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Free and controlled motion of a body with moving internal mass though a fluid in the presence of circulation around the body
Doklady Physics, 2016, vol. 466, no. 3, pp. 293-297
Abstract
pdf (300.44 Kb)
In this paper, we study the free and controlled motion of an arbitrary two-dimensional body with a moving internal material point through an ideal fluid in presence of constant circulation around the body. We perform bifurcation analysis of free motion (with fixed internal mass). We show that by changing the position of the internal mass the body can be made to move to a specified point. There are a number of control problems associated with the nonzero drift of the body in the case of fixed internal mass.
Citation:
Vetchanin E. V., Kilin A. A., Free and controlled motion of a body with moving internal mass though a fluid in the presence of circulation around the body, Doklady Physics, 2016, vol. 466, no. 3, pp. 293-297
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
On a mechanical lens
International Journal of Non-Linear Mechanics, 2016, vol. 79, pp. 115?121
Abstract
pdf (548.99 Kb)
In this paper, we consider the dynamics of a heavy homogeneous ball moving under the influence of dry friction on a fixed horizontal plane. We assume the ball to slide without rolling. We demonstrate that the plane may be divided into two regions, each characterized by a distinct coefficient of friction, so that balls with equal initial linear and angular velocity will converge upon the same point from different initial locations along a certain segment. We construct the boundary between the two regions explicitly and discuss possible applications to real physical systems.
Keywords:
Dry friction; Variable coefficient of friction; Dynamics of a ball; Bowling ball hook
Citation:
Ivanov A. P., Erdakova N. N., On a mechanical lens, International Journal of Non-Linear Mechanics, 2016, vol. 79, pp. 115?121
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.
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
Homogeneous systems with quadratic integrals, Lie?Poisson quasi-brackets, and the Kovalevskaya method
Sbornik: Mathematics, 2015, vol. 206, no. 12, pp. 29?54
Abstract
pdf (481.65 Kb)
We consider differential equations with quadratic right-hand sides which admit two quadratic first integrals, one of which is a positive definite quadratic form. We present general conditions under which a linear change of variables reduces this system to some "canonical" form. Under these conditions the system turns out to be nondivergent and is reduced to Hamiltonian form, however, the corresponding linear Lie–Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case the equations are reduced to the classical equations of the Euler top, and in the four-dimensional space the system turns out to be superintegrable and coincides with the Euler–Poincare? equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplication with which the Poisson bracket satisfies the Jacobi identity. In the general case, we prove that there is no reducing multiplier for $n>5$. As an example, we consider a system of Lotka–Volterra type with quadratic right-hand sides, which was studied already by Kovalevskaya from the viewpoint of the conditions for uniqueness of its solutions as functions of complex time.
Keywords:
first integrals, conformally Hamiltonian system, Poisson bracket, Kovalevskaya system, dynamical systems with quadratic right-hand sides
Citation:
Bizyaev I. A., Kozlov V. V., Homogeneous systems with quadratic integrals, Lie?Poisson quasi-brackets, and the Kovalevskaya method, Sbornik: Mathematics, 2015, vol. 206, no. 12, pp. 29?54
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
Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra?
Journal of Geometry and Physics, 2015, vol. 87, pp. 61-75
Abstract
pdf (754.64 Kb)
The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
Bolsinov A. V., Kilin A. A., Kazakov A. O., Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: Pro or contra? , Journal of Geometry and Physics, 2015, vol. 87, pp. 61-75
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
Influence of rolling friction on the controlled motion of a robot wheel
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 583-592
Abstract
pdf (395.52 Kb)
This paper presents an experimental investigation of the influence of rolling friction on the dynamics of a robot wheel. The robot is set in motion by changing the proper gyrostatic momentum using the controlled rotation of a rotor installed in the robot. The problem is considered under the assumption that the center of mass of the system does not coincide with its geometric center. In this paper we derive equations describing the dynamics of the system and give an example of the controlled motion of a wheel by specifying a constant angular acceleration of the rotor. A description of the design of the robot wheel is given and a method for experimentally determining the rolling friction coefficient is proposed. For the verification of the proposed mathematical model, experimental studies of the controlled motion of the robot wheel are carried out. We show that the theoretical results qualitatively agree with the experimental ones, but are quantitatively different.
Keywords:
robot-wheel, rolling friction, displacement of the center of mass
Citation:
Pivovarova E. N., Klekovkin A. V., Influence of rolling friction on the controlled motion of a robot wheel, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 583-592
Experimental determination of the added masses by method of towing
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 568-582
Abstract
pdf (1.88 Mb)
This paper is concerned with the experimental determination of the added masses of bodies completely or partially immersed in a fluid. The paper presents an experimental setup, a technique of the experiment and an underlying mathematical model. The method of determining the added masses is based on the towing of the body with a given propelling force. It is known (from theory) that the concept of an added mass arises under the assumption concerning the potentiality of flow over the body. In this context, the authors have performed PIV visualization of flows generated by the towed body, and defined a part of the trajectory for which the flow can be considered as potential. For verification of the technique, a number of experiments have been performed to determine the added masses of a spheroid. The measurement results are in agreement with the known reference data. The added masses of a screwless freeboard robot have been defined using the developed technique.
Keywords:
added mass, movement on a free surface, hydrodynamic resistance, method of towing
Citation:
Klenov A. I., Vetchanin E. V., Kilin A. A., Experimental determination of the added masses by method of towing, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 568-582
Optical measurement of a fluid velocity field around a falling plate
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 554-567
Abstract
pdf (4.06 Mb)
The paper is devoted to the experimental verification of the Andersen?Pesavento?Wang model describing the falling of a heavy plate through a resisting medium. As a main research method the authors have used video filming of a falling plate with PIV measurement of the velocity of surrounding fluid flows. The trajectories of plates and streamlines were determined and oscillation frequencies were estimated using experimental results. A number of experiments for plates of various densities and sizes were performed. The trajectories of plates made of plastic are qualitatively similar to the trajectories predicted by the Andersen?Pesavento?Wang model. However, measured and computed frequencies of oscillations differ significantly. For a plate made of high carbon steel the results of experiments are quantitatively and qualitatively in disagreement with computational results.
Keywords:
PIV ? Particle Image Velocimetry, Maxwell problem, model of Andersen?Pesavento?Wang
Citation:
Vetchanin E. V., Klenov A. I., Optical measurement of a fluid velocity field around a falling plate, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 554-567
A model of a screwless underwater robot
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
Abstract
pdf (308.24 Kb)
The paper is devoted to the development of a model of an underwater robot actuated by inner rotors. This design has no moving elements interacting with an environment, which minimizes a negative impact on it, and increases noiselessness of the robot motion in a liquid. Despite numerous discussions on the possibility and efficiency of motion by means of internal masses' movement, a large number of works published in recent years confirms a relevance of the research. The paper presents an overview of works aimed at studying the motion by moving internal masses. A design of a screwless underwater robot that moves by the rotation of inner rotors to conduct theoretical and experimental investigations is proposed. In the context of theoretical research a robot model is considered as a hollow ellipsoid with three rotors located inside so that the axes of their rotation are mutually orthogonal. For the proposed model of a screwless underwater robot equations of motion in the form of classical Kirchhoff equations are obtained.
Keywords:
mobile robot, screwless underwater robot, movement in ideal fluid
Citation:
Vetchanin E. V., Karavaev Y. L., Kalinkin A. A., Klekovkin A. V., Pivovarova E. N., A model of a screwless underwater robot, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
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
Sequential Dynamics in the Motif of Excitatory Coupled Elements
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 701-715
Abstract
pdf (769.83 Kb)
In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of the generalized Lotka?Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized three-dimensional Lotka?Volterra model.
Korotkov A. G., Kazakov A. O., Osipov G. V., Sequential Dynamics in the Motif of Excitatory Coupled Elements, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 701-715
Spherical Robot of Combined Type: Dynamics and Control
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728
Abstract
pdf (306.92 Kb)
This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.
Kilin A. A., Pivovarova E. N., Ivanova T. B., Spherical Robot of Combined Type: Dynamics and Control, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728
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 Dynamics of Systems with Servoconstraints. II
Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 401-427
Abstract
pdf (861.95 Kb)
This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servoconstraint, which implies that the projection of the body?s angular velocity on some body-fixed direction is zero.
Keywords:
servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides, vakonomic systems
Citation:
Kozlov V. V., The Dynamics of Systems with Servoconstraints. II, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 401-427
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
The Dynamics of Systems with Servoconstraints. I
Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 205-224
Abstract
pdf (810.79 Kb)
The paper discusses the dynamics of systems with Béghin?s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin?s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.
Keywords:
servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides
Citation:
Kozlov V. V., The Dynamics of Systems with Servoconstraints. I, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 205-224
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
The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform
Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 134-152
Abstract
pdf (1.38 Mb)
This paper deals with the problem of a spherical robot propelled by an internal omniwheel platform and rolling without slipping on a plane. The problem of control of spherical robot motion along an arbitrary trajectory is solved within the framework of a kinematic model and a dynamic model. A number of particular cases of motion are identified, and their stability is investigated. An algorithm for constructing elementary maneuvers (gaits) providing the transition from one steady-state motion to another is presented for the dynamic model. A number of experiments have been carried out confirming the adequacy of the proposed kinematic model.
Karavaev Y. L., Kilin A. A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 134-152
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
Experimental research of dynamic of spherical robot of combined type
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 721?734
Abstract
pdf (761.73 Kb)
This paper presents the results of experimental investigations for the rolling of a spherical robot of combined type actuated by an internal wheeled vehicle with rotor on a horizontal plane. The control of spherical robot based on nonholonomic dynamical by means of gaits. We consider the motion of the spherical robot in case of constant control actions, as well as impulse control. A number of experiments have been carried out confirming the importance of rolling friction.
Keywords:
spherical robot of combined type, dynamic model, control by means of gaits, rolling friction
Citation:
Kilin A. A., Karavaev Y. L., Experimental research of dynamic of spherical robot of combined type, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 721?734
The contol of the motion through an ideal fluid of a rigid body by means of two moving masses
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 633?645
Abstract
pdf (413.58 Kb)
In this paper we consider the problem of motion of a rigid body in an ideal fluid with two material points moving along circular trajectories. The controllability of this system on the zero level set of first integrals is shown. Elementary ?gaits? are presented which allow the realization of the body?s motion from one point to another. The existence of obstacles to a controlled motion of the body along an arbitrary trajectory is pointed out.
Keywords:
ideal fluid, Kirchhoff equations, controllability of gaits
Citation:
Kilin A. A., Vetchanin E. V., The contol of the motion through an ideal fluid of a rigid body by means of two moving masses, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 633?645
The dynamics of systems with servoconstraints. II
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 579-611
Abstract
pdf (560.42 Kb)
This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body?s angular velocity on some body-fixed direction is zero.
Keywords:
servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides, vakonomic systems
Citation:
Kozlov V. V., The dynamics of systems with servoconstraints. II, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 579-611
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
The dynamics of systems with servoconstraints. I
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 353-376
Abstract
pdf (505.17 Kb)
The paper discusses the dynamics of systems with Béghin?s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint ? the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) ? and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin?s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.
Keywords:
servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides
Citation:
Kozlov V. V., The dynamics of systems with servoconstraints. I, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 353-376
The dynamic of a spherical robot with an internal omniwheel platform
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 187-204
Abstract
pdf (530.7 Kb)
The dynamic model for a spherical robot with an internal omniwheel platform is presented. Equations of motion and first integrals according to the non-holonomic model are given. We consider particular solutions and their stability. The algorithm of control of spherical robot for movement along a given trajectory are presented.
Karavaev Y. L., Kilin A. A., The dynamic of a spherical robot with an internal omniwheel platform, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 187-204
Principles of dynamics and servo-constraints
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 1, pp. 169-178
Abstract
pdf (316.31 Kb)
It is well known that in the Béghin? Appel theory servo-constraints are realized using controlled external forces. In this paper an expansion of the Béghin?Appel theory is given in the case where
servo-constraints are realized using controlled change of the inertial properties of a dynamical system. The analytical mechanics of dynamical systems with servo-constraints of general form is discussed. The key principle of the approach developed is to appropriately determine virtual displacements of systems with constraints.
Geometrisation of Chaplygin's reducing multiplier theorem
Nonlinearity, 2015, vol. 28, no. 7, pp. 2307?2318
Abstract
pdf (164.17 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 (616.56 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
Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator
arXiv:1511.02655v1, 2015, 14 pp.
Abstract
pdf (535.3 Kb)
This paper investigates the possibility of the motion control of a ball with a pendulum mechanism with non-holonomic constraints using gaits — the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.
Citation:
Ivanova T. B., Pivovarova E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, arXiv:1511.02655v1, 2015, 14 pp.
Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top
Doklady Physics, 2014, vol. 90, no. 2, pp. 631?634
Abstract
pdf (203.47 Kb)
Citation:
Bizyaev I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Doklady Physics, 2014, vol. 90, no. 2, pp. 631?634
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 (923.35 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
On the dynamics of point vortices in an annular region
Fluid Dynamics Research, 2014, vol. 46, no. 3, 031420, 7 pp.
Abstract
pdf (205.4 Kb)
This paper reviews the results of stability analysis for polygonal configurations of a point vortex system in an annular region depending on the ratio of the inner and outer radii of the annulus. Conditions are found for linear stability of Thomson?s configurations for the case $N<7$. The paper also shows that a system of two vortices between parallel walls is a limiting case of a two-vortex system in an annular region, as the radii of the annulus tend to infinity.
Citation:
Erdakova N. N., Mamaev I. S., On the dynamics of point vortices in an annular region , Fluid Dynamics Research, 2014, vol. 46, no. 3, 031420, 7 pp.
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
On Rational Integrals of Geodesic Flows
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 601-606
Abstract
pdf (145.78 Kb)
This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.
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
Remarks on Integrable Systems
Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 145-161
Abstract
pdf (186.79 Kb)
The problem of integrability conditions for systems of differential equations is discussed. Darboux?s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension $\geqslant n$. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.
Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II"
Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 140-143
Abstract
pdf (170.77 Kb)
In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.
Ivanova T. B., Pivovarova E. N., Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II", Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 140-143
The Dynamics of a Rigid Body with a Sharp Edge in Contact with an Inclined Surface in the Presence of Dry Friction
Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 116-139
Abstract
pdf (735.75 Kb)
In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.
Keywords:
rod, Painlevé paradox, dry friction, loss of contact, frictional impact
Citation:
Mamaev I. S., Ivanova T. B., The Dynamics of a Rigid Body with a Sharp Edge in Contact with an Inclined Surface in the Presence of Dry Friction, Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 116-139
Paul Painlev? 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.
The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 497-511
Abstract
pdf (986.08 Kb)
The kinematic control model for a spherical robot with an internal omniwheel platform is presented. We consider singularities of control of spherical robot with an unbalanced internal omniwheel platform. The general algorithm of control of spherical robot according to the kinematical quasi-static model and controls for simple trajectories (a straight line and in a circle) are presented. Experimental investigations have been carried out for all introduced control algorithms.
Keywords:
spherical robot, kinematic model, nonholonomic constraint, omniwheel, displacement of center of mass
Citation:
Kilin A. A., Karavaev Y. L., The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 497-511
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
Control of a Vehicle with Omniwheels on a Plane
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 473-481
Abstract
pdf (520.34 Kb)
The problem of motion of a vehicle in the form of a platform with an arbitrary number of Mecanum wheels fastened on it is considered. The controllability of this vehicle is discussed within the framework of the nonholonomic rolling model. An explicit algorithm is presented for calculating the control torques of the motors required to follow an arbitrary trajectory. Examples of controls for executing the simplest maneuvers are given.
On Rational Integrals of Geodesic Flows
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 439-445
Abstract
pdf (302.05 Kb)
This paper is concerned with the problem of first integrals of the equations of geodesics on twodimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy?Kovalevskaya theorem.
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
Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 329-343
Abstract
pdf (5.62 Mb)
This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.
Keywords:
point vortices, nonintegrability, bifurcations, chart of dynamical regimes
Citation:
Vetchanin E. V., Kazakov A. O., Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 329-343
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
On a generalization of systems of Calogero type
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, pp. 209-212
Abstract
pdf (237.84 Kb)
This paper isљconcerned with aљthree-body system onљaљstraight line inљaљpotential field proposed byљTsiganov. The Liouville integrability ofљthis system isљshown. Reduction and separation ofљvariables are performed.
Keywords:
Calogero systems, reduction, integrable systems, Jacobi problem
Citation:
Bizyaev I. A., On a generalization of systems of Calogero type, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, pp. 209-212
Comment on the paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev ?How to control the Chaplygin ball using rotors. II?
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 127-131
Abstract
pdf (234.11 Kb)
Inљthis paper weљconsider the control ofљaљdynamically asymmetric balanced ball onљaљplane inљthe case ofљslipping atљthe contact point. Necessary conditions under which aљcontrol isљpossible are obtained. Specific algorithms ofљcontrol along aљgiven trajectory are constructed.
Ivanova T. B., Pivovarova E. N., Comment on the paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev ?How to control the Chaplygin ball using rotors. II?, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 127-131
Kinematic control of a high manoeuvrable mobile spherical robot with internal omni-wheeled platform
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 113-126
Abstract
pdf (3.43 Mb)
Inљthis article aљkinematic model ofљthe spherical robot isљconsidered, which isљset inљmotion byљthe internal platform with omni-wheels. Itљhas been introduced aљdescription ofљconstruction, algorithm ofљtrajectory planning according toљdeveloped kinematic model, itљhas been realized experimental research for typical trajectories: moving along aљstraight line and moving along aљcircle.
Kilin A. A., Karavaev Y. L., Klekovkin A. V., Kinematic control of a high manoeuvrable mobile spherical robot with internal omni-wheeled platform, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 113-126
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 isљconcerned with the figures ofљequilibrium ofљaљself-gravitating ideal fluid with density stratification and aљsteady-state velocity field. Asљinљthe classical setting, itљisљassumed that the figure orљits layers uniformly rotate about anљaxis fixed inљspace. Asљisљwell known, when there isљnoљrotation, only aљball can beљaљfigure ofљequilibrium.
Itљisљshown that the ellipsoid ofљrevolution (spheroid) with confocal stratification, inљwhich each layer rotates with inherent constant angular velocity, isљatљequilibrium. Expressions are obtained for the gravitational potential, change inљthe angular velocity and pressure, and the conclusion isљdrawn that the angular velocity onљthe outer surface isљthe same asљthat ofљthe Maclaurin spheroid. Weљnote that the solution found generalizes aљpreviously known solution for piecewise constant density distribution. For comparison, weљalso present aљsolution, due toљChaplygin, for aљhomothetic density stratification.
Weљconclude byљconsidering aљhomogeneous spheroid inљthe space ofљconstant positive curvature. Weљshow that inљthis case the spheroid cannot rotate asљaљrigid body, since the angular velocity distribution ofљfluid particles depends onљthe distance toљthe 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
In this paper we describe an inertiameter, which is an experimental facility for determining the inertia tensor components and the position of the center of mass of compound bodies. An algorithm for determining these dynamical properties is presented. Using the algorithm obtained, the displacement of the center of mass and the tensor of inertia are determined experimentally for a spherical robot of combined type.
Keywords:
inertiameter, spherical robot, moment of inertia, center of mass
Citation:
Alalykin S. S., Bogatyrev A. V., Ivanova T. B., Pivovarova E. N., , Bulletin of Udmurt University. Physics and Chemistry, 2014, no. 4, pp. 79-86
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.
On the Routh sphere problem
Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, no. 8, 085202, 11 pp.
Abstract
pdf (169.11 Kb)
We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on six-dimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra $e^*$(3). It allows us to relate the nonholonomic Routh system with the Hamiltonian system on a cotangent bundle to the sphere with a canonical Poisson structure.
Citation:
Bizyaev I. A., Tsiganov A. V., On the Routh sphere problem , Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, no. 8, 085202, 11 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
Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone
Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 521-538
Abstract
pdf (3.11 Mb)
We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
Gonchenko A. S., Gonchenko S. V., Kazakov A. O., Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 521-538
Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane
Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 508-520
Abstract
pdf (1.65 Mb)
We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.
Kazakov A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 508-520
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 (825.09 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 Euler?Jacobi?Lie Integrability Theorem
Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 329-343
Abstract
pdf (377.18 Kb)
This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of $n$ differential equations is proved, which admits $n?2$ independent symmetry fields and an invariant volume $n$-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.
Keywords:
symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics
Citation:
Kozlov V. V., The Euler?Jacobi?Lie Integrability Theorem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 4, pp. 329-343
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 Self-propulsion of a Body with Moving Internal Masses in a Viscous Fluid
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 100-117
Abstract
pdf (1.71 Mb)
An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier–Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.
Keywords:
finite-volume numerical method, Navier–Stokes equations, variable internal mass distribution, motion control
Citation:
Vetchanin E. V., Mamaev I. S., Tenenev V. A., The Self-propulsion of a Body with Moving Internal Masses in a Viscous Fluid, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 100-117
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 ofљthe reducing multiplier for aљspecial class ofљnonholonomic dynamical systems, when the resulting nonlinear Poisson structure isљreduced toљthe Lie?Poisson bracket ofљthe algebra $e(3)$. Asљanљillustration, the Chaplygin ball rolling problem and the Veselova system are considered. Inљaddition, anљintegrable gyrostatic generalization ofљthe Veselova system isљobtained.
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 rigid body whose sharp edge is in contact with a inclined surface with dry friction
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 567-594
Abstract
pdf (886.59 Kb)
Inљthis paper weљconsider the dynamics ofљrigid body whose sharp edge isљinљcontact with aљrough plane. The body can move soљthat its contact point does not move orљslips orљloses touch with the support. Inљthis paper, the dynamics ofљthe system isљconsidered within three mechanical models that describe different modes ofљmotion. The boundaries ofљdefinition range ofљeach model are given, the possibility ofљtransitions from one mode toљanother and their consistency with different coefficients ofљfriction onљthe horizontal and inclined surfaces isљdiscussed.
Mamaev I. S., Ivanova T. B., The dynamics of rigid body whose sharp edge is in contact with a inclined surface with dry friction, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 567-594
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)
Inљthis paper weљinvestigate two systems consisting ofљaљspherical shell rolling onљaљplane without slipping and aљmoving rigid body fixed inside the shell byљmeans ofљtwo different mechanisms. Inљthe former case the rigid body isљfixed atљthe center ofљthe ball onљaљspherical hinge. Weљshow anљisomorphism between the equations ofљmotion for the inner body with those for the ball moving onљaљsmooth plane. Inљthe latter case the rigid body isљfixed byљmeans ofљthe nonholonomic hinge. The equations ofљmotion for this system have been obtained and new integrable cases found. Aљspecial feature ofљthe set ofљtensor invariants ofљthis system isљthat itљleads toљthe Euler?Jacobi?Lie theorem, which isљaљnew integration mechanism inљnonholonomic 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 dynamics of a body with an axisymmetric base sliding on a rough plane
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 521-545
Abstract
pdf (612.94 Kb)
Inљthis paper weљinvestigate the dynamics ofљaљbody with aљflat base sliding onљaљhorizontal plane under the assumption ofљlinear pressure distribution ofљthe body onљthe plane asљthe simplest dynamically consistent friction model.
For analysis weљuse the descriptive function method similar toљthe methods used inљthe problems ofљHamiltonian dynamics with one degree ofљfreedom and allowing aљqualitative analysis ofљthe system toљbeљmade without explicit integration ofљequations ofљmotion. Inљaddition, weљgive aљsystematic review ofљthe well-known experimental and theoretical results inљthis area.
Keywords:
dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation:
Erdakova N. N., Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 521-545
Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 507-520
Abstract
pdf (582.48 Kb)
This paper investigates the possibility ofљthe motion control ofљaљball with aљpendulum mechanism with non-holonomic constraints using gaitsљ? the simplest motions such asљacceleration and deceleration during the motion inљaљstraight line, rotation through aљgiven angle and their combination. Also, the controlled motion ofљthe system along aљstraight line with aљconstant acceleration isљconsidered. For this problem the algorithm for calculating the control torques isљgiven and itљisљshown that the resulting reduced system has the first integral ofљmotion.
Keywords:
non-holonomic constraint, control, spherical shell, integral of motion
Citation:
Ivanova T. B., Pivovarova E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 507-520
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
Notes on integrable systems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 459-478
Abstract
pdf (375.2 Kb)
The problem ofљintegrability conditions for systems ofљdifferential equations isљdiscussed. Darboux?s classical results onљthe integrability ofљlinear non-autonomous systems with anљincomplete set ofљparticular solutions are generalized. Special attention isљpaid toљlinear Hamiltonian systems. The paper discusses the general problem ofљintegrability ofљthe systems ofљautonomous differential equations inљanљ$n$-dimensional space which permit the algebra ofљsymmetry fields ofљdimension $\geqslant n$. Using aљmethod due toљLiouville, this problem isљreduced toљinvestigating the integrability conditions for Hamiltonian systems with Hamiltonians linear inљthe momentums inљphase space ofљdimension that isљtwice asљlarge. Inљconclusion, the integrability ofљanљautonomous system inљthree-dimensional space with two independent non-trivial symmetry fields isљproved. Itљshould beљemphasized that noљadditional conditions are imposed onљthese fields.
Keywords:
integrability by quadratures, adjoint system, Hamilton equations, Euler?Jacobi theorem, Lie theorem, symmetries
Citation:
Kozlov V. V., Notes on integrable systems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 459-478
Chaotic dynamics phenomena in the rubber rock-n-roller on a plane problem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 309-325
Abstract
pdf (5.28 Mb)
Inљthis paper weљstudy aљproblem ofљrolling ofљthe dynamically asymmetric ball with displacement center ofљgravity onљaљplane without slipping and vertical rotating. Itљisљshown that the dynamics ofљthe ball isљsignificantly affected byљthe type ofљreversibility. Depending onљthe type ofљthe reversibility weљfound two different types ofљdynamical chaos: strange attractors and mixed chaotic dynamics. Inљthis paper weљdescribe aљstrange attractor development, and then its basic properties. Aљset ofљcriteria byљwhich inљnumerical experiments mixed dynamics may beљdistinguished from other types ofљdynamical chaos are given.
Kazakov A. O., Chaotic dynamics phenomena in the rubber rock-n-roller on a plane problem, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 309-325
Integrability and stochastic behavior in some nonholonomic dynamics problems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 257-265
Abstract
pdf (2.27 Mb)
Inљthis paper, weљinvestigate the dynamics ofљsystems describing the rolling without slipping and spinning (rubber rolling) ofљanљellipsoid onљaљplane and aљsphere. Weљresearch these problems using Poincare maps, which investigation helps toљdiscover aљnew integrable case.
Keywords:
nonholonomic constraint, invariant measure, first integral, Poincare map, integrability and chaos
Citation:
Bizyaev I. A., Kazakov A. O., Integrability and stochastic behavior in some nonholonomic dynamics problems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 257-265
The Euler?Jacobi?Lie integrability theorem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 229-245
Abstract
pdf (377.18 Kb)
This paper addresses aљclass ofљproblems associated with the conditions for exact integrability ofљaљsystem ofљordinary differential equations expressed inљterms ofљthe properties ofљtensor invariants. The general theorem ofљintegrability ofљthe system ofљ$n$љdifferential equations isљproved, which admits $n ? 2$љindependent symmetry fields and anљinvariant volume $n$-form (integral invariant). General results are applied toљthe study ofљsteady motions ofљaљcontinuous medium with infinite conductivity.
Keywords:
symmetry field, integral invariant, nilpotent group, magnetic hydrodynamics
Citation:
Kozlov V. V., The Euler?Jacobi?Lie integrability theorem, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 229-245
Topological monodromy in nonholonomic systems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 203-227
Abstract
pdf (890.26 Kb)
The phenomenon ofљaљtopological monodromy inљintegrable Hamiltonian and nonholonomic systems isљdiscussed. Anљefficient method for computing and visualizing the monodromy isљdeveloped. The comparative analysis ofљthe topological monodromy isљgiven for the rolling ellipsoid ofљrevolution problem inљtwo cases, namely, onљaљsmooth and onљaљrough plane. The first ofљthese systems isљHamiltonian, the second isљnonholonomic. Weљshow that, from the viewpoint ofљmonodromy, there isљnoљdifference between the two systems, and thus disprove the conjecture byљCushman and Duistermaat stating that the topological monodromy gives aљtopological obstruction for Hamiltonization ofљthe rolling ellipsoid ofљrevolution onљaљrough plane.
Bolsinov A. V., Kilin A. A., Kazakov A. O., Topological monodromy in nonholonomic systems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 203-227
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)
Inљthis paper, weљinvestigate the dynamics ofљsystems describing the rolling without slipping and spinning (rubber rolling) ofљvarious rigid bodies onљaљplane and aљsphere. Itљisљshown that aљhierarchy ofљpossible types ofљdynamical behavior arises depending onљthe body?s surface geometry and mass distribution. New integrable cases and cases ofљexistence ofљanљinvariant measure are found. Inљaddition, these systems are used toљillustrate that the existence ofљseveral nontrivial involutions inљreversible dissipative systems leads toљquasi-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
Deviation based discrete control algorithm for omni-wheeled mobile robot
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 91-100
Abstract
pdf (434.7 Kb)
The paper deals with deviation based control algorithm for trajectory following ofљomni-wheeled mobile robot. The kinematic model and the dynamics ofљthe robot actuators are described.
Keywords:
omni-wheeled mobile robot, discrete algorithm, deviation based control, linearization, feedback
Citation:
Karavaev Y. L., Trefilov S. A., Deviation based discrete control algorithm for omni-wheeled mobile robot, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 1, pp. 91-100
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)
Inљour earlier paper [2] weљexamined the problem ofљcontrol ofљaљbalanced dynamically nonsymmetric sphere with rotors with no-slip condition atљthe point ofљcontact. Inљthis paper weљinvestigate the controllability ofљaљball inљthe presence ofљfriction. Weљalso study the problem ofљthe existence and stability ofљsingular dissipation-free periodic solutions for aљfree ball inљthe presence ofљfriction forces. The issues ofљconstructive realization ofљthe 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
Motion control of a rigid body in viscous fluid
Computer Research and Modeling, 2013, vol. 5, no. 4, pp. 659-675
Abstract
pdf (359.11 Kb)
We consider the optimal motion control problem for a mobile device with an external rigid shell moving along a prescribed trajectory in a viscous fluid. The mobile robot under consideration possesses the property of self-locomotion. Self-locomotion is implemented due to back-and-forth motion of an internal material point. The optimal motion control is based on the Sugeno fuzzy inference system. An approach based on constructing decision trees using the genetic algorithm for structural and parametric synthesis has been proposed to obtain the base of fuzzy rules.
Vetchanin E. V., Tenenev V. A., Shaura A. S., Motion control of a rigid body in viscous fluid, Computer Research and Modeling, 2013, vol. 5, no. 4, pp. 659-675
On the mathematical simulation of the impact of a double pendulum against an obstacle
Proceedings of MITP, 2013, vol. 5, no. 2, pp. 134-141
Abstract
pdf (572.94 Kb)
An algorithm for the investigation of a mathematical model of the impact of a double pendulum against an obstacle is constructed and realized by computer. This algorithm allows calculation of impact loads and coefficients of restitution at the contact point and hinges. The main goal of this investigation is to minimize negative impact conditions in the hingings.
Keywords:
double pendulum, equations of motion, mathematical model, numerical investigation
Citation:
Erdakova N. N., Ivanov A. P., On the mathematical simulation of the impact of a double pendulum against an obstacle, Proceedings of MITP, 2013, vol. 5, no. 2, pp. 134-141
An Extended Hamilton?Jacobi Method
Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 580-596
Abstract
pdf (216.54 Kb)
We develop a new method for solving Hamilton?s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton–Jacobi method.
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 J. 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 (252.64 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
On Invariant Manifolds of Nonholonomic Systems
Regular and Chaotic Dynamics, 2012, vol. 17, no. 2, pp. 131-141
Abstract
pdf (239.09 Kb)
Invariant manifolds of equations governing the dynamics of conservative nonholonomic systems are investigated. These manifolds are assumed to be uniquely projected onto configuration space. The invariance conditions are represented in the form of generalized Lamb?s equations. Conditions are found under which the solutions to these equations admit a hydrodynamical description typical of Hamiltonian systems. As an illustration, nonholonomic systems on Lie groups with a left-invariant metric and left-invariant (right-invariant) constraints are considered.
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)
Aљnew integrable system describing the rolling ofљaљrigid body with aљspherical cavity over aљspherical base isљconsidered. Previously the authors found the separation ofљvariables for this system atљthe zero level ofљaљlinear (inљangular velocity) first integral, whereas inљthe general case itљisљnot possible toљseparate the variables. Inљthis paper weљshow that the foliation into invariant tori inљthis problem isљequivalent toљthe corresponding foliation inљthe Clebsch integrable system inљrigid body dynamics (for which noљreal separation ofљvariables has been found either). Inљparticular, aљfixed point ofљfocus type isљpossible for this system, which can serve asљaљtopological obstacle toљthe real separation ofљvariables.
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
The motion of a body with variable mass geometry in a viscous fluid
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 815-836
Abstract
pdf (15.9 Mb)
Anљinvestigation ofљthe characteristics ofљmotion ofљaљrigid body with variable internal mass distribution inљaљviscous fluid isљcarried out onљthe basis ofљaљjoint numerical solution ofљthe Navier?Stokes equations and equations ofљmotion. Aљnon-stationary three-dimensional solution toљthe problem isљfound. The motion ofљaљsphere and aљdrop-shaped body inљaљviscous fluid, which isљcaused byљthe motion ofљinternal material points, inљaљgravitational field isљexplored. The possibility ofљmotion ofљaљbody inљanљarbitrary given direction isљshown.
Keywords:
finite-volume numerical method, Navier-Stokes equations, variable internal mass distribution, motion control
Citation:
Vetchanin E. V., Mamaev I. S., Tenenev V. A., The motion of a body with variable mass geometry in a viscous fluid, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 815-836
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)
Inљthis paper weљinvestigate various kinematic properties ofљrolling ofљone rigid body onљanother both for the classical model ofљrolling without slipping (the velocities ofљbodies atљthe point ofљcontact coincide) and for the model ofљrubber-rolling (with the additional condition that the spinning ofљthe bodies relative toљeach other beљexcluded). Furthermore, inљthe case where both bodies are bounded byљspherical surfaces and one ofљthem isљfixed, the equations ofљmotion for aљmoving ball are represented inљthe form ofљthe Chaplygin system. When the center ofљmass ofљthe moving ball coincides with its geometric center, the equations ofљmotion are represented inљconformally Hamiltonian form, and inљthe case where the radii ofљthe moving and fixed spheres coincides, they are written inљHamiltonian 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
On a mechanical lens
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 773-781
Abstract
pdf (380.35 Kb)
The problem ofљdynamics ofљheavy uniform ball moving onљthe fixed rough plane under its own inertia and forces ofљdry friction isљconsidered. Assuming that friction coefficient isљvariable, the switching curve for change the value ofљfriction coefficient isљconstructed. Using this curve toљchange the value ofљfriction coefficient weљhave shown that the bundle ofљequal balls starting from one interval with equal linear and angular velocities should gather atљone point.
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)
Inљthe paper weљconsider aљsystem ofљaљball that rolls without slipping onљaљplane. The ball isљassumed toљbeљinhomogeneous and its center ofљmass does not necessarily coincide with its geometric center. Weљhave proved that the governing equations can beљrecast into aљsystem ofљsix ODEs that admits four integrals ofљmotion. Thus, the phase space ofљthe system isљfoliated byљinvariant 2-tori; moreover, this foliation isљequivalent toљthe Liouville foliation encountered inљthe case ofљEuler ofљthe rigid body dynamics. However, the system cannot beљsolved inљterms ofљquadratures because there isљnoљinvariant measure which weљproved byљfinding 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
On the final motion of cylindrical solids on a rough plane
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 585-603
Abstract
pdf (623.55 Kb)
The problem ofљaљuniform straight cylinder (disc) sliding onљaљhorizontal plane under the action ofљdry friction forces isљconsidered. The contact patch between the cylinder and the plane coincides with the base ofљthe cylinder. Weљconsider axisymmetric discs, i.e. weљassume that the base ofљthe cylinder isљsymmetric with respect toљthe axis lying inљthe plane ofљthe base. The focus isљonљthe qualitative properties ofљthe dynamics ofљdiscs whose circular base, triangular base and three points are inљcontact with aљrough plane.
Keywords:
Amontons?Coulomb law, dry friction, disc, final dynamics, stability
Citation:
Treschev D. V., Erdakova N. N., Ivanova T. B., On the final motion of cylindrical solids on a rough plane, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 585-603
On the Routh sphere
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 569-583
Abstract
pdf (299.77 Kb)
Weљdiscuss anљembedding ofљthe vector field associated with the nonholonomic Routh sphere inљsubgroup ofљthe commuting Hamiltonian vector fields associated with this system. Weљprove that the corresponding Poisson brackets are reduced toљcanonical ones inљthe region without ofљhomoclinic trajectories.
On some new aspects of Celtic stone chaotic dynamics
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 507-518
Abstract
pdf (26.28 Mb)
Weљstudy chaotic dynamics ofљaљnonholonomic model ofљceltic stone movement onљthe plane. Scenarious ofљappearance and development ofљchaos are investigated.
Gonchenko A. S., Gonchenko S. V., Kazakov A. O., On some new aspects of Celtic stone chaotic dynamics, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 507-518
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)
Inљthe paper weљstudy control ofљaљbalanced dynamically nonsymmetric sphere with rotors. The no-slip condition atљthe point ofљcontact isљassumed. The algebraic contrability isљshown and the control inputs providing motion ofљthe ball along aљgiven trajectory onљthe 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)
Weљconsider the problem ofљthe motion ofљaxisymmetric vortex rings inљanљideal incompressible fluid. Using the topological approach, weљpresent aљmethod for complete qualitative analysis ofљthe dynamics ofљaљsystem ofљtwo vortex rings. Inљparticular, weљcompletely solve the problem ofљdescribing the conditions for the onset ofљleapfrogging motion ofљvortex rings. Inљaddition, for the system ofљtwo vortex rings weљfind new families ofљmotions inљwhich the mutual distances remain finite (weљcall them pseudo-leapfrogging). Weљalso find solutions for the problem ofљthree vortex rings, which describe both the regular and chaotic leapfrogging motion ofљvortex 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)
Weљconsider the problem ofљrolling ofљaљball with anљellipsoidal cavity filled with anљideal fluid, which executes aљuniform vortex motion, onљanљabsolutely rough plane. Weљpoint out the case ofљexistence ofљanљinvariant measure and show that there isљaљparticular case ofљintegrability under conditions ofљaxial 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
On invariant manifolds of nonholonomic systems
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 57-69
Abstract
pdf (329.1 Kb)
Invariant manifolds ofљequations governing the dynamics ofљconservative nonholonomic systems are investigated. These manifolds are assumed toљbeљuniquely projected onto configuration space. The invariance conditions are represented inљthe form ofљgeneralized Lamb?s equations. Conditions are found under which the solutions toљthese equations admit aљhydrodynamical description typical ofљHamiltonian systems. Asљanљillustration, nonholonomic systems onљLie groups with aљleft-invariant metric and left-invariant (right-invariant) constraints are considered.
Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics
Journal of Applied Nonlinear Dynamics, 2012, 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, 2012, vol. 2, no. 2, pp. 161-173
Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 4, pp. 146-155
Abstract
pdf (688.03 Kb)
In the paper we study the stability of a spherical shell rolling on a horizontal plane with Lagrange?s gyroscope inside. A linear stability analysis is made for the upper and lower position of a top. A bifurcation diagram of the system is constructed. The trajectories of the contact point for di?erent values of the integrals of motion are constructed and analyzed.
Keywords:
rolling motion, stability, Lagrange?s gyroscope, bifurcational diagram
Citation:
Pivovarova E. N., Ivanova T. B., Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 4, pp. 146-155
On detachment conditions of a top on an absolutely rough support
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 3, pp. 103-113
Abstract
pdf (292.26 Kb)
The classical problem about the motion of a heavy symmetric rigid body (top) with a fixed point on the horizontal plane is discussed. Due to the unilateral nature of the contact, detachments (jumps) are possible under certain conditions. We know two scenarios of detachment related to changing the sign of the normal reaction or the sign of the normal acceleration, and the mismatch of these conditions leads to a paradox. To determine the nature of paradoxes an example of the pendulum (rod) within the limitations of the real coefficient of friction was studied in detail. We showed that in the case of the first type of the paradox (detachment is impossible and contact is impossible) the body begins to slide on the support. In the case of the paradox of the second type (detachment is possible and contact is possible) contact is retained up to the sign change of the normal reaction, and then at the detachment the normal acceleration is non-zero.
Keywords:
friction, Lagrange top, paradox, detachment
Citation:
Ivanov A. P., Shuvalov N. D., Ivanova T. B., On detachment conditions of a top on an absolutely rough support, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 3, pp. 103-113
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
The Vlasov Kinetic Equation, Dynamics of Continuum and Turbulence
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 602-622
Abstract
pdf (323.98 Kb)
We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.
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
Statistical Irreversibility of the Kac Reversible Circular Model
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 536-549
Abstract
pdf (202.6 Kb)
The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M.Kac formulated necessary conditions for irreversibility over "short" time intervals to take place and demonstrated Boltzmann?s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the "zeroth" law of thermodynamics based on the analysis of weak convergence of probability distributions.
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
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 isљconcerned with the use ofљbifurcation analysis and the Conley index inљHamiltonian dynamical systems. Weљgive the proof ofљthe theorem onљthe appearance (disappearance) ofљfixed points inљthe case ofљthe Morse index change. New relative equilibria inљthe problem ofљthe motion ofљpoint vortices ofљequal intensity inљaљcircle 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
The Lorentz force and its generalizations
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 627-634
Abstract
pdf (359.22 Kb)
The structure ofљthe Lorentz force and the related analogy between electromagnetism and inertia are discussed. The problem ofљinvariant manifolds ofљthe equations ofљmotion for aљcharge inљanљelectromagnetic field and the conditions for these manifolds toљbeљLagrangian are considered.
A rigid cylinder on a viscoelastic plane
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 601-625
Abstract
pdf (499.7 Kb)
The paper considers two two-dimensional dynamical problems for anљabsolutely rigid cylinder interacting with aљdeformable flat base (the motion ofљanљabsolutely rigid disk onљaљbase which inљnon-deformed condition isљaљstraight line). The base isљaљsufficiently stiff viscoelastic medium that creates aљnormal pressure $p(x) = kY(x)+?\dot{Y}(x)$, where $x$љisљaљcoordinate onљthe straight line, $Y(x)$ isљaљnormal displacement ofљthe pointљ$x$, and $k$љand $?$љare elasticity and viscosity coefficients (the Kelvin?Voigt medium). Weљare also ofљthe opinion that during deformation the base generates friction forces, which are subject toљCoulomb?s law. Weљconsider the phenomenon ofљimpact that arises during anљarbitrary fall ofљthe disk onto the straight line and investigate the disk?s motion ?along the straight line? including the stages ofљsliding and rolling.
Kuleshov A. S., Treschev D. V., Ivanova T. B., Naymushina O. S., A rigid cylinder on a viscoelastic plane, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 601-625
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 toљChaplygin, which describe the rolling ofљaљspherical shell onљaљplane, with aљball orљLagrange?s gyroscope inside. All necessary first integrals and anљinvariant measure are found. The reduction toљquadratures isљgiven.
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 aљcollection ofљV.A. Steklov?s selected works, which isљbeing prepared for publication and isљentitled ?Works onљMechanics 1902?1909: Translations from French?. The collection isљbased onљV.A. Steklov?s papers onљmechanics published inљFrench journals from 1902 toљ1909.
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)
Weљconsider the problem ofљexplicit integration and bifurcation analysis for two systems ofљnonholonomic mechanics. The first one isљthe Chaplygin?s problem onљno-slip rolling ofљaљbalanced dynamically non-symmetrical ball onљaљhorizontal plane. The second problem isљonљthe motion ofљrigid body inљaљspherical support. Weљexplicitly integrate this problem byљgeneralizing the transformation which Chaplygin applied toљthe integration ofљthe problem ofљthe rolling ball atљaљnon-zero constant ofљareas. Weљconsider the geometric interpretation ofљthis transformation from the viewpoint ofљaљtrajectory isomorphism between two systems atљdifferent levels ofљthe energy integral. Generalization ofљthis transformation for the case ofљdynamics inљaљspherical support allowsљus toљintegrate the equations ofљmotion explicitly inљquadratures and, inљaddition, toљindicate periodic solutions and analyze their stability. Weљalso show that adding aљgyrostat does not lead toљthe loss ofљintegrability.
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 aљtopological approach toљthe search and stability analysis ofљrelative equilibria ofљthree point vortices ofљequal intensities. Itљisљshown that the equations ofљmotion can beљreduced byљone degree ofљfreedom. Weљhave 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
Statistical irreversibility of the Kac reversible circular model
Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 101-117
Abstract
pdf (419.07 Kb)
The Kac circular model isљaљdiscrete dynamical system which has the property ofљrecurrence and reversibility. Within the framework ofљthis model M.Kac formulated necessary conditions for irreversibility over ?short? time intervals toљtake place and demonstrated Boltzmann?s most important exploration methods and ideas, outlining their advantages and limitations. Weљstudy the circular model within the realm ofљthe theory ofљGibbs ensembles and offer aљnew approach toљaљrigorous proof ofљthe ?zeroth? law ofљthermodynamics basing onљthe analysis ofљweak convergence ofљprobability distributions.
Kozlov V. V., Statistical irreversibility of the Kac reversible circular model, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 1, pp. 101-117
Motion control simulating in a viscous liquid of a body with variable geometry of weights
Computer Research and Modeling, 2011, vol. 3, no. 4, pp. 371-381
Abstract
pdf (594.85 Kb)
Statement of a problem of management of movement of a body in a viscous liquid is given. Movement bodies it is induced by moving of internal material points. On a basis the numerical decision of the equations of movement of a body and the hydrodynamic equations approximating dependencies for viscous forces are received. With application approximations the problem of optimum control of body movement dares on the set trajectory with application of hybrid genetic algorithm. Possibility of the directed movement of a body under action is established back and forth motion of an internal point. Optimum control movement direction it is carried out by motion of other internal point on circular trajectory with variable speed
Keywords:
optimum control, the equations of movement, Navier?Stokes equations, numerical methods, fuzzy decision trees, genetic algorithm
Citation:
Vetchanin E. V., Tenenev V. A., Motion control simulating in a viscous liquid of a body with variable geometry of weights, Computer Research and Modeling, 2011, vol. 3, no. 4, pp. 371-381
Figures of equilibrium of liquid self-gravitating inhomogeneous mass
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2011, no. 3, pp. 142-153
Abstract
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We consider the inhomogeneous self-gravitating liquid spheroid with confocal stratification which rotates around the minor semiaxis and is in equilibrium. General relationships for pressure, angular velocity and gravitational potential of the spheroid with any density function are obtained. Special cases of piecewise constant and continuous density functions are analyzed.
Bizyaev I. A., Ivanova T. B., Figures of equilibrium of liquid self-gravitating inhomogeneous mass, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2011, no. 3, pp. 142-153
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)
Weљconsider aљnovel mechanical system consisting ofљtwo spherical bodies rolling over each other, which isљaљnatural extension ofљthe famous Chaplygin problem ofљrolling motion ofљaљball onљaљplane. Inљcontrast toљthe previously explored non-holonomic systems, this one has aљhigher dimension and isљconsiderably more complicated. One remarkable property ofљour system isљthe existence ofљ?clandestine? linear inљmomenta first integrals. For aљmore trivial integrable system, their counterparts were discovered byљChaplygin. Weљhave also found aљfew cases ofљintegrability.
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
Lagrangian mechanics and dry friction
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 4, pp. 855-868
Abstract
pdf (265.11 Kb)
Aљgeneralization ofљAmantons? law ofљdry friction for constrained Lagrangian systems isљformulated. Under aљchange ofљgeneralized coordinates the components ofљthe dry-friction force transform according toљthe covariant rule and the force itself satisfies the Painlev? condition. Inљparticular, the pressure ofљthe system onљaљconstraint isљindependent ofљthe anisotropic-friction tensor. Such anљapproach provides anљinsight into the Painlev? dry-friction paradoxes. Asљanљexample, the general formulas for the sliding friction force and torque and the rotation friction torque onљaљbody contacting with aљsurface are obtained.
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, isљconsidered. This question isљimportant for qualitative analysis ofљsuch systems and allows one toљdetermine possible dynamical effects. The first part isљdevoted toљthe representation ofљintegrable systems inљaљconformally Hamiltonian form. Inљthe second part, the existence ofљaљconformally Hamiltonian representation inљaљneighbourhood ofљaљperiodic solution isљproved for anљarbitrary measure preserving system (including integrable). General consructions are always illustrated byљexamples 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)
Weљconsider figures ofљequilibrium and stability ofљaљliquid self-gravitating elliptic cylinder. The flow within the cylinder isљassumed toљbeљdew toљanљelliptic perturbation. Aљbifurcation diagram isљplotted and conditions for steady solutions toљexist 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
On the dynamics of two point vortices in an annular region
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 531-547
Abstract
pdf (411.62 Kb)
Inљthis paper, the system ofљtwo vortices inљanљannular region isљshown toљbeљintegrable inљthe sense ofљLiouville. Aљfew methods for analysis ofљthe dynamics ofљintegrable systems are discussed and these methods are then applied toљthe study ofљpossible motions ofљtwo vortices ofљequal inљmagnitude intensities. Using the previously established fact ofљthe existence ofљrelative choreographies, the absolute motions ofљthe vortices are classified inљrespect toљthe corresponding regions inљthe phase portrait ofљthe reduced system.
Keywords:
point vortex, reduction, equations of motion, bifurcational diagram, relative choreographies, vortex pair
Citation:
Vaskin V. V., Erdakova N. N., On the dynamics of two point vortices in an annular region, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 531-547
Dynamic advection
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 521-530
Abstract
pdf (10.3 Mb)
Aљnew concept ofљdynamic advection isљintroduced. The model ofљdynamic advection deals with the motion ofљmassive particles inљaљ2Dљflow ofљanљideal incompressible liquid. Unlike the standard advection problem, which isљwidely treated inљthe modern literature, our equations ofљmotion account not only for particles? kinematics, governed byљthe Euler equations, but also for their dynamics (which isљobviously neglected ifљthe mass ofљparticles isљtaken toљbeљzero). Aљfew 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
The Vlasov kinetic equation, dynamics of continuum and turbulence
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 489-512
Abstract
pdf (276.41 Kb)
Weљconsider aљcontinuum ofљinteracting particles whose evolution isљgoverned byљthe Vlasov kinetic equation. Anљinfinite sequence ofљequations ofљmotion for this medium (inљthe Eulerian description) isљderived and its general properties are explored. Anљimportant example isљaљcollisionless gas, which exhibits irreversible behavior. Though individual particles interact via aљpotential, the dynamics ofљthe continuum bears dissipative features. Applicability ofљthe Vlasov equations toљthe modeling ofљsmall-scale turbulence isљdiscussed.
Keywords:
The Vlasov kinetic equation, dynamics of continuum and turbulence
Citation:
Kozlov V. V., The Vlasov kinetic equation, dynamics of continuum and turbulence, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 3, pp. 489-512
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)
Inљthis paper weљdevelop aљnew model ofљnon-holonomic billiard that accounts for the intrinsic rotation ofљthe billiard ball. This model isљaљlimit case ofљthe problem ofљrolling without slipping ofљaљball without slipping over aљquadric surface. The billiards between two parallel walls and inside aљcircle are studied inљdetail. Using the three-dimensional-point-map technique, the non-integrability ofљthe non-holonomic billiard within anљellipse isљshown.
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
Problems of stability and asymptotic behavior of vortex patches on the plane
Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, pp. 327-343
Abstract
pdf (685.19 Kb)
With the help ofљmathematical modelling, weљstudy the dynamics ofљmany point vortices system onљthe plane. For this system, weљconsider the following cases:
?љvortex rings with outer radius $r = 1$љand variable inner radius $r_0$,
?љvortex ellipses with semiaxesљ$a$, $b$.
The emphasis isљonљthe analysis ofљthe asymptotic $(t ? ?)$ behavior ofљthe system and onљthe verification ofљthe stability criteria for vorticity continuous distributions.
Keywords:
vortex dynamics, point vortex, hydrodynamics, asymptotic behavior
Citation:
Vaskin V. V., Vaskina A. V., Mamaev I. S., Problems of stability and asymptotic behavior of vortex patches on the plane, Russian Journal of Nonlinear Dynamics, 2010, vol. 6, no. 2, pp. 327-343
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)
Weљconsider the problems ofљHamiltonian representation and integrability ofљthe nonholonomic Suslov system and its generalization suggested byљS.љA.љChaplygin. These aspects are very important for understanding the dynamics and qualitative analysis ofљthe system. Inљparticular, they are related toљthe nontrivial asymptotic behaviour (i.љe. toљsome scattering problem). The paper presents aљgeneral approach based onљthe study ofљthe hierarchy ofљdynamical behaviour ofљnonholonomic 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
Construction of bifurcation diagram and analysis of stability of self-gravitating fluid elliptical cylinder with internal flow
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, pp. 77-86
Abstract
pdf (588.63 Kb)
Figures of equilibrium are considered and the stability of liquid self-gravitating elliptic cylinder with an internal flow in a class of elliptic indignations are researched. The bifurcation diagram of given system is constructed, areas of existence of the stationary solutions are specified.
Ivanova T. B., Construction of bifurcation diagram and analysis of stability of self-gravitating fluid elliptical cylinder with internal flow, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, pp. 77-86
Thomson?s configurations in dynamics of two vortices in an annular region
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, pp. 71-76
Abstract
pdf (146.57 Kb)
For the system of two point vortices in anulus the Hamiltonian is expressed in terms of elliptic functions. The stability of the Thomson configuration is studied.
Keywords:
point vortex, Thomson configuration, elliptic functions, stability
Citation:
Erdakova N. N., Thomson?s configurations in dynamics of two vortices in an annular region, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2010, no. 4, pp. 71-76
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.
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
Kinetics of collisionless gas: Equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox
Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 535-540
Abstract
pdf (172.17 Kb)
The Poincaré model for dynamics of a collisionless gas in a rectangular parallelepiped with mirror walls is considered. The question on smoothing of the density and the temperature of this gas and conditions for the monotone growth of the coarse-grained entropy are discussed. All these effects provide a new insight of the classical paradox of mixing of gases.
Kozlov V. V., Kinetics of collisionless gas: Equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 535-540
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 particle?interaction 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
Statistical mechanics of relativistic gas in a one-dimensional tube
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 4, pp. 561-567
Abstract
pdf (436.68 Kb)
This study isљthe continuation ofљthe computer experiment [1] with particles ofљgas inљaљone-dimensional tube, described earlier. Inљthis paper weљgive investigation results for the statistical properties ofљaљrelativistic gas inљaљone-dimensional tube. Itљisљshown that this system reaches the state ofљthermodynamical equilibrium whose distribution function isљdetermined byљthe relativistic energy ofљparticles. The system ofљparticles inљaљone-dimensional tube isљdescribed byљanalogy with the billiards inљaљpolygon.
Keywords:
relativistic gas, thermodynamical equilibrium, gas in a one-dimensional tube, Boltzmann distribution
Citation:
Vaskin V. V., Erdakova N. N., Statistical mechanics of relativistic gas in a one-dimensional tube, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 4, pp. 561-567
New superintegrable system on a sphere
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 4, pp. 455-462
Abstract
pdf (214.58 Kb)
Weљconsider the motion ofљaљmaterial point onљthe surface ofљaљsphere inљthe field ofљ2n+1 identical Hooke centers (singularities with elastic potential) lying onљaљgreat circle. Our main result isљthat this system isљsuperintegrable. The property ofљsuperintegrability for this system has been conjectured byљus inљ[3], where the structure ofљaљsuperintegral ofљarbitrarily high odd degree inљmomemnta was outlined. Weљalso indicate anљisomorphism between this system and the one-dimensional N-particle system discussed inљthe recent paper [13] and show that for the latter system anљanalogous superintegral can beљconstructed.
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
Statistical mechanics of nonlinear dynamical systems
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 385-402
Abstract
pdf (896.55 Kb)
With the help ofљmathematical modeling, weљstudy the behavior ofљaљgas ($\sim10^6$ particles) inљaљone-dimensional tube. For this dynamical system, weљconsider the following cases:
?љcollisionless gas (with and without gravity) inљaљtube with both ends closed, the particles ofљthe gas bounce elastically between the ends,
?љcollisionless gas inљaљtube with its left end vibrating harmonically inљaљprescribed manner,
?љcollisionless gas inљaљtube with aљmoving piston, the piston?s mass isљcomparable toљthe mass ofљaљparticle.
The emphasis isљonљthe analysis ofљthe asymptotic ($t??$)) behavior ofљthe system and specifically onљthe transition toљthe state ofљstatistical orљthermal equilibrium. This analysis allows preliminary conclusions onљthe nature ofљrelaxation processes.
Atљthe end ofљthe paper the numerical and theoretical results obtained are discussed. Itљshould beљnoted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works onљthe subject.
Vaskin V. V., Erdakova N. N., Mamaev I. S., Statistical mechanics of nonlinear dynamical systems, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 385-402
Kinetics of collisionless gas: equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 377-383
Abstract
pdf (208.37 Kb)
The Poincar? model for dynamics ofљaљcollisionless gas inљaљrectangular parallelepiped with mirror walls isљconsidered. The question onљsmoothing ofљthe density and the temperature ofљthis gas and conditions for the monotone growth ofљthe coarse-grained entropy are discussed. All these effects provide aљnew insight ofљthe classical paradox ofљmixing ofљgases.
Kozlov V. V., Kinetics of collisionless gas: equalization of temperature, growth of the coarse-grained entropy and the Gibbs paradox, Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 3, pp. 377-383
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 isљconcerned with aљclass ofљproblems which involves the dynamical interaction ofљaљrigid body with point vortices onљthe surface ofљaљtwo-dimensional sphere. The general approach toљthe 2Dљhydrodynamics isљfurther developed. The problem ofљmotion ofљaљdynamically symmetric circular body interacting with aљsingle vortex isљshown toљbeљintegrable. Mass vortices onљ$S^2$ are introduced and the related issues (such asљequations ofљmotion, integrability, partial solutions, etc.) are discussed. This paper isљaљnatural progression ofљthe author?s previous research onљinteraction ofљrigid bodies and point vortices inљaљplane.
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)
Weљconsider several well-known isomorphisms between Jacobi?s geodesic problem and some integrable cases from rigid body dynamics (the cases ofљClebsch and Brun). Aљrelationship between these isomorphisms isљindicated. The problem ofљcompactification for geodesic flows onљnoncompact surfaces isљstated. This problem isљhypothesized toљbeљintimately connected with the property ofљintegrability.
The Jacobi problem on a plane
Russian Journal of Nonlinear Dynamics, 2009, vol. 5, no. 1, pp. 83-86
Abstract
pdf (137.41 Kb)
3-particle systems with aљparticle-interaction homogeneous potential ofљdegree $?=-2$љis considered. Aљconstructive procedure ofљreduction ofљthe system byљ2љdegrees ofљfreedom isљperformed. The nonintegrability ofљthe systems isљshown using the Poincare mapping.
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 ofљmaterial points interacting both with one another and with anљexternal field are considered inљEuclidean space. For the case ofљarbitrary binary interaction depending solely onљthe mutual distance between the bodies, new integrals are found, which form aљGalilean momentum vector.
Aљcorresponding algebra ofљintegrals constituted byљthe integrals ofљmomentum, angular momentum, and Galilean momentum isљpresented. Particle systems with aљparticle-interaction potential homogeneous ofљdegree $?=-2$ are considered. The most general form ofљthe additional integral ofљmotion, which weљterm the Jacobi integral, isљpresented for such systems. Aљnew nonlinear algebra ofљintegrals including the Jacobi integral isљfound. Aљsystematic description isљgiven toљaљnew reduction procedure and possibilities ofљapplying itљtoљdynamics with the aim ofљlowering the order ofљHamiltonian systems.
Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations ofљthe Lagrangian identity for systems with aљparticle-interaction potential homogeneous ofљdegree $?=-2$ are presented. Inљaddition, computational experiments are used toљprove the nonintegrability ofљthe Jacobi problem onљaљplane.
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
Generalized model of kinetics of formation of a new phase
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2009, no. 2, pp. 110-117
Abstract
pdf (724.17 Kb)
The generalized model of formation of a new phase is considered. The basic stages of process of growth are gathered in a model at phase transition of the first sort. The numerical solution of the kinetic equation of Fokker–Planck is received. Dependence of the solution on parametres of system is investigated. Areas of applicability of assumptions made by Zeldovich, Lifshits and Slezov are revealed. Also it is shown, that depending on parametres of system it is possible to reserve both equilibrium distribution, and automodelling distribution of Lifshits–Slezov. At some values of parametres the equation has the oscillatory solution.
Keywords:
Generalized model of kinetics of formation of a new phase
Citation:
Ivanova T. B., Vaskin V. V., Generalized model of kinetics of formation of a new phase, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2009, no. 2, pp. 110-117
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
Gauss Principle and Realization of Constraints
Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 431-434
Abstract
pdf (144.15 Kb)
The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces of viscous friction our statement transforms into the common Gauss principle for systems with constraints.
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
Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 141-154
Abstract
pdf (192.61 Kb)
The paper develops an approach to the proof of the "zeroth" law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.
Kozlov V. V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 141-154
Lagrange?s Identity and Its Generalizations
Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 71-80
Abstract
pdf (144.77 Kb)
The famous Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through the kinetic energy and homogeneous potential energy. The paper presents various extensions of this brilliant result to the case 1) of constrained mechanical systems, 2) when the potential energy is quasi-homogeneous in coordinates and 3) of continuum of interacting particles governed by the well-known Vlasov kinetic equation.
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 onљthe three-dimensional sphere. Canonical variables for the reduced system are constructed both onљtwo-dimensional and three-dimensional spheres. The method isљillustrated byљapplying itљtoљthe two-body problem onљaљsphere (the bodies are assumed toљinteract with aљpotential that depends only onљthe geodesic distance between them) and the three-vortex problem onљaљtwo-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 onљthe dynamics ofљliquid and gas self-gravitating ellipsoids. Equations ofљmotion are given inљLagrangian and Hamiltonian form, inљparticular, the Hamiltonian formalism onљLie algebras isљpresented. Problems ofљnonintegrability and chaotical behavior ofљthe system are formulated and studied. Weљalso classify all known integrable cases and give some hypotheses about nonintegrability inљthe general case. Results ofљnumerical modelling are presented, which can beљconsidered asљaљcomputer proof ofљnonintegrability.
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
Gauss Principle and Realization of Constraints
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 3, pp. 281-285
Abstract
pdf (78.44 Kb)
The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces ofљviscous friction our statement transforms into the common Gauss principle for systems with constraints.
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 beљregarded asљaљcontinuation ofљour previous work [70,71] onљthe hierarchy ofљthe dynamical behavior ofљnonholonomic systems. Weљconsider different mechanical systems with nonholonomic constraints; inљparticular, weљexamine the existence ofљtensor invariants (laws ofљconservation) and their connection with the behavior ofљaљsystem. Considerable attention isљgiven toљthe possibility ofљconformally Hamiltonian representation ofљthe equations ofљmotion, which isљmainly used for the integration ofљthe 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
Lagrange?s identity and its generalizations
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 2, pp. 157-168
Abstract
pdf (128.42 Kb)
The famous Lagrange identity expresses the second derivative ofљthe moment ofљinertia ofљaљsystem ofљmaterial points through the kinetic energy and homogeneous potential energy. The paper presents various extensions ofљthis brilliant result toљthe caseљ1) ofљconstrained mechanical systems, 2) when the potential energy isљquasi-homogeneous inљcoordinates andљ3) ofљcontinuumof interacting particles governed byљthe well-known Vlasov kinetic equation.
Generalization of Lagrange?s identity and new integrals of motion
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2008, no. 3, pp. 69-74
Abstract
pdf (154.14 Kb)
We discuss system of material points in Euclidean space interacting both with each other and with external field. In particular we consider systems of particles whose interacting is described by homogeneous potential of degree of homogeneity $\alpha=-2$. Such systems were first considered by Newton and—more systematically—by Jacobi). For such systems there is an extra hidden symmetry, and corresponding first integral of motion which we call Jacobi integral. This integral was given in different papers starting with Jacobi, but we present in general. Furthermore, we construct a new algebra of integrals including Jacobi integral. A series of generalizations of Lagrange's identity for systems with homogeneous potential of degree of homogeneity $\alpha=-2$ is given. New integrals of motion for these generalizations are found.
Keywords:
Lagrange?s identity, many-particle system, first integral, integrability, algebra of integrals
Citation:
Kilin A. A., Generalization of Lagrange?s identity and new integrals of motion, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2008, no. 3, pp. 69-74
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 ofљthe equations ofљmotion for two spheres inљanљunbounded volume ofљideal and incompressible fluid inљ3DљEuclidean space. Reduction ofљorder, based onљthe use ofљnew variables that form aљLie algebra, isљoffered. Aљtrivial case ofљintegrability isљindicated.
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)
Weљconsider two problems from the rigid body dynamics and use new methods ofљstability and asymptotic behavior analysis for their solution. The first problem deals with motion ofљaљrigid body inљanљunbounded volume ofљideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, isљconcerned with motion ofљaљsleigh onљanљinclined plane. The equations ofљmotion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. Aљcomprehensive survey ofљreferences isљgiven and new problems connected with falling motion ofљheavy bodies inљfluid 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 ofљanљantipodal vortex onљaљsphere (aљpoint vortex plus its antipode with opposite circulation) isљconsidered. Itљisљshown that the system ofљnљantipodal vortices can beљreduced byљfour dimensions (two degrees ofљfreedom). The cases n=2,3 are explored inљgreater detail both analytically and numerically. Weљdiscuss Thomson, collinear and isosceles configurations ofљantipodal 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
Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat
Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 123-140
Abstract
pdf (263.66 Kb)
The paper develops anљapproach toљthe proof ofљthe ?zeroth? law ofљthermodynamics. The approach isљbased onљthe analysis ofљweak limits ofљsolutions toљthe Liouville equation asљtime grows infinitely. Aљclass ofљlinear oscillating systems isљindicated for which the average energy becomes eventually uniformly distributed among the degrees ofљfreedom for any initial probability density functions. Anљexample ofљsuch systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number ofљdegrees ofљfreedom toљconverge inљaљweak sense toљthe state where the average energies ofљthe interacting subsystems are the same. Some issues related toљstatistical models ofљthe thermostat are discussed.
Kozlov V. V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Russian Journal of Nonlinear Dynamics, 2007, vol. 3, no. 2, pp. 123-140
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)
Weљconsider
trajectory isomorphisms between various integrable
systems onљanљ$n$-dimensional sphere $S^n$ and aљEuclidean space $R^n$.
Some ofљthe systems are classical integrable problems ofљCelestial Mechanics
inљplane and curved spaces. All the systems under consideration have anљadditional
first integral quadratic inљmomentum and can beљintegrated analytically byљusing
the separation ofљvariables. Weљshow that
some integrable problems inљconstant curvature spaces are not essentially new from the viewpoint ofљthe
theory ofљintegration, and they can beљanalyzed using known results ofљclassical 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 ofљrolling aљdynamically asymmetric balanced ball (the Chaplygin ball) over aљsphere. Suppose that the contact point has zero velocity and the projection ofљthe angular velocity toљthe normal vector ofљthe sphere equals zero. This model ofљrolling differs from the classical one. Itљcan beљrealized, inљsome approximation, ifљthe ball isљrubber coated and the sphere isљabsolutely rough. Recently, Koiller and Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure weљconstruct anљisomorphism between this problem and the problem ofљthe motion ofљaљpoint onљaљsphere inљsome 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
Vorticity equation ofљ2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 425-434
Abstract
pdf (152.1 Kb)
The issues discussed inљthis paper relate toљthe description ofљdeveloped two-dimensional turbulence, when the mean values ofљcharacteristics ofљsteady flow stabilize. More exactly, the problem ofљaљweak limit ofљvortex distribution inљtwo-dimensional flow ofљanљideal fluid atљtime tending toљinfinity isљconsidered. Relations between the vorticity equation and the well-known Vlasov equation are discussed.
Kozlov V. V., Vorticity equation ofљ2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence, Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 4, pp. 425-434
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)
Weљhave discovered aљnew first integral inљthe problem ofљmotion ofљaљdynamically symmetric ball, subject toљgravity, onљthe surface ofљaљparaboloid. Using this integral, weљhave obtained conditions for stability (inљthe Lyapunov sense) ofљsteady rotations ofљthe ball inљthe 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)
Weљpresent aљreduction-of-order procedure inљthe problem ofљmotion ofљtwo bodies onљthe Lobatchevsky plane $H^2$. The bodies interact with aљpotential that depends only onљthe distance between the bodies (this holds for anљanalog ofљthe Newtonian potential). Inљearlier works, this reduction procedure was used toљanalyze the motion ofљtwo bodies onљthe 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)
Weљconsider the interaction ofљtwo vortex patches (elliptic Kirchhoff vortices) which move inљanљunbounded volume ofљanљideal incompressible fluid. Aљmoment second-order model isљused toљdescribe the interaction. The case ofљintegrability ofљaљKirchhoff vortex and aљpoint vortex byљthe variable separation method isљqualitatively analyzed. Aљnew case ofљintegrability ofљtwo Kirchhoff vortices isљfound. Aљreduced form ofљequations for two Kirchhoff vortices isљproposed and used toљanalyze 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
New integral in nonholonomic Painleve-Chaplygin problem
Russian Journal of Nonlinear Dynamics, 2006, vol. 2, no. 2, pp. 193-198
Abstract
pdf (132.6 Kb)
Inљthe paper weљpresent aљnew integral ofљmotion inљthe problem ofљrolling motion ofљaљheavy symmetric sphere onљthe surface ofљaљparaboloid. Weљuse this integral toљstudy the Lyapunov stability ofљsome trivial steady rotations.
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 ofљtwo vortex rings onљaљsphere isљconsidered. This motion generalizes the well-known centrally symmetrical solution ofљthe equations ofљpoint vortex dynamics onљaљplane derived byљD.N. Goryachev and H.љAref. The equations ofљmotion inљthis case are shown toљbeљLiouville integrable, and anљexplicit reduction toљaљHamiltonian system with one degree ofљfreedom isљdescribed. Two particular cases inљwhich 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 ofљfour vortices onљaљsphere.
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 ofљpoint vortices onљaљplane and aљsphere
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 233-246
Abstract
pdf (473.55 Kb)
Weљoffer aљnew method ofљreduction for aљsystem ofљpoint vortices onљaљplane and aљsphere. This method isљsimilar toљthe classical node elimination procedure. However, asљapplied toљthe vortex dynamics, itљrequires substantial modification. Reduction ofљfour vortices onљaљsphere isљgiven inљmore detail. Weљalso use the Poincare surface-of-section technique toљperform the reduction aљfour-vortex system onљaљsphere.
Keywords:
reduction, point vortex, equations of motion, Poincare map
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Reduction and chaotic behavior ofљpoint vortices onљaљplane and aљsphere, Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 233-246
Chaos inљaљrestricted problem ofљrotation ofљaљrigid body with aљfixed point
Russian Journal of Nonlinear Dynamics, 2005, vol. 1, no. 2, pp. 191-207
Abstract
pdf (650.7 Kb)
The paper deals with aљtransition toљchaos inљthe phase-plane portrait ofљaљrestricted problem ofљrotation ofљaљrigid body with aљfixed point. Two interrelated mechanisms responsible for chaotisation have been indicated: 1) growth ofљthe homoclinic structure andљ2) development ofљcascades ofљperiod doubling bifurcations. Onљthe zero level ofљthe integral ofљareas, anљadiabatic behavior ofљthe system (asљthe energy tends toљzero) has been noticed. Meander tori induced byљthe breakdown ofљthe torsion property ofљthe 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 inљaљrestricted problem ofљrotation ofљaљrigid body with aљfixed 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
pdf (401.63 Kb)
For the classical problem ofљmotion ofљaљrigid body about aљfixed point with zero integral ofљareas, the paper presents aљfamily ofљsolutions which are periodic inљthe absolute space. Such solutions are known asљchoreographies. The family includes the famous Delaunay solution inљthe case ofљKovalevskaya, some particular solutions inљthe Goryachev-Chaplygin case and Steklov?s solution. The ?genealogy? ofљthe solutions ofљthe family, arising naturally from the energy continuation, and their connection with the Staude rotations are considered.
Itљisљshown that ifљthe integral ofљareas isљzero, the solutions are periodic but with respect toљaљcoordinate 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)
Inљthis paper weљconsider the system ofљtwo 2Dљrigid circular cylinders immersed inљanљunbounded volume ofљinviscid perfect fluid. The circulations around the cylinders are assumed toљbeљequal inљmagnitude and opposite inљsign. Special cases ofљthis system (the cylinders move along the line through their centers and the circulation around each cylinder isљzero) are considered. Aљsimilar system ofљtwo interacting spheres was originally considered inљclassical works ofљCarl and Vilhelm Bjerknes, G.љLamb and N.E. Joukowski.
Byљmaking the radii ofљthe cylinders infinitesimally small, weљhave obtained aљnew mechanical system which consists ofљtwo regular point vortices but with non-zero masses. The study ofљthis system can beљreduced toљthe study ofљthe motion ofљaљparticle subject toљpotential and gyroscopic forces. Aљnew integrable case isљfound. The Hamiltonian equations ofљmotion for this system have been generalized toљthe case ofљanљarbitrary number ofљmass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations ofљmotion for nљpoint 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.
Selected Problems on Nonholonomic Mechanics
Moscow–Izhevsk: Institute of Computer Science, 2005, 290 pp.
Abstract
pdf (5.92 Mb)
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Selected Problems on Nonholonomic Mechanics, Moscow–Izhevsk: Institute of Computer Science, 2005, 290 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
Hamiltonization of nonholonomic systems
arXiv:nlin/0509036v1, 2005, 24 pp.
Abstract
pdf (321.8 Kb)
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
Billiards, invariant measures, and equilibrium thermodynamics. II
Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 91-100
Abstract
pdf (283.7 Kb)
The kinetics of collisionless continuous medium is studied in a bounded region on a curved manifold. We have assumed that in statistical equilibrium, the probability distribution density depends only on the total energy. It is shown that in this case, all the fundamental relations for a multi-dimensional ideal gas in thermal equilibrium hold true.
Citation:
Kozlov V. V., Billiards, invariant measures, and equilibrium thermodynamics. II, Regular and Chaotic Dynamics, 2004, vol. 9, no. 2, pp. 91-100
Notes on diffusion in collisionless medium
Regular and Chaotic Dynamics, 2004, vol. 9, no. 1, pp. 29-34
Abstract
pdf (148.96 Kb)
A collisionless continuous medium in Euclidean space is discussed, i.e. a continuum of free particles moving inertially, without interacting with each other. It is shown that the distribution density of such medium is weakly converging to zero as time increases indefinitely. In the case of Maxwell's velocity distribution of particles, this density satisfies the well-known diffusion equation, the diffusion coefficient increasing linearly with time.
Citation:
Kozlov V. V., Notes on diffusion in collisionless medium, Regular and Chaotic Dynamics, 2004, vol. 9, no. 1, pp. 29-34
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.
Two Integrable System on a Two-dimensional Sphere
Doklady Physics, 2003, vol. 389, no. 3, pp.
Abstract
pdf (160.12 Kb)
In their paper "Kepler's problem in constant curvature spaces" (Cel. Mech. And Dyn. Ast., v. 54, 1992. p. 393-399) Kozlov V.V. and Harin А.О. showed that the Euler problem of planar motion of a particle attracted by two fixed Newtonian centers has an integrable analogue on a two-dimensional sphere $S^2$. The integrability was shown using the separation of variables. In the book "Poisson structures and Lie algebras in Hamiltonian mechanics" by Borisov A.V. and Mamaev I.S. the integrability of a spatial analogue of the problem was proved. The proof is based on the reduction by using a cyclic variable. As a result of this reduction, the 'spatial' problem on $S^3$ becomes a 'plane' one on $S^2$. In this case, however, an additional Hook's center appears at the pole on the perpendicular to an equatorial plane of the two former centers. In this paper we present algebraic integrals for a more general situation when a particle is attracted by two Newtonian centers and three mutually orthogonal Hook's centers of which two together with the Newtonian centers lie in a plane, and the third one is on the perpendicular to the plane.
Citation:
Mamaev I. S., Two Integrable System on a Two-dimensional Sphere, Doklady Physics, 2003, vol. 389, no. 3, 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
New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface
Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 331-335
Abstract
pdf (148.37 Kb)
Some new cases when the invariant measure and an additional first integral exist in the problem of a rigid body rolling on a sphere and on an ellipsoid are discussed in the paper. These cases generalize the results obtained previously by V.A.Yaroshchuk and A.V.Borisov, I.S.Mamaev, A.A.Kilin.
Citation:
Mamaev I. S., New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 331-335
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.
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 Integration Theory of Equations of Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 161-176
Abstract
pdf (456.43 Kb)
The paper deals with the problem of integration of equations of motion in nonholonomic systems. By means of well-known theory of the differential equations with an invariant measure the new integrable systems are discovered. Among them there are the generalization of Chaplygin's problem of rolling nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body with a fixed point. The structure of dynamics of systems on the invariant manifold in the integrable problems is shown. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are suggested. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. The existence conditions of invariant measure with analytical density for the differential equations of nonholonomic mechanics is given.
Citation:
Kozlov V. V., On the Integration Theory of Equations of Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 161-176
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
On Justification of Gibbs Distribution
Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 1-10
Abstract
pdf (324.27 Kb)
The paper develop a new approach to the justification of Gibbs canonical distribution for Hamiltonian systems with finite number of degrees of freedom. It uses the condition of nonintegrability of the ensemble of weak interacting Hamiltonian systems.
Citation:
Kozlov V. V., On Justification of Gibbs Distribution, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 1-10
Nonholonomic Dynamical Systems
Moscow–Izhevsk: Institute of Computer Science, 2002, 324 pp.
Abstract
pdf (3.62 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
The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis
Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291-306
Abstract
pdf (1.16 Mb)
The motion of Chaplygin ball with and without gyroscope in the absolute space is analyzed. In particular, the trajectories of the point of contact are studied in detail. We discuss the motions in the absolute space, that correspond to the different types of motion in the moving frame of reference related to the body. The existence of the bounded trajectories of the ball's motion is shown by means of numerical methods in the case when the problem is reduced to a certain Hamiltonian system.
Citation:
Kilin A. A., The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291-306
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
Kinetics of Collisionless Continuous Medium
Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 235-251
Abstract
pdf (468.67 Kb)
In this article we develop Poincar? ideas about a heat balance of ideal gas considered as a collisionless continuous medium. We obtain the theorems on diffusion in nondegenerate completely integrable systems. As a corollary we show that for any initial distribution the gas will be eventually irreversibly and uniformly distributed over all volume, although every particle during this process approaches arbitrarily close to the initial position indefinitely many times. However, such individual returnability is not uniform, which results in diffusion in a reversible and conservative system. Balancing of pressure and internal energy of ideal gas is proved, the formulas for limit values of these quantities are given and the classical law for ideal gas in a heat balance is deduced. It is shown that the increase of entropy of gas under the adiabatic extension follows from the law of motion of a collisionless continuous medium.
Citation:
Kozlov V. V., Kinetics of Collisionless Continuous Medium, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 235-251
First Integral in the Problem of Motion of a Circular Cylinder and a Point Vortex in Unbounded Ideal Fluid
Regular and Chaotic Dynamics, 2001, vol. 6, no. 2, pp. 233-234
Abstract
pdf (139.78 Kb)
In the paper Motion of a circular cylinder and a vortex in an ideal fluid (Reg. & Chaot. Dyn. V. 6. 2001. No 1. P. 33-38) Ramodanov S.M. showed the integrability of the problem of motion of a circular cylinder and a point vortex in unbounded ideal fluid. In the present paper we find additional first integral and invariant measure of motion equations.
Citation:
Kilin A. A., First Integral in the Problem of Motion of a Circular Cylinder and a Point Vortex in Unbounded Ideal Fluid, Regular and Chaotic Dynamics, 2001, vol. 6, no. 2, pp. 233-234
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
Billiards, Invariant Measures, and Equilibrium Thermodynamics
Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 129-138
Abstract
pdf (207.72 Kb)
The questions of justification of the Gibbs canonical distribution for systems with elastic impacts are discussed. A special attention is paid to the description of probability measures with densities depending on the system energy.
Citation:
Kozlov V. V., Billiards, Invariant Measures, and Equilibrium Thermodynamics, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 129-138
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:
Mamaev I. S., Borisov A. V., 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
The restricted two-body problem and the kepler problem in the constant curvature spaces
Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, pp. 112-124
Abstract
pdf (807.85 Kb)
In this work we carry out the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$, and construct the analogues of Delaunau variables. We consider the problem of motion of a mass point in the field of moving Newtonian center on $S^2$ and $L^2$. The perihelion deviation is derived by the method of perturbation theory under the small curvature, and a numerical investigation is made, using anology of this problem with rigid body dynamics.
Citation:
Chernoivan V. A., Mamaev I. S., The restricted two-body problem and the kepler problem in the constant curvature spaces, Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, pp. 112-124
Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom
Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, pp. 44-54
Abstract
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Traditional derivation of Gibbs canonical distribution and the justification of thermodynamics are based on the assumption concerning an isoenergetic ergodicity of a system of n weakly interacting identical subsystems and passage to the limit $n \to\infty$. In the presented work we develop another approach to these problems assuming that n is fixed and $n \geqslant 2$. The ergodic hypothesis (which frequently is not valid due to known results of the KAM-theory) is substituted by a weaker assumption that the perturbed system does not have additional first integrals independent of the energy integral. The proof of nonintegrability of perturbed Hamiltonian systems is based on the Poincare method. Moreover, we use the natural Gibbs assumption concerning a thermodynamic equilibrium of bsystems at vanishing interaction. The general results are applied to the system of the weakly connected pendula. The averaging with respect to the Gibbs measure allows to pass from usual dynamics of mechanical systems to the classical thermodynamic model.
Citation:
Kozlov V. V., Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom, Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, pp. 44-54
Libration points in spaces $S^2$ and $L^2$
Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 91-103
Abstract
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We consider two-body problem and restricted three-body problem in spaces $S^2$ and $L^2$. For two-body problem we have showed the absence of exponential instability of partiбular solutions relevant to roundabout motion on the plane. New libration points are found, and the dependence of their positions on parameters of a system is explored. The regions of existence of libration points in space of parameters were constructed. Basing on a examination of the Hill's regions we found the qualitative estimation of stability of libration points was produced.
Citation:
Kilin A. A., Libration points in spaces $S^2$ and $L^2$, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 91-103
Lie algebras in vortex dynamics and celestial mechanics — IV
Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 23-50
Abstract
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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
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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
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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
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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
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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
Integrability of the Problems of Motion of a Particle in Constant Curvature Spaces in the Presence of Magnetic Monopole and in the Presence of Two Fixed Newtonian Centers
Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, pp. 75-78
Abstract
pdf (4.6 Mb)
Citation:
Mamaev I. S., Integrability of the Problems of Motion of a Particle in Constant Curvature Spaces in the Presence of Magnetic Monopole and in the Presence of Two Fixed Newtonian Centers, Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, pp. 75-78
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
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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
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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
Averaging in a neighborhood of stable invariant tori
Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 41-46
Abstract
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We analyse the operation of averaging of smooth functions along exact trajectories of dynamic systems in a neighborhood of stable nonresonance invariant tori. It is shown that there exists the first integral after the averaging; however in the typical situation the mean value is discontinuous or even not everywhere defind. If the temporal mean were a smooth function it would take its stationary values in the points of nondegenerate invariant tori. We demonstrate that this result can be properly derived if we change the operations of averaging and differentiating with respect to the initial data by their places. However, in general case for nonstable tori this property is no longer preserved. We also discuss the role of the reducibility condition of the invariant tori and the possibility of the generalization for the case of arbitrary compact invariant manifolds on which the initial dynamic system is ergodic.
Citation:
Kozlov V. V., Averaging in a neighborhood of stable invariant tori, Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 41-46
Adiabatic Chaos in Rigid Body Dynamics
Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 65-78
Abstract
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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
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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
Closed Orbits and Chaotic Dynamics of a Charged Particle in a Periodic Electromagnetic Field
Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 3-12
Abstract
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We study motion of a charged particle on the two dimensional torus in a constant direction magnetic field. This analysis can be applied to the description of electron dynamics in metals, which admit a $2$-dimensional translation group (Bravais crystal lattice). We found the threshold magnetic value, starting from which there exist three closed Larmor orbits of a given energy. We demonstrate that if there are n lattice atoms in a primitive Bravais cell then there are $4+n$ different Larmor orbits in the nondegenerate case. If the magnetic field is absent the electron dynamics turns out to be chaotic, dynamical systems on the corresponding energy shells possess positive entropy in the case that the total energy is positive.
Citation:
Kozlov V. V., Closed Orbits and Chaotic Dynamics of a Charged Particle in a Periodic Electromagnetic Field, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 3-12
Necessary and Sufficient Conditions of Kirchhoff Equation Integrability
Regular and Chaotic Dynamics, 1996, vol. 1, no. 2, pp. 61-76
Abstract
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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
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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
Symmetries and Regular Behavior of Hamilton's Systems
Regular and Chaotic Dynamics, 1996, vol. 1, no. 1, pp. 3-14
Abstract
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The paper discusses relationship between regular behavior of Hamilton's systems and the existence a sufficient number of fields of symmetry. Some properties of quite regular schemes and their relationship with various characteristics of stochastic behavior are studied.
Citation:
Kozlov V. V., Symmetries and Regular Behavior of Hamilton's Systems, Regular and Chaotic Dynamics, 1996, vol. 1, no. 1, pp. 3-14
On two modified integrable problems of dynamics
Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1995, no. 6, pp. 102-105
Abstract
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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
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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.
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.