Mb)
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.
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, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 141-202
How to control the Chaplygin ball using rotors. II
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 59-76
Abstract
pdf (2.71 Mb)
In our earlier paper [2] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.
Borisov A. V., Kilin A. A., Mamaev I. S., How to control the Chaplygin ball using rotors. II, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 59-76
Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics
Journal of Applied Nonlinear Dynamics, 2013, vol. 2, no. 2, pp. 161-173
Abstract
pdf (269.77 Kb)
In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.
Keywords:
Rolling without slipping, Nonholonomic constraint, Chaplygin system, Conformally Hamiltonian system
Citation:
Borisov A. V., Mamaev I. S., Treschev D. V., Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics, Journal of Applied Nonlinear Dynamics, 2013, vol. 2, no. 2, pp. 161-173
Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 571-579
Abstract
pdf (402.12 Kb)
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.
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, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 571-579
Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
Abstract
pdf (4.23 Mb)
We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.
Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
The Bifurcation Analysis and the Conley Index in Mechanics
Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 457-478
Abstract
pdf (614.32 Kb)
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.
Bolsinov A. V., Borisov A. V., Mamaev I. S., The Bifurcation Analysis and the Conley Index in Mechanics, Regular and Chaotic Dynamics, 2012, vol. 17, no. 5, pp. 457-478
How to Control Chaplygin’s Sphere Using Rotors
Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 258-272
Abstract
pdf (242.89 Kb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., How to Control Chaplygin’s Sphere Using Rotors, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 258-272
Two Non-holonomic Integrable Problems Tracing Back to Chaplygin
Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 191-198
Abstract
pdf (150.32 Kb)
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.
Borisov A. V., Mamaev I. S., Two Non-holonomic Integrable Problems Tracing Back to Chaplygin, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 191-198
Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support
Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 170-190
Abstract
pdf (484.82 Kb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 170-190
Topological analysis of one integrable system related to the rolling of a ball over a sphere
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 957-975
Abstract
pdf (796.84 Kb)
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.
Borisov A. V., Mamaev I. S., Topological analysis of one integrable system related to the rolling of a ball over a sphere, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 5, pp. 957-975
Rolling of a rigid body without slipping and spinning: kinematics and dynamics
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 783-797
Abstract
pdf (347.06 Kb)
In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.
Keywords:
rolling without slipping, nonholonomic constraint, Chaplygin system, conformally Hamiltonian system
Citation:
Borisov A. V., Mamaev I. S., Treschev D. V., Rolling of a rigid body without slipping and spinning: kinematics and dynamics, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 783-797
Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 605-616
Abstract
pdf (328.96 Kb)
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.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 605-616
How to control the Chaplygin sphere using rotors
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 2, pp. 289-307
Abstract
pdf (400.44 Kb)
In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic contrability is shown and the control inputs providing motion of the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.
Borisov A. V., Kilin A. A., Mamaev I. S., How to control the Chaplygin sphere using rotors, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 2, pp. 289-307
Viatcheslav Vladimirovich Meleshko (07.10.1951–14.11.2011)
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 179-182
Abstract
pdf (1.73 Mb)
Citation:
Grinchenko V. T., Krasnopolskaya T. S., Borisov A. V., van Heijst G. J., Viatcheslav Vladimirovich Meleshko (07.10.1951–14.11.2011), Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 179-182
The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 113-147
Abstract
pdf (1.03 Mb)
We consider the problem of the 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 in which the mutual 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.
Borisov A. V., Kilin A. A., Mamaev I. S., The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 113-147
The dynamics of the Chaplygin ball with a fluid-filled cavity
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 103-111
Abstract
pdf (305.43 Kb)
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.
Borisov A. V., Mamaev I. S., The dynamics of the Chaplygin ball with a fluid-filled cavity, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 103-111
Hassan Aref (1950–2011)
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Abstract
pdf (777.79 Kb)
Citation:
Borisov A. V., Meleshko V. V., Stremler M. A., van Heijst G. J., Hassan Aref (1950–2011), Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
On the Model of Non-holonomic Billiard
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 653-662
Abstract
pdf (199.9 Kb)
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.
Keywords:
billiard, impact, point map, nonintegrability, periodic solution, nonholonomic constraint, integral of motion
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., On the Model of Non-holonomic Billiard, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 653-662
Hassan Aref (1950–2011)
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Abstract
pdf (777.79 Kb)
Citation:
Borisov A. V., Meleshko V. V., Stremler M. A., van Heijst G. J., Hassan Aref (1950–2011), Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 465-483
Abstract
pdf (643.15 Kb)
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.
Keywords:
nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 465-483
Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 443-464
Abstract
pdf (425.83 Kb)
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.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 443-464
Hamiltonicity and integrability of the Suslov problem
Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 104-116
Abstract
pdf (239.81 Kb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., Hamiltonicity and integrability of the Suslov problem, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 104-116
An omni-wheel vehicle on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 4, pp. 785-801
Abstract
pdf (726.7 Kb)
We consider a nonholonomic model of the dynamics of an omni-wheel vehicle on a plane and a sphere. An elementary derivation of equations is presented, the dynamics of a free system is investigated, a relation to control problems is shown.
Borisov A. V., Kilin A. A., Mamaev I. S., An omni-wheel vehicle on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 4, pp. 785-801
The bifurcation analysis and the Conley index in mechanics
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 649-681
Abstract
pdf (782.35 Kb)
The paper is concerned with the use of bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We give the proof of the theorem on the appearance (disappearance) of fixed points in the case of the Morse index change. New relative equilibria in the problem of the motion of point vortices of equal intensity in a circle are found.
Bolsinov A. V., Borisov A. V., Mamaev I. S., The bifurcation analysis and the Conley index in mechanics, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 649-681
Two non-holonomic integrable systems of coupled rigid bodies
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 559-568
Abstract
pdf (404.52 Kb)
The paper considers two new integrable systems due to Chaplygin, which describe the rolling of a spherical shell on a plane, with a ball or Lagrange’s gyroscope inside. All necessary first integrals and an invariant measure are found. The reduction to quadratures is given.
Borisov A. V., Mamaev I. S., Two non-holonomic integrable systems of coupled rigid bodies, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 559-568
On V.A. Steklov’s legacy in classical mechanics
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, pp. 389-403
Abstract
pdf (368.21 Kb)
This paper has been written for a collection of V.A. Steklov’s selected works, which is being prepared for publication and is entitled «Works on Mechanics 1902–1909: Translations from French». The collection is based on V.A. Steklov’s papers on mechanics published in French journals from 1902 to 1909.
Citation:
Borisov A. V., Gazizullina L., Mamaev I. S., On V.A. Steklov’s legacy in classical mechanics, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, pp. 389-403
Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, pp. 313-338
Abstract
pdf (1.78 Mb)
We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a non-zero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of integrability.
Borisov A. V., Kilin A. A., Mamaev I. S., Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 2, pp. 313-338
Stability of new relative equilibria of the system of three point vortices in a circular domain
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 119-138
Abstract
pdf (1.2 Mb)
This paper presents a topological approach to the search and stability analysis of relative equilibria of three point vortices of equal intensities. It is shown that the equations of motion can be reduced by one degree of freedom. We have found two new stationary configurations (isosceles and non-symmetrical collinear) and studied their bifurcations and stability.
Keywords:
point vortex, reduction, bifurcational diagram, relative equilibriums, stability, periodic solutions
Citation:
Borisov A. V., Mamaev I. S., Vaskina A. V., Stability of new relative equilibria of the system of three point vortices in a circular domain, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 119-138
Topology and stability of integrable systems
Russian Mathematical Surveys, 2010, vol. 65, no. 2, pp. 259–318
Abstract
pdf (1.12 Mb)
In this paper a general topological approach is proposed for the study of stability of periodic solutions of integrable dynamical systems with two degrees of freedom. The methods developed are illustrated by examples of several integrable problems related to the classical Euler–Poisson equations, the motion of a rigid body in a fluid, and the dynamics of gaseous expanding ellipsoids. These topological methods also enable one to find non-degenerate periodic solutions of integrable systems, which is especially topical in those cases where no general solution (for example, by separation of variables) is known.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and stability of integrable systems, Russian Mathematical Surveys, 2010, vol. 65, no. 2, pp. 259–318
Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface
Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 440-461
Abstract
pdf (298.75 Kb)
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.
Keywords:
hydrodynamics on a sphere, coupled body-vortex system, mass vortex, equations of motion, integrability
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 440-461
Rolling of a homogeneous ball over a dynamically asymmetric sphere
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 869-889
Abstract
pdf (486.45 Kb)
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.
Keywords:
nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Rolling of a homogeneous ball over a dynamically asymmetric sphere, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 869-889
Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 829-854
Abstract
pdf (398.78 Kb)
Hamiltonisation problem for non-holonomic systems, both integrable and non-integrable, is considered. This question is important for qualitative analysis of such systems and allows one to determine possible dynamical effects. The first part is devoted to the representation of integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighbourhood of a periodic solution is proved for an arbitrary measure preserving system (including integrable). General consructions are always illustrated by examples from non-holonomic mechanics.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Hamiltonisation of non-holonomic systems in the neighborhood of invariant manifolds, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 829-854
Stability of a liquid self-gravitating elliptic cylinder with intrinsic rotation
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 807-822
Abstract
pdf (902.16 Kb)
We consider figures of equilibrium and stability of a liquid self-gravitating elliptic cylinder. The flow within the cylinder is assumed to be dew to an elliptic perturbation. A bifurcation diagram is plotted and conditions for steady solutions to exist are indicated.
Keywords:
self-gravitating liquid, elliptic cylinder, bifurcation point, stability, Riemann equations
Citation:
Borisov A. V., Mamaev I. S., Ivanova T. B., Stability of a liquid self-gravitating elliptic cylinder with intrinsic rotation, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 807-822
Dynamic advection
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 521-530
Abstract
pdf (10.3 Mb)
A new concept of dynamic advection is introduced. The model of dynamic advection deals with the motion of massive particles in a 2D flow of an ideal incompressible liquid. Unlike the standard advection problem, which is widely treated in the modern literature, our equations of motion account not only for particles’ kinematics, governed by the Euler equations, but also for their dynamics (which is obviously neglected if the mass of particles is taken to be zero). A few simple model problems are considered.
Keywords:
advection, mixing, point vortex, coarse-grained impurities, bifurcation complex
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamic advection, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 521-530
Valery Vasilievich Kozlov. On his 60th birthday
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 461-488
Abstract
pdf (25.39 Mb)
Citation:
Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V., Valery Vasilievich Kozlov. On his 60th birthday, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 461-488
On the model of non-holonomic billiard
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, pp. 373-385
Abstract
pdf (237.96 Kb)
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.
Keywords:
billiard, impact, point mapping, nonintegrability, periodic solution, nonholonomic constraint, integral of motion
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., On the model of non-holonomic billiard, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, pp. 373-385
Hamiltonian representation and integrability of the Suslov problem
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 1, pp. 127-142
Abstract
pdf (654.76 Kb)
We consider the problems of Hamiltonian representation and integrability of the nonholonomic Suslov system and its generalization suggested by S. A. Chaplygin. These aspects are very important for understanding the dynamics and qualitative analysis of the system. In particular, they are related to the nontrivial asymptotic behaviour (i. e. to some scattering problem). The paper presents a general approach based on the study of the hierarchy of dynamical behaviour of nonholonomic systems.
Borisov A. V., Kilin A. A., Mamaev I. S., Hamiltonian representation and integrability of the Suslov problem, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 1, pp. 127-142
Dynamics of a wheeled carriage on a plane
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, pp. 39-48
Abstract
pdf (385.82 Kb)
The paper deals with the problem of motion of a wheeled carriage on a plane in the case where one of the wheeled pairs is fixed. In addition, the case of motion of a wheeled carriage on a plane in the case of two free wheeled pairs is considered.
Keywords:
nonholonomic constraint, dynamics of the system, wheeled carriage
Citation:
Borisov A. V., Lutsenko S. G., Mamaev I. S., Dynamics of a wheeled carriage on a plane, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, pp. 39-48
Dynamics of liquid and gas ellipsoids
Izhevsk: Regular and Chaotic Dynamics, 2010, 364 pp.
Abstract
pdf (3.89 Mb)
This book is a collection of the most significant classical results on the dynamics of liquid and gaseous ellipsoids, starting with the fundamental investigations of Dirichlet and Riemann.
The papers of the collection deal primarily with the derivation of various forms of the equations of motion and the investigation of qualitative properties of the dynamics of ellipsoidal figures.
The book addresses specialists and graduate students interested in mechanics, mathematical physics and the history of science.
Citation:
Borisov A. V., Mamaev I. S., Dynamics of liquid and gas ellipsoids, Izhevsk: Regular and Chaotic Dynamics, 2010, 364 pp.
Lyapunov A.M. Works on theoretical mechanics. From the 1882-1894 handwritten heritage
Izhevsk: Regular and Chaotic Dynamics, 2010, pp. 6-20
Abstract
pdf (174.49 Kb)
Citation:
Borisov A. V., Mamaev I. S., Tsiganov A. V., Lyapunov A.M. Works on theoretical mechanics. From the 1882-1894 handwritten heritage, Izhevsk: Regular and Chaotic Dynamics, 2010, pp. 6-20
The dynamics of a Chaplygin sleigh
Journal of Applied Mathematics and Mechanics, 2009, vol. 73, no. 2, pp. 156-161
Abstract
pdf (263.77 Kb)
The problem of the motion of a Chaplygin sleigh on horizontal and inclined surfaces is considered. The possibility of representing the equations of motion in Hamiltonian form and of integration using Liouville’s theorem (with a redundant algebra of integrals) is investigated. The asymptotics for the rectilinear uniformly accelerated sliding of a sleigh along the line of steepest descent are determined in the case of an inclined plane. The zones in the plane of the initial conditions, corresponding to a different behaviour of the sleigh, are constructed using numerical calculations. The boundaries of these domains are of a complex fractal nature, which enables a conclusion to be drawn concerning the probable character from of the dynamic behaviour.
Citation:
Borisov A. V., Mamaev I. S., The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 2009, vol. 73, no. 2, pp. 156-161
Superintegrable system on a sphere with the integral of higher degree
Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 615-620
Abstract
pdf (125.27 Kb)
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.
Keywords:
superintegrable systems, systems with a potential, Hooke center
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Superintegrable system on a sphere with the integral of higher degree, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 615-620
Isomorphisms of geodesic flows on quadrics
Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 455-465
Abstract
pdf (376.58 Kb)
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.
The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids
Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 179-217
Abstract
pdf (885.59 Kb)
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).
Keywords:
liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids, Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 179-217
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 18-41
Abstract
pdf (472.45 Kb)
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.
Keywords:
multiparticle systems, Jacobi integral
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 18-41
New superintegrable system on a sphere
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 4, pp. 455-462
Abstract
pdf (214.58 Kb)
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 [3], 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 [13] and show that for the latter system an analogous superintegral can be constructed.
Keywords:
superintegrable systems, systems with a potential, Hooke center
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., New superintegrable system on a sphere, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 4, pp. 455-462
Coupled motion of a rigid body and point vortices on a sphere
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 319-343
Abstract
pdf (429.33 Kb)
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.
Keywords:
hydrodynamics on a sphere, coupled body-vortex system, mass vortex, equations of motion, integrability
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Coupled motion of a rigid body and point vortices on a sphere, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 319-343
Isomorphisms of geodesic flows on quadrics
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 2, pp. 145-158
Abstract
pdf (532.83 Kb)
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.
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 1, pp. 53-82
Abstract
pdf (508.81 Kb)
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 particle-interaction potential homogeneous of degree $α=-2$ 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 $α=-2$ are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.
Keywords:
multiparticle systems, Jacobi integral
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 1, pp. 53-82
The Clebsch System. Separation of Variables and Explicit Integration?
Izhevsk: Regular and Chaotic Dynamics, 2009, 288 pp.
Abstract
pdf (1.68 Mb)
This book is a collection of the main classical works concerned with the Clebsch problem and related integrable systems. These papers, written by outstanding mathematicians of XIX centry, had a substantial influence on the development of many fields of modern mathematics and physics. The study of the Clebsch case and equivalent systems is far from completion, therefore, this problem remains one of the central in the theory of integrable systems.
The book addresses researchers, undergraduate and graduate students interested in theoretical mechanics, mathematical physics and the history of science.
Citation:
Borisov A. V., Tsiganov A. V., The Clebsch System. Separation of Variables and Explicit Integration?, Izhevsk: Regular and Chaotic Dynamics, 2009, 288 pp.
Explicit integration of one problem in nonholonomic mechanics
Doklady Physics, 2008, vol. 53, no. 10, pp. 525-528
Abstract
pdf (229.05 Kb)
Citation:
Borisov A. V., Mamaev I. S., Marikhin V. G., Explicit integration of one problem in nonholonomic mechanics, Doklady Physics, 2008, vol. 53, no. 10, pp. 525-528
Chaplygin ball over a fixed sphere: an explicit integration
Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 557-571
Abstract
pdf (282.96 Kb)
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.
Borisov A. V., Fedorov Y. N., Mamaev I. S., Chaplygin ball over a fixed sphere: an explicit integration, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 557-571
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 443-490
Abstract
pdf (508.23 Kb)
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.
Keywords:
nonholonomic systems, implementation of constraints, conservation laws, hierarchy of dynamics, explicit integration
Citation:
Borisov A. V., Mamaev I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems , Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 443-490
Stability of Steady Rotations in the Nonholonomic Routh Problem
Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 239-249
Abstract
pdf (392.42 Kb)
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.
Borisov A. V., Kilin A. A., Mamaev I. S., Stability of Steady Rotations in the Nonholonomic Routh Problem, Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 239-249
Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 221-233
Abstract
pdf (491.62 Kb)
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.
Keywords:
motion of a rigid body, phase portrait, mechanism of chaotization, bifurcations
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 221-233
Absolute and Relative Choreographies in Rigid Body Dynamics
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 204-220
Abstract
pdf (447.19 Kb)
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).
Keywords:
rigid-body dynamics, periodic solutions, continuation by a parameter, bifurcation
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and Relative Choreographies in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 204-220
E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 497-513
Abstract
pdf (286.99 Kb)
Citation:
Borisov A. V., Gazizullina L., Ramodanov S. M., E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 497-513
Algebraic reduction of systems on two- and three-dimensional spheres
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 407-416
Abstract
pdf (180.6 Kb)
The paper develops further the algebraic-reduction method for $SO(4)$-symmetric systems on the three-dimensional sphere. Canonical variables for the reduced system are constructed both on two-dimensional and three-dimensional spheres. The method is illustrated by applying it to the two-body problem on a sphere (the bodies are assumed to interact with a potential that depends only on the geodesic distance between them) and the three-vortex problem on a two-dimensional sphere.
Borisov A. V., Mamaev I. S., Ramodanov S. M., Algebraic reduction of systems on two- and three-dimensional spheres, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 407-416
Hamiltonian Dynamics of Liquid and Gas Self-Gravitating Ellipsoids
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 363-406
Abstract
pdf (994.54 Kb)
The paper contains the review and original results on the dynamics of liquid and gas self-gravitating ellipsoids. Equations of motion are given in Lagrangian and Hamiltonian form, in particular, the Hamiltonian formalism on Lie algebras is presented. Problems of nonintegrability and chaotical behavior of the system are formulated and studied. We also classify all known integrable cases and give some hypotheses about nonintegrability in the general case. Results of numerical modelling are presented, which can be considered as a computer proof of nonintegrability.
Keywords:
liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Hamiltonian Dynamics of Liquid and Gas Self-Gravitating Ellipsoids, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 363-406
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, pp. 223-280
Abstract
pdf (634.39 Kb)
This paper can be regarded as a continuation of our previous work [70,71] 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.
Keywords:
nonholonomic systems, implementation of constraints, conservation laws, hierarchy of dynamics, explicit integration
Citation:
Borisov A. V., Mamaev I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, pp. 223-280
Dynamics of Two Rings of Vortices on a Sphere
in IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Springer, 2008, vol. 6, pp. 445–458
Abstract
pdf (7.9 Mb)
The motion of two vortex rings on a sphere is considered. This motion generalizes the well-known centrally symmetrical solution of the equations of point vortex dynamics on a plane derived by D.N. Goryachev, N.S. Vasiliev and H. Aref. The equations of motion in this case are shown to be Liouville integrable, and an explicit reduction to a Hamiltonian system with one degree of freedom is described. Two particular cases in which the solutions are periodical are presented. Explicit quadratures are given for these solutions. Phase portraits are described and bifurcation diagrams are shown for centrally symmetrical motion of four vortices on a sphere.
Keywords:
Vortices, Hamiltonian, motion on a sphere, phase portrait
Citation:
Borisov A. V., Mamaev I. S., Dynamics of Two Rings of Vortices on a Sphere, in IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Springer, 2008, vol. 6, pp. 445–458
A New Integrable Problem of Motion of Point Vortices on the Sphere
in IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Springer, 2008, vol. 6, pp. 39–53
Abstract
pdf (10.34 Mb)
The dynamics of an antipodal vortex on a sphere (a point vortex plus its antipode with opposite circulation) is considered. It is shown that the system of n antipodal vortices can be reduced by four dimensions (two degrees of freedom). The cases $n = 2, 3$ are explored in greater detail both analytically and numerically. We discuss Thomson, collinear and isosceles configurations of antipodal vortices and study their bifurcations.
Keywords:
Hydrodynamics, ideal fluid, vortex dynamics, point vortex, reduction, bifurcation analysis
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., A New Integrable Problem of Motion of Point Vortices on the Sphere, in IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Springer, 2008, vol. 6, pp. 39–53
Isomorphism and Hamilton representation of some nonholonomic systems
Siberian Mathematical Journal, 2007, vol. 48, no. 1, pp. 26-36
Abstract
pdf (154.1 Kb)
We consider some questions connected with the Hamiltonian form of two problems of nonholonomic mechanics, namely the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.
Borisov A. V., Mamaev I. S., Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Mathematical Journal, 2007, vol. 48, no. 1, pp. 26-36
Asymptotic stability and associated problems of dynamics of falling rigid body
Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 531-565
Abstract
pdf (1.81 Mb)
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.
Keywords:
rigid body, ideal fluid, non-holonomic mechanics
Citation:
Borisov A. V., Kozlov V. V., Mamaev I. S., Asymptotic stability and associated problems of dynamics of falling rigid body, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 531-565
Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting
Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 153-159
Abstract
pdf (189.47 Kb)
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.
Borisov A. V., Mamaev I. S., Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting, Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 153-159
Interaction between Kirchhoff vortices and point vortices in an ideal fluid
Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 68-80
Abstract
pdf (358.32 Kb)
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.
Keywords:
vortex patch, point vortex, integrability
Citation:
Borisov A. V., Mamaev I. S., Interaction between Kirchhoff vortices and point vortices in an ideal fluid, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 68-80
Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 4, pp. 411-422
Abstract
pdf (182.63 Kb)
The paper deals with the derivation of the equations of motion for two spheres in an unbounded volume of ideal and incompressible fluid in 3D Euclidean space. Reduction of order, based on the use of new variables that form a Lie algebra, is offered. A trivial case of integrability is indicated.
Keywords:
motion of two spheres, ideal fluid, reduction, integrability
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 4, pp. 411-422
Asymptotic stability and associated problems of dynamics of falling rigid body
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 3, pp. 255-296
Abstract
pdf (1.62 Mb)
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.
Keywords:
nonholonomic mechanics, rigid body, ideal fluid, resisting medium
Citation:
Borisov A. V., Kozlov V. V., Mamaev I. S., Asymptotic stability and associated problems of dynamics of falling rigid body, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 3, pp. 255-296
A New Integrable Problem of Motion of Point Vortices on the Sphere
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 211-223
Abstract
pdf (298.41 Kb)
The dynamics of an antipodal vortex on a sphere (a point vortex plus its antipode with opposite circulation) is considered. It is shown that the system of n antipodal vortices can be reduced by four dimensions (two degrees of freedom). The cases n=2,3 are explored in greater detail both analytically and numerically. We discuss Thomson, collinear and isosceles configurations of antipodal vortices and study their bifurcations.
Keywords:
hydrodynamics, ideal fluid, vortex dynamics, point vortex, reduction, bifurcation analysis
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., A New Integrable Problem of Motion of Point Vortices on the Sphere, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 211-223
On isomorphisms of some integrable systems on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 1, pp. 49-56
Abstract
pdf (166.52 Kb)
We consider
trajectory isomorphisms between various integrable
systems on an $n$-dimensional sphere $S^n$ and a Euclidean space $R^n$.
Some of the systems are classical integrable problems of Celestial Mechanics
in plane and curved spaces. All the systems under consideration have an additional
first integral quadratic in momentum and can be integrated analytically by using
the separation of variables. We show that
some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the
theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.
Borisov A. V., Mamaev I. S., On isomorphisms of some integrable systems on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 1, pp. 49-56
Dynamic Interaction of Point Vortices and a Two-Dimensional Cylinder
Journal of Mathematical Physics, 2007, vol. 48, no. 6, 065403, 9 pp.
Abstract
pdf (126.99 Kb)
In this paper we consider the system of an arbitrary two-dimensional cylinder interacting with point vortices in a perfect fluid. We present the equations of motion and discuss their integrability. Simulations show that the system of an elliptic cylinder (with nonzero eccentricity) and a single point vortex already exhibits chaotic features and the equations of motion are nonintegrable. We suggest a Hamiltonian form of the equations. The problem we study here, namely, the equations of motion, the Hamiltonian structure for the interacting system of a cylinder of arbitrary cross-section shape, with zero circulation around it, and $N$ vortices, has been addressed by Shashikanth [Regular Chaotic Dyn. 10, 1 (2005)]. We slightly generalize the work by Shashikanth by allowing for nonzero circulation around the cylinder and offer a different approach than that by Shashikanth by using classical complex variable theory.
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamic Interaction of Point Vortices and a Two-Dimensional Cylinder, Journal of Mathematical Physics, 2007, vol. 48, no. 6, 065403, 9 pp.
Relations between Integrable Systems in Plane and Curved Spaces
Celestial Mechanics and Dynamical Astronomy, 2007, vol. 99, no. 4, pp. 253–260
Abstract
pdf (151.07 Kb)
We consider trajectory isomorphisms between various integrable systems on an $n$-dimensional sphere $S^n$ and a Euclidean space $\mathbb{R}^n$. Some of the systems are classical integrable problems of Celestial Mechanics in plane and curved spaces. All the systems under consideration have an additional first integral quadratic in momentum and can be integrated analytically by using the separation of variables. We show that some integrable problems in constant curvature spaces are not essentially new from the viewpoint of the theory of integration, and they can be analyzed using known results of classical Celestial Mechanics.
Keywords:
Integrable systems, Euclidean spaces
Citation:
Borisov A. V., Mamaev I. S., Relations between Integrable Systems in Plane and Curved Spaces, Celestial Mechanics and Dynamical Astronomy, 2007, vol. 99, no. 4, pp. 253–260
Dynamics of Two Interacting Circular Cylinders in Perfect Fluid
Discrete and Continuous Dynamical Systems - Series A, 2007, vol. 19, no. 2, pp. 235-253
Abstract
pdf (350.69 Kb)
In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. We also explore some special cases of this system assuming that the cylinders move along the line through their centers and the circulation around each cylinder is zero. A similar system of two interacting spheres was originally considered in the classical works of Carl and Vilhelm Bjerknes, H. Lamb and N.E. Joukowski. By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for n point vortices.
Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamics of Two Interacting Circular Cylinders in Perfect Fluid, Discrete and Continuous Dynamical Systems - Series A, 2007, vol. 19, no. 2, pp. 235-253
New effects in dynamics of rattlebacks
Doklady Physics, 2006, vol. 408, no. 2, pp. 192-195
Abstract
pdf (214.19 Kb)
The paper considers the dynamics of a rattleback as a model of a heavy balanced ellipsoid of revolution rolling without slippage on a fixed horizontal plane. Central ellipsoid of inertia is an ellipsoid of revolution as well. In presence of the angular displacement between two ellipsoids, there occur dynamical effects somewhat similar to the reverse fenomena in earlier models. However, unlike a customary rattleback model (a truncated biaxial paraboloid) our system allows the motions which are superposition of the reverse motion (reverse of the direction of spinning) and the turn over (change of the axis of rotation). With appropriate values of energies and mass distribution, this effect (reverse + turn over) can occur more than once. Such motions as repeated reverse or repeated turn over are also possible.
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., New effects in dynamics of rattlebacks, Doklady Physics, 2006, vol. 408, no. 2, pp. 192-195
Transition to chaos in dynamics of four point vortices on a plane
Doklady Physics, 2006, vol. 51, no. 5, pp. 262-267
Abstract
pdf (249.72 Kb)
The paper considers the process of transition to chaos in the problem of four point vortices on a plane. A new method for constructive reduction of the order for a system of vortices on a plane is presented. Existence of the cascade of period doubling bifurcations in the given problem is indicated.
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Transition to chaos in dynamics of four point vortices on a plane, Doklady Physics, 2006, vol. 51, no. 5, pp. 262-267
Motion of Chaplygin ball on an inclined plane
Doklady Physics, 2006, vol. 51, no. 2, pp. 73-76
Abstract
pdf (212.42 Kb)
The rolling motion of a dynamically nonsymmetric balanced ball (Chaplygin ball) on an inclined plane is studied. For the case of a horizontal plane, Chaplygin demonstrated this problem to be integrable. For a nonzero slope, the system is integrable only if the motion starts from a state of rest (E.N. Kharlamova). It is shown that, in the general case, the system exhibits a rather simple asymptotic behavior.
Citation:
Borisov A. V., Mamaev I. S., Motion of Chaplygin ball on an inclined plane, Doklady Physics, 2006, vol. 51, no. 2, pp. 73-76
An Integrable System with a Nonintegrable Constraint
Mathematical Notes, 2006, vol. 80, no. 1, pp. 127-130
Abstract
pdf (98.56 Kb)
The paper considers a general case of rolling motion of a rigid body with sharp edge on an icy sphere in a field of gravity. Cases of integrability are indicated and probability of a body fall is analyzed.
On a Nonholonomic Dynamical Problem
Mathematical Notes, 2006, vol. 79, no. 5, pp. 734-740
Abstract
pdf (189.22 Kb)
Rolling (without slipping) of a homogeneous ball on an oblique cylinder in different potential fields and the integrability of the equations of motion are considered. We examine also if the equations can be reduced to a Hamiltonian form. We prove the theorem stated that if there is a gravity (and the cylinder is oblique), the ball moves without any vertical shift, on the average.
Keywords:
nonholonomic dynamics, rolling motion without slipping, nonholonomic constraints, quasiperiodic oscillations
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., On a Nonholonomic Dynamical Problem, Mathematical Notes, 2006, vol. 79, no. 5, pp. 734-740
On the fall of a heavy rigid body in an ideal fluid
Proceedings of the Steklov Institute of Mathematics, 2006, vol. 12, no. 1, pp. S24-S47
Abstract
pdf (1.06 Mb)
We consider a problem about the motion of a heavy rigid body in an unbounded volume of an ideal irrotational incompressible uid. This problem generalizes a classical Kirchhoff problem describing the inertial motion of a rigid body in a uid. We study dfferent special statements of the problem: the plane motion and the motion of an axially symmetric body. In the general case of motion of a rigid body, we study the stability of partial solutions and point out limiting behaviors of the motion when the time increases innitely. Using numerical computations on the plane of initial conditions, we construct domains corresponding to different types of the asymptotic behavior. We establish the fractal nature of the boundary separating these domains.
Citation:
Borisov A. V., Kozlov V. V., Mamaev I. S., On the fall of a heavy rigid body in an ideal fluid, Proceedings of the Steklov Institute of Mathematics, 2006, vol. 12, no. 1, pp. S24-S47
On the problem of motion of vortex sources on a plane
Regular and Chaotic Dynamics, 2006, vol. 11, no. 4, pp. 455-466
Abstract
pdf (377.53 Kb)
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.
Keywords:
vortex sources, integrability, Hamiltonian, point vortex
Citation:
Borisov A. V., Mamaev I. S., On the problem of motion of vortex sources on a plane , Regular and Chaotic Dynamics, 2006, vol. 11, no. 4, pp. 455-466
Rolling of a heterotgeneous ball over a sphere without sliding and spinning
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 445-452
Abstract
pdf (162.92 Kb)
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, Koiller and 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.
Keywords:
Chaplygin ball, rolling model, Hamiltonian structure
Citation:
Borisov A. V., Mamaev I. S., Rolling of a heterotgeneous ball over a sphere without sliding and spinning, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 445-452
Stability of steady rotations in the non-holonomic Routh problem
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 333-345
Abstract
pdf (398.23 Kb)
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 in the upmost, downmost and saddle point.
Borisov A. V., Kilin A. A., Mamaev I. S., Stability of steady rotations in the non-holonomic Routh problem, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 333-345
Reduction in the two-body problem on the Lobatchevsky plane
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 279-285
Abstract
pdf (148.66 Kb)
We present a reduction-of-order procedure in the problem of motion of two bodies on the Lobatchevsky plane $H^2$. The bodies interact with a potential that depends only on the distance between the bodies (this holds for an analog of the Newtonian potential). In earlier works, this reduction procedure was used to analyze the motion of two bodies on the sphere
Keywords:
Lobatchevsky plane, first integral, reduction-of-order procedure, potential of interaction
Citation:
Borisov A. V., Mamaev I. S., Reduction in the two-body problem on the Lobatchevsky plane, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 3, pp. 279-285
Interaction between Kirchhoff vortices and point vortices in an ideal fluid
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 199-213
Abstract
pdf (535.61 Kb)
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 by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.
Keywords:
Kirchhoff vortices, integrability, Hamiltonian, stability, point vortex
Citation:
Borisov A. V., Mamaev I. S., Interaction between Kirchhoff vortices and point vortices in an ideal fluid, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 199-213
Dynamics of two vortex rings on a sphere
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 181-192
Abstract
pdf (321.21 Kb)
The motion of two vortex rings on a sphere is considered. This motion generalizes the well-known centrally symmetrical solution of the equations of point vortex dynamics on a plane derived by D.N. Goryachev and H. Aref. The equations of motion in this case are shown to be Liouville integrable, and an explicit reduction to a Hamiltonian system with one degree of freedom is described. Two particular cases in which the solutions are periodical are presented. Explicit quadratures are given for these solutions. Phase portraits are described and bifurcation diagrams are shown for centrally symmetrical motion of four vortices on a sphere.
Keywords:
vortex, Hamiltonian, motion on a sphere, phase portrait
Citation:
Borisov A. V., Mamaev I. S., Dynamics of two vortex rings on a sphere, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 181-192
On the motion of a heavy rigid body in an ideal fluid with circulation
Chaos, 2006, vol. 16, no. 1, 013118, 7 pp.
Abstract
pdf (303.93 Kb)
We consider Chaplygin's equations [Izd. Akad. Nauk SSSR 3(3), 1933] describing the planar motion of a rigid body in an unbounded volume of an ideal fluid while circulation around the body is not zero. Hamiltonian structures and new integrable cases are revealed; certain remarkable partial solutions are found and their stability is examined. The nonintegrability of the system describing the motion of a body in the field of gravity is proved and the chaotic behavior of the system is illustrated.
Citation:
Borisov A. V., Mamaev I. S., On the motion of a heavy rigid body in an ideal fluid with circulation, Chaos, 2006, vol. 16, no. 1, 013118, 7 pp.
The restricted two-body problem in constant curvature spaces
Celestial Mechanics and Dynamical Astronomy, 2006, vol. 96, no. 1, pp. 1-17
Abstract
pdf (479.18 Kb)
The bifurcation analysis of the Kepler problem on $\mathbb{S}^3$ and $\mathbb{H}^3$ is performed. An analogue of the Delaunay variables is introduced and the motion of a point mass in the field of the Newtonian center moving along a geodesic on $\mathbb{S}^2$ and $\mathbb{H}^2$ (the restricted two-body problem) is investigated. When the curvature is small, the pericenter shift is computed using the perturbation theory. We also present the results of the numerical analysis based on the analogy with the motion of rigid body.
Borisov A. V., Mamaev I. S., The restricted two-body problem in constant curvature spaces, Celestial Mechanics and Dynamical Astronomy, 2006, vol. 96, no. 1, pp. 1-17
Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow August 25-30, 2006)
Dordrecht Springer, 2006, 512 pp.
Abstract
pdf (10.44 Mb)
This work brings together previously unpublished notes contributed by participants of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence held in Moscow, August 25-30, 2006. The study of vortex motion is of great interest to fluid and gas dynamics; since all real flows are vortical in nature, applications of the vortex theory are extremely diverse, many of them (e.g. aircraft dynamics, atmospheric and ocean phenomena) being especially important. The last few decades have shown that serious possibilities for progress in the research of real turbulent vortex motions are essentially related to the combined use of mathematical methods, computer simulation and laboratory experiments. These approaches have led to a series of interesting results which allow us to study these processes from new perspectives. Based on this principle, the papers collected in this proceedings volume present new results on theoretical and applied aspects and processes of formation and evolution of various flows, wave and coherent structures in gas and fluid. Much attention is given to the studies of nonlinear regular and chaotic regimes of vortex interactions, advective and convective motions. The contributors are leading scientists engaged in fundamental and applied aspects of the above mentioned fields.
Citation:
Borisov A. V., Kozlov V. V., Mamaev I. S., Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow August 25-30, 2006), Dordrecht Springer, 2006, 512 pp.
Classification of Birkhoff-Integrable generalized Toda lattices
in "Topological Methods in the Theory of Integrable Systems", Cambridge: Cambridge Scientific Publishers Ltd., 2006, pp. 69-79
Abstract
pdf (560.25 Kb)
This paper presents the most complete classification of Birkhoff-integrable generalized Toda lattices and considers new integrable lattices.
Citation:
Borisov A. V., Mamaev I. S., Classification of Birkhoff-Integrable generalized Toda lattices, in "Topological Methods in the Theory of Integrable Systems", Cambridge: Cambridge Scientific Publishers Ltd., 2006, pp. 69-79
Absolute and Relative Choreographies in the Problem of the Motion of Point Vortices in a Plane
Doklady Physics, 2005, vol. 71, no. 1, pp. 139-144
Abstract
pdf (168.54 Kb)
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and Relative Choreographies in the Problem of the Motion of Point Vortices in a Plane, Doklady Physics, 2005, vol. 71, no. 1, pp. 139-144
The Nonexistence of an Invariant Measure for an Inhomogeneous Ellipsoid Rolling on a Plane
Mathematical Notes, 2005, vol. 77, no. 6, pp. 855-857
Abstract
pdf (74.38 Kb)
This note presents new conditions for nonexistance of an invariant measure for an inhomogeneous ellipsoid with the special mass distribution rolling on an absolutely rough plane. This work supplements results on the nonexistence of the measure in the rolling of a rattleback.
Keywords:
invariant measure, rolling ellipsoid, Liouville equation, Celtic stone
Citation:
Borisov A. V., Mamaev I. S., The Nonexistence of an Invariant Measure for an Inhomogeneous Ellipsoid Rolling on a Plane, Mathematical Notes, 2005, vol. 77, no. 6, pp. 855-857
Superintegrable systems on a sphere
Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 257-266
Abstract
pdf (312.81 Kb)
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 $L^3$.
Keywords:
spaces of constant curvature, Kepler problem, integrability
Citation:
Borisov A. V., Mamaev I. S., Superintegrable systems on a sphere , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 257-266
Reduction and chaotic behavior of point vortices on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 233-246
Abstract
pdf (473.55 Kb)
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
Keywords:
reduction, point vortex, equations of motion, Poincare map
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Reduction and chaotic behavior of point vortices on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 233-246
Chaos in a restricted problem of rotation of a rigid body with a fixed point
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 191-207
Abstract
pdf (650.7 Kb)
The paper deals with a transition to chaos in the phase-plane portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotisation have been indicated: 1) growth of the homoclinic structure and 2) development of cascades of period doubling bifurcations. On the zero level of the integral of areas, an adiabatic behavior of the system (as the energy tends to zero) has been noticed. Meander tori induced by the breakdown of the torsion property of the mapping have been found.
Keywords:
motion of a rigid body, phase-plane portrait, mechanism of chaotisation, bifurcations
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Chaos in a restricted problem of rotation of a rigid body with a fixed point, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 191-207
Absolute and relative choreographies in rigid body dynamics
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, pp. 123-141
Abstract
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For the classical problem of motion of a rigid body about a fixed point with zero integral of areas, the paper presents a family of solutions which are periodic in the absolute space. Such solutions are known as choreographies. The family includes the famous Delaunay solution in the case of Kovalevskaya, some particular solutions in the Goryachev-Chaplygin case and Steklov’s solution. The «genealogy» of the solutions of the family, arising naturally 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 but with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
Keywords:
rigid body dynamics, periodic solutions, continuation by a parameter, bifurcation
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and relative choreographies in rigid body dynamics, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, pp. 123-141
Interaction of two circular cylinders in a perfect fluid
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, pp. 3-21
Abstract
pdf (401.14 Kb)
In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. Special cases of this system (the cylinders move along the line through their centers and the circulation around each cylinder is zero) are considered. A similar system of two interacting spheres was originally considered in classical works of Carl and Vilhelm Bjerknes, G. Lamb and N.E. Joukowski.
By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for n point vortices.
Borisov A. V., Mamaev I. S., Ramodanov S. M., Interaction of two circular cylinders in a perfect fluid, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 1, pp. 3-21
Rigid body dynamics. Hamiltonian methods, integrability, chaos
Moscow–Izhevsk: Institute of Computer Science, 2005, 576 pp.
Abstract
pdf (21.11 Mb)
In this book we discuss the main forms of equations of motion of rigid-body systems, such as motion of rigid bodies in potential fields, in fluid (Kirchhoff equation) and rigid bodies with fluid-filled cavities. All the systems considered in the book can be described within the framework of the Hamiltonian formalism. Almost all known integrable cases and methods of explicit integration are included. Compared to the previous volume, new sections dealing with non-integrability analysis and chaos in various problems of rigid body dynamics are added. Computer-based visualization of motion is widely used. Some results are obtained by the authors.
Citation:
Borisov A. V., Mamaev I. S., Rigid body dynamics. Hamiltonian methods, integrability, chaos, Moscow–Izhevsk: Institute of Computer Science, 2005, 576 pp.
Mathematical methods in the dynamics of vortex structures
Moscow–Izhevsk: Institute of Computer Science, 2005, 368 pp.
Abstract
pdf (7.6 Mb)
The book describes the main mathematical methods of investigation of vortex structures in an ideal incompressible fluid. All the methods of analysis of integrability and non-integrable systems are based on a systematic use of the Hamiltonian formalism and qualitative analysis. Some topics discussed in the book are: motion of point vortices on a plane and a sphere, interaction of vortex patches and some fresh issues concerned with dynamical interaction between rigid bodies and vortex structures in an ideal fluid. The appendices contain some new results, obtained by the authors in cooperation with their students and colleagues.
Citation:
Borisov A. V., Mamaev I. S., Mathematical methods in the dynamics of vortex structures, Moscow–Izhevsk: Institute of Computer Science, 2005, 368 pp.
Dynamics of a circular cylinder interacting with point vortices
Discrete and Continuous Dynamical Systems - Series B, 2005, vol. 5, no. 1, pp. 35-50
Abstract
pdf (210.01 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Dynamics of a circular cylinder interacting with point vortices, Discrete and Continuous Dynamical Systems - Series B, 2005, vol. 5, no. 1, pp. 35-50
Generalized problem of two and four Newtonian centers
Celestial Mechanics and Dynamical Astronomy, 2005, vol. 92, no. 4, pp. 371-380
Abstract
pdf (291.25 Kb)
We consider integrable spherical analogue of the Darboux potential, which appear in the problem (and its generalizations) of the planar motion of a particle in the field of two and four fixed Newtonian centers. The obtained results can be useful when constructing a theory of motion of satellites in the field of an oblate spheroid in constant curvature spaces.
Keywords:
spherical two (and four) centers problem, Newtonian potential, sphero-conical coordinates, separation of variables
Citation:
Borisov A. V., Mamaev I. S., Generalized problem of two and four Newtonian centers, Celestial Mechanics and Dynamical Astronomy, 2005, vol. 92, no. 4, pp. 371-380
We consider some aspects of Hamiltonianicity of two problems of nonholonomic mechanics, namely, the Chaplygin's ball problem and the Veselova problem. Representations for these two problems have been found in the form of generalized Chaplygin systems, integrable with the method of reducing multiplier. We also specify the algebraic form of the Poisson brackets, with which, after appropriate time substitution, the equations of motion for the stated problems can be represented. We consider generalizations of the two stated problems and offer new realizations of nonholonomic constraints. Some nonholonomic systems are shown, which have the invariant measure and a sufcient number of rst integrals; for such systems, the question of Hamiltonianicity is still open, even after the time substitution.
Citation:
Borisov A. V., Mamaev I. S., Hamiltonization of nonholonomic systems, arXiv:nlin/0509036v1, 2005, 24 pp.
Reduction and chaotic behavior of point vortices on a plane and a sphere
Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 100-109
Abstract
pdf (360.5 Kb)
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincare surface-of-section technique to perform the reduction a four-vortex system on a sphere.
Keywords:
Vortex dynamics, reduction, Poincaré map, point vortices
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Reduction and chaotic behavior of point vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 100-109
New periodic solutions for three or four identical vortices on a plane and a sphere
Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 110-120
Abstract
pdf (194.7 Kb)
In this paper we describe new classes of periodic solutions for point vortices on a plane and a sphere. They correspond to similar solutions (so-called choreographies) in celestial mechanics.
Borisov A. V., Mamaev I. S., Kilin A. A., New periodic solutions for three or four identical vortices on a plane and a sphere, Discrete and Continuous Dynamical Systems - Series B (Supplement Volume devoted to the 5th AIMS International Conference on Dynamical Systems and Differential Equations (Pomona, California, USA, June 2004)), 2005, pp. 110-120
Necessary and Sufficient Conditions for the Polynomial Integrability of Generalized Toda Chains
Doklady Physics, 2004, vol. 69, no. 1, pp. 131-135
Abstract
pdf (280.72 Kb)
We present rather complete classification of the Birkhoff integrable generalized Toda lattices and consider new cases of integrable lattices.
Citation:
Borisov A. V., Mamaev I. S., Necessary and Sufficient Conditions for the Polynomial Integrability of Generalized Toda Chains, Doklady Physics, 2004, vol. 69, no. 1, pp. 131-135
Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid
Mathematical Notes, 2004, vol. 75, no. 1, pp. 19-22
Abstract
pdf (79.87 Kb)
In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and $n$ point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case $n = 1$ (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.
Keywords:
ideal fluid, motion of a circular cylinder in an ideal fluid, point vortices, Poisson structure, Poisson bracket, Casimir function
Citation:
Borisov A. V., Mamaev I. S., Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid, Mathematical Notes, 2004, vol. 75, no. 1, pp. 19-22
Two-body problem on a sphere. Reduction, stochasticity, periodic orbits
Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 265-279
Abstract
pdf (13.58 Mb)
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.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 265-279
Absolute and relative choreographies in the problem of point vortices moving on a plane
Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 101-111
Abstract
pdf (389.11 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Absolute and relative choreographies in the problem of point vortices moving on a plane, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 101-111
Classical dynamics in non-Eucledian spaces
Moscow–Izhevsk: Institute of Computer Science, 2004, 348 pp.
Abstract
pdf (3.5 Mb)
The book is a collecton of recent and classical works of dynamics in spaces of constant curvature. The Kepler problem and its extentions, the two- and three-body problem and the rigid body dinamics in curved spaces are considered. Many classical works by W. Killing, H.Liebmann, etc. have been practically not available for a wide audience and almost forgotten. The recent papers collected here discuss stochasticity and integrability issues, various generalizations of some results from classical and celestial mechanics, and the Newton potential theory.
The book is for undergraduate and postgraduate students and specialists in dynamical systems. We hope that it will be interesting for scientific historians.
Citation:
Borisov A. V., Mamaev I. S., Classical dynamics in non-Eucledian spaces, Moscow–Izhevsk: Institute of Computer Science, 2004, 348 pp.
Strange Attractors in Rattleback Dynamics
Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393-403
Abstract
pdf (484.78 Kb)
This review is dedicated to the dynamics of the rattleback, a phenomenon with curious physical properties that is
studied in nonholonomic mechanics. All known analytical results are collected here, and some results of our numerical
simulation are presented. In particular, three-dimensional Poincare maps associated with dynamical systems are systematically investigated for the first time. It is shown that the loss of
stability of periodic and quasiperiodic solutions, which gives rise
to strange attractors, is typical of the three-dimensional maps related to rattleback dynamics. This explains some newly discovered properties of the rattleback related to the transition from regular to chaotic solutions at certain values of the physical parameters.
Citation:
Borisov A. V., Mamaev I. S., Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393-403
The Hess case in the dynamics of a rigid body
Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 2, pp. 227-235
Abstract
pdf (605.38 Kb)
In the paper we consider modifications of the Hess integral suitable for various forms of the equations of motion for rigid body. This integral occurs due to some additional symmetry properties of the equations of motion. Moreover, we discuss the general conditions under which the integral exists. Assuming that these conditions are satisfied, we discuss the reduction of the order of the equations, their explicit integration and a qualitative analysis of motion. For the first time, the paper indicates new counterparts of the Hess case for the problem of a gyroscope fixed in a gimbal suspension and for Chaplygin's equations describing the motion of a heavy rigid body in an ideal fluid.
Citation:
Borisov A. V., Mamaev I. S., The Hess case in the dynamics of a rigid body, Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 2, pp. 227-235
Motion of a circular cylinder and $n$ point vortices in a perfect fluid
Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 449-462
Abstract
pdf (585.43 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 449-462
Dynamics of rolling disk
Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 201-212
Abstract
pdf (648.4 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Dynamics of rolling disk, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 201-212
An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid
Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 163-166
Abstract
pdf (90.36 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 163-166
Fundamental and Applied Problems in the Theory of Vortices
Moscow–Izhevsk: Institute of Computer Science, 2003, 704 pp.
Abstract
pdf (16.02 Mb)
The book includes works of national and abroad authors examining the dynamics of vortex structures in fluid. The collected articles show this domain of study as an intensively evolving branch of fluid dynamics. With this aim in view, there are given both the results that have become well known, and the last achievements of the authors. The first part of the book is dedicated to the statement and solution of problems formulated in the frames of the classic hydrodynamic theory. Vortex problems of the geophysical hydrodynamics are stated in the second part of the book.
This collection of articles will be useful for the specialists in the field of dynamic systems and hydrodynamics, for lecturers, post-graduates and students in this scientific branch.
Citation:
Borisov A. V., Mamaev I. S., Sokolovskiy M. A., Fundamental and Applied Problems in the Theory of Vortices, Moscow–Izhevsk: Institute of Computer Science, 2003, 704 pp.
Nonintegrability of a system of interacting particles with Dyson potential
Fundamental and applied problems in vortex theory, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2003, pp. 387–391
Abstract
Citation:
Borisov A. V., Kozlov V. V., Nonintegrability of a system of interacting particles with Dyson potential, Fundamental and applied problems in vortex theory, Ser. Komp'yut. Mat. Fiz. Biol., Izhevsk: Institute of Computer Science, 2003, pp. 387–391
Modern Methods of the Theory of Integrable Systems
Moscow–Izhevsk: Institute of Computer Science, 2003, 296 pp.
Abstract
pdf (1.77 Mb)
The book studies integrable systems of the Hamiltonian mechanics within the context of the Lax representation and the explicit integration procedures. The authors introduce new methods of separation of variables and formulate the universal algorithm for constructing L-A pairs based on bi-hamiltonianity. In the book are also discussed multidimensional analogues of the integrable problems in the rigid body dynamics, generalized Toda lattices, geodesic flows and other problems in mechanics and geometry.
Citation:
Borisov A. V., Mamaev I. S., Modern Methods of the Theory of Integrable Systems, Moscow–Izhevsk: Institute of Computer Science, 2003, 296 pp.
Obstacle to the Reduction of Nonholonomic Systems to the Hamiltonian Form
Doklady Physics, 2002, vol. 47, no. 12, pp. 892-894
Abstract
pdf (36.81 Kb)
In the paper, classical Chaplygin's problem on an unbalanced ball that is rolling without slipping on a plane is considered. Using numerical simulations, we have shown the possibility of mixing integrable non-holonomic systems on invariant tori. Therein lies the obstacle for this system to be Hamiltonian. It should be noted, that, nevertheless, in such systems there is an invariant measure and conservation of energy.
Citation:
Borisov A. V., Mamaev I. S., Obstacle to the Reduction of Nonholonomic Systems to the Hamiltonian Form, Doklady Physics, 2002, vol. 47, no. 12, pp. 892-894
A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid
Doklady Physics, 2002, vol. 47, no. 7, pp. 544-547
Abstract
pdf (85.97 Kb)
The problem of rolling motion without slipping of an unbalanced ball on 1) an arbitrary ellipsoid and 2) an ellipsoid of revolution is considered. In his famous treatise E. Routh showed that the problem of rolling motion of a body on a surface of revolution even in the presence of axisymmetrical potential fields is integrable. In case 1, we present a new integral of motion. New solutions expressed in elementary functions are found in case 2.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid, Doklady Physics, 2002, vol. 47, no. 7, pp. 544-547
Compatible Poisson Brackets on Lie Algebras
Mathematical Notes, 2002, vol. 72, no. 1, pp. 10-30
Abstract
pdf (244.55 Kb)
We discuss the relationship between the representation of an integrable system as an $L-A$-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.
The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 201-219
Abstract
pdf (628.29 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 201-219
The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 177-200
Abstract
pdf (784.15 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 177-200
On the History of the Development of the Nonholonomic Dynamics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 43-47
Abstract
pdf (183.63 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., On the History of the Development of the Nonholonomic Dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 43-47
Generalization of the Goryachev–Chaplygin Case
Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 21-30
Abstract
pdf (289.22 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., Generalization of the Goryachev–Chaplygin Case, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 21-30
Nonholonomic Dynamical Systems
Moscow–Izhevsk: Institute of Computer Science, 2002, 324 pp.
Abstract
pdf (3.86 Mb)
The book is a collection of papers concerned with dynamical effects in the motion of nonholonomic systems. Most of the contributions have been exclusively written for this book by leading Russian experts and involve novel results. These include, among others, new geometrical images of dynamics and various hierarchies of systems behavior. Also three-dimensional mappings in the problems of rolling motion of bodies are investigated numerically.
Citation:
Borisov A. V., Mamaev I. S., Nonholonomic Dynamical Systems, Moscow–Izhevsk: Institute of Computer Science, 2002, 324 pp.
A New Integrable Case on $so(4)$
Doklady Physics, 2001, vol. 46, no. 12, pp. 888-889
Abstract
pdf (28.12 Kb)
In his paper "New integrable Case of Kirchoff's equations" (Theor. and Math. Physics. 2001) V.V. Sokolov proposed a new integrable case with additional integral of fourth degree. We have shown that the integral can be written in a more natural form and consider its generalization to a bundle of Poisson brackets.
Citation:
Borisov A. V., Mamaev I. S., Sokolov V. V., A New Integrable Case on $so(4)$, Doklady Physics, 2001, vol. 46, no. 12, pp. 888-889
Chaplygin's Ball Rolling Problem Is Hamiltonian
Mathematical Notes, 2001, vol. 70, no. 5, pp. 720-723
Abstract
pdf (339.65 Kb)
In this paper we introduce a new nonlinear Poisson bracket in the problem of rolling motion of a Chaplygin's ball. Thus, upon some change of time, the equations of motion become Hamiltonian. We have also established that the trajectories of this system are isomorphic to the trajectories of the Braden system which describes the motion of a point on a two-dimensional sphere in a potential field; using the isomorphism, we have shown that in the Chaplygin problem the variables are separable.
Keywords:
Chaplygin's ball rolling problem, potential force field, Poisson bracket, Euler\,--\,Jacobi theorem
Citation:
Borisov A. V., Mamaev I. S., Chaplygin's Ball Rolling Problem Is Hamiltonian, Mathematical Notes, 2001, vol. 70, no. 5, pp. 720-723
Euler–Poisson Equations and Integrable Cases
Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 253-276
Abstract
pdf (1.41 Mb)
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.
Citation:
Borisov A. V., Mamaev I. S., Euler–Poisson Equations and Integrable Cases, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 253-276
Kovalevskaya top and generalizations of integrable systems
Regular and Chaotic Dynamics, 2001, vol. 6, no. 1, pp. 1-16
Abstract
pdf (286.68 Kb)
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.
Citation:
Borisov A. V., Mamaev I. S., Kholmskaya A. G., Kovalevskaya top and generalizations of integrable systems, Regular and Chaotic Dynamics, 2001, vol. 6, no. 1, pp. 1-16
Dynamics of the Rigid Body
Izhevsk: Regular and Chaotic Dynamics, 2001, 384 pp.
Abstract
pdf (4.8 Mb)
Here the fundamental forms of the equations of motion for rigid body are discussed. These involve motion in potential fields and in fluid (the Kirchhoff equations), motion of a body with cavities filled with fluid. Conditions under which the reduction of the order of these equations is possible and cyclic variables exist are given. In addition, the book collects almost all known to date integrable cases along with methods of their explicit integration. Throughout the book results of computer simulations are copiously presented to help visualize motions' peculiarities. Most results discussed in the book are obtained by the authors.
Citation:
Borisov A. V., Mamaev I. S., Dynamics of the Rigid Body, Izhevsk: Regular and Chaotic Dynamics, 2001, 384 pp.
Stability of Thomson's Configurations of Vortices on a Sphere
Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200
Abstract
pdf (284.39 Kb)
In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated.
Citation:
Borisov A. V., Kilin A. A., Stability of Thomson's Configurations of Vortices on a Sphere, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200
Some comments to the paper by A.M.Perelomov "A note on geodesics on ellipsoid" RCD 2000 5(1) 89-91
Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 92-94
Abstract
pdf (178.43 Kb)
Citation:
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, Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 92-94
The Kovalevskaya case and new integrable systems of dynamics
Vestnik molodyh uchenyh. "Prikladnaya matematika i mehanika", 2000, no. 4, pp. 13-25
Abstract
pdf (910.69 Kb)
Citation:
Borisov A. V., Mamaev I. S., Kholmskaya A. G., The Kovalevskaya case and new integrable systems of dynamics, Vestnik molodyh uchenyh. "Prikladnaya matematika i mehanika", 2000, no. 4, pp. 13-25
Nonintegrability of a System of Interacting Particles with the Dyson Potential
Doklady Physics, 1999, vol. 59, no. 3, pp. 485-486
Abstract
pdf (162.58 Kb)
Citation:
Borisov A. V., Kozlov V. V., Nonintegrability of a System of Interacting Particles with the Dyson Potential, Doklady Physics, 1999, vol. 59, no. 3, pp. 485-486
Kovalevskaya Exponents and Poisson Structures
Regular and Chaotic Dynamics, 1999, vol. 4, no. 3, pp. 13-20
Abstract
pdf (248.75 Kb)
We consider generalizations of pairing relations for Kovalevskaya exponents in quasihomogeneous systems with quasihomogeneous tensor invariants. The case of presence of a Poisson structure in the system is investigated in more detail. We give some examples which illustrate general theorems.
Citation:
Borisov A. V., Dudoladov S. L., Kovalevskaya Exponents and Poisson Structures, Regular and Chaotic Dynamics, 1999, vol. 4, no. 3, pp. 13-20
Lie algebras in vortex dynamics and celestial mechanics — IV
Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 23-50
Abstract
pdf (1.16 Mb)
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.
Citation:
Bolsinov A. V., Borisov A. V., Mamaev I. S., Lie algebras in vortex dynamics and celestial mechanics — IV, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 23-50
Poisson Structures and Lie Algebras in Hamiltonian Mechanics
Izhevsk: Izd. UdSU, 1999, 464 pp.
Abstract
pdf (5.91 Mb)
The book is about Poisson structures and their application to various problems of Hamiltonian mechanics arising in a number of areas, such as dynamics of rigid body, celestial mechanics, the theory of vortices, cosmological models. The equations governing the motion of such systems can be written in a convenient polynomial (algebraic) form. This form is in close connection with the possibility of representing the equations of motion as a set of Hamiltonian equations with linear Poisson structure associated with some Lie algebra. The authors also discuss nonlinear Poisson structures defined by infinite-dimensional Lie algebras and consider most typical situations in which such structures occur. The equations obtained are studied using the Painleve–Kovalevskaya method. Also the book presents some new integrable cases and establishes isomorphisms between integrable problems.
Citation:
Borisov A. V., Mamaev I. S., Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Izhevsk: Izd. UdSU, 1999, 464 pp.
Dynamics of Three Vortices on a Plane and a Sphere — III. Noncompact Case. Problems of Collaps and Scattering
Regular and Chaotic Dynamics, 1998, vol. 3, no. 4, pp. 74-86
Abstract
pdf (657.83 Kb)
In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and scattering of vortices and obtained the conditions of realization. We completed the bifurcation analysis and investigated the dependence of stability in linear approximation and frequency of rotation in relative coordinates for collinear and Thomson's configurations from value of a full moment and indicated the geometric interpretation for characteristic situations. We constructed a phase portrait and geometric projection for an integrable configuration of four vortices on a plane.
Citation:
Borisov A. V., Lebedev V. G., Dynamics of Three Vortices on a Plane and a Sphere — III. Noncompact Case. Problems of Collaps and Scattering, Regular and Chaotic Dynamics, 1998, vol. 3, no. 4, pp. 74-86
Dynamics of three vortices on a plane and a sphere — II. General compact case
Regular and Chaotic Dynamics, 1998, vol. 3, no. 2, pp. 99-114
Abstract
pdf (493.96 Kb)
Integrable problem of three vorteces on a plane and sphere are considered. The classification of Poisson structures is carried out. We accomplish the bifurcational analysis using the variables introduced in previous part of the work.
Citation:
Borisov A. V., Lebedev V. G., Dynamics of three vortices on a plane and a sphere — II. General compact case, Regular and Chaotic Dynamics, 1998, vol. 3, no. 2, pp. 99-114
Dynamics and statics of vortices on a plane and a sphere - I
Regular and Chaotic Dynamics, 1998, vol. 3, no. 1, pp. 28-38
Abstract
pdf (223.68 Kb)
In the present paper a description of a problem of point vortices on a plane and a sphere in the "internal" variables is discussed. The hamiltonian equations of motion of vortices on a plane are built on the Lie–Poisson algebras, and in the case of vortices on a sphere on the quadratic Jacobi algebras. The last ones are obtained by deformation of the corresponding linear algebras. Some partial solutions of the systems of three and four vortices are considered. Stationary and static vortex configurations are found.
Citation:
Borisov A. V., Pavlov A. E., Dynamics and statics of vortices on a plane and a sphere - I, Regular and Chaotic Dynamics, 1998, vol. 3, no. 1, pp. 28-38
A degenerate Poisson Structure and Lie algebras in the Two Problems of Hamiltonian Dynamics
Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, pp. 71-74
Abstract
pdf (3.81 Mb)
Citation:
Borisov A. V., Mamaev I. S., A degenerate Poisson Structure and Lie algebras in the Two Problems of Hamiltonian Dynamics, Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, pp. 71-74
Kovalevskaya's method in rigid body dynamics
Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 1, pp. 27-32
Abstract
pdf (701.12 Kb)
An example from the field of rigid body dynamics, possessing a natural physical justification, is presented. The behaviour of the solutions of the equations of motion in the real domain, whatever the initial data, is regular; nevertheless, depending on the values of a certain control parameter, the solution of the system may branch in the complex time plane, and the system will have multi-valued first integrals. A denumerable sequence of single-valued polynomial integrals of arbitrarily high even degree is found (unlike Kovalevskaya's case, in which the degree of the first integral of the Euler–Poisson equations is four). As an extension, a system from non-holonomic mechanics is considered.
Citation:
Borisov A. V., Tsygvintsev A. V., Kovalevskaya's method in rigid body dynamics, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 1, pp. 27-32
Non-linear Poisson brackets and isomorphisms in dynamics
Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 72-89
Abstract
pdf (1.69 Mb)
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.
Citation:
Borisov A. V., Mamaev I. S., Non-linear Poisson brackets and isomorphisms in dynamics, Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 72-89
Adiabatic Chaos in Rigid Body Dynamics
Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 65-78
Abstract
pdf (1.71 Mb)
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.
Citation:
Borisov A. V., Mamaev I. S., Adiabatic Chaos in Rigid Body Dynamics, Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 65-78
Period Doubling Bifurcation in Rigid Body Dynamics
Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 64-74
Abstract
pdf (926.92 Kb)
Taking a classical problem of motion of a rigid body in a gravitational field as an example, we consider Feigenbaum's script for transition to stochasticity. Numerical results are obtained using Andoyer-Deprit's canonical variables. We calculate universal constants describing "doubling tree" self-duplication scaling. These constants are equal for all dynamical systems, which can be reduced to the study of area-preserving mappings of a plan onto itself. We show that stochasticity in Euler-Poisson equations can progress according to Feigenbaum's script under some restrictions on the parameters of our system.
Citation:
Borisov A. V., Simakov N. N., Period Doubling Bifurcation in Rigid Body Dynamics, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 64-74
Necessary and Sufficient Conditions of Kirchhoff Equation Integrability
Regular and Chaotic Dynamics, 1996, vol. 1, no. 2, pp. 61-76
Abstract
pdf (809.02 Kb)
The problem of motion of a 1-connected solid on interia in an infinite volume of irrotational ideal incompressible liquid in Kirchhoff setting [1-3] is considered in the paper. As it is known, the equations of this problem are structurally analogous to motion equations for the classical problem of motion of a heavy solid around a fixed point. In general case these equations are not integrable as well, and one more additional integral is needed for their integrability. Classical cases of integrability were found by A. Klebsch, V.A. Steklov, A.M. Lyapunov, S.A. Chaplygin in the previous century. It has been shown in [4] that Kirchhoff problems are not integrable in general case, and necessary conditions of integrability, which in some cases are sufficient, have been found there. In the present paper necessary and sufficient conditions of Kirchhoff equations integrability from the view-point of existence of additional analytical and single-valued integrals (in a complex meaning) are investigated.
Analytical results are illustrated with a numerical construction of Poincare mapping and of perturbed asymptotic surfaces (separatrices). Transversal intersection of separatrices may serve as a numerical proof of non-integrability, for great values of pertubing parameter as well.
Citation:
Borisov A. V., Necessary and Sufficient Conditions of Kirchhoff Equation Integrability, Regular and Chaotic Dynamics, 1996, vol. 1, no. 2, pp. 61-76
Kowalewski exponents and integrable systems of classic dynamics. I, II
Regular and Chaotic Dynamics, 1996, vol. 1, no. 1, pp. 15-37
Abstract
pdf (1.16 Mb)
In the frame of this work the Kovalevski exponents (KE) have been found for various problems arising in rigid body dynamics and vortex dynamics. The relations with the parameters of the system are shown at which KE are integers. As it is shown earlier the power of quasihomogeneous integral in quasihomogeneous systems of differential equations is equal to one of IKs. That let us to find the power of an additional integral for the dynamical systems studied in this work and then find it in the explicit form for one of the classic problems of rigid body dynamics. This integral has an arbitrary even power relative to phase variables and the highest complexity among all the first integrals found before in classic dynamics (in Kovalevski case the power of the missing first integral is equal to four). The example of a many-valued integral in one of the dynamic systems is given.
Citation:
Borisov A. V., Tsygvintsev A. V., Kowalewski exponents and integrable systems of classic dynamics. I, II, Regular and Chaotic Dynamics, 1996, vol. 1, no. 1, pp. 15-37
On two modified integrable problems of dynamics
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1995, no. 6, pp. 102-105
Abstract
pdf (55.18 Kb)
In the paper two integrable systems are considered. These systems are modifications of the classical Brun (Clebsh) problems and of the Chaplygin problem, that is, the right-hand sides of the Poisson equations are multiplied by -1. Integrability of some other systems that can be obtained from these classical systems via modifications of more general type is discussed.
Citation:
Borisov A. V., Fedorov Y. N., On two modified integrable problems of dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1995, no. 6, pp. 102-105
Nonintegrability and Stochasticity in Rigid Body Dynamics
Izhevsk: Publishing House of Udmurt State University, 1995, 57 pp.
Abstract
pdf (2.37 Mb)
In this paper we present analytical and numerical data on nonintegrability of the equation of motion and the stochastic behaviour of trajectories in a range of classical and modern problems of dynamics of rigid body. There are considered analytical methods of proof of non-integrability in the systems close to integrable and their relation with data of numerical simulation. The observed spliting of separatrixes leads to absence of the additional first integrals and formation of stochastic layers within chaos in dynamics of rigid body.
Citation:
Borisov A. V., Emel'yanov K. V., Nonintegrability and Stochasticity in Rigid Body Dynamics, Izhevsk: Publishing House of Udmurt State University, 1995, 57 pp.
Nonintegrability of the Classical Homogeneous Three-Component Yang–Mills Field
in "Numerical Modelling in the Problems of Mechanics", Moscow.: Izd. Mosk. Uiver., 1991, pp. 157-166
Abstract
pdf (717.17 Kb)
A finite-dimensional system of equations, corresponding to the classical Yang–Mills equations, is investigated. In the three-component case, the Yang–Mills equations are studied both analytically and numerically. A computer-based proof of nonintegrability of these equations is given. A close relation between the nonintegrability and the presence of stochastic regions in the phase space is marked.
Citation:
Dovbysh S. A., Borisov A. V., Nonintegrability of the Classical Homogeneous Three-Component Yang–Mills Field, in "Numerical Modelling in the Problems of Mechanics", Moscow.: Izd. Mosk. Uiver., 1991, pp. 157-166
On the Liouville Problem
in "Numerical Modelling in the Problems of Mechanics", Moscow.: Izd. Mosk. Uiver., 1991, pp. 110-118
Abstract
pdf (403.08 Kb)
In the paper the Liouville problem of motion of a deformable elastic body is considered. Necessary conditions for the problem to be integrable are studied using the separatrix split technique. In addition, some integrable cases are found.
Citation:
Borisov A. V., On the Liouville Problem, in "Numerical Modelling in the Problems of Mechanics", Moscow.: Izd. Mosk. Uiver., 1991, pp. 110-118
Non-integrability of the Kirchhoff Equations
in "Mathematical Methods in Mechanics", Moscow: Izd. Mosk. Univer., 1990, pp. 13-18
Abstract
pdf (380.89 Kb)
We study the integrability of the Kirchoff equations by expanding perturbations in a Fourier series in the angle-variables. Conditions for an additional single-valued integral to exist are investigated numerically. Considering a perturbed Steklov's case, we have shown, by the use of numerical simulations, that the system exhibits stochastic and non-integrable behavior.
Citation:
Borisov A. V., Kiryanov A. I., Non-integrability of the Kirchhoff Equations, in "Mathematical Methods in Mechanics", Moscow: Izd. Mosk. Univer., 1990, pp. 13-18
Non-integrability of the Kirchhoff equations and related problems in rigid body dynamics
VINITI RAN, 1989, no. 5037-В89. М., pp.
Abstract
Citation:
Barkin Y. V., Borisov A. V., Non-integrability of the Kirchhoff equations and related problems in rigid body dynamics, VINITI RAN, 1989, no. 5037-В89. М., pp.