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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
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
Notes on new friction models and nonholonomic mechanics
Physics-Uspekhi, 2015, vol. 58, no. 12, pp. 1220-1222
Abstract
pdf (262.98 Kb)
This is a reply to the comment by V.F. Zhuravlev (see Usp. Fiz. Nauk 185 1337 (2015) [Phys. Usp. 58 (12) (2015)]) on the inadequacy of the nonholonomic model when applied to the rolling of rigid bodies. The model of nonholonomic mechanics is discussed. Using recent results as examples, it is shown that the validity and potential of the nonholonomic model are not inferior to those of other dynamics and friction models.
Keywords:
nonholonomic model, dry friction, rattleback, rolling motion of a rigid body
Citation:
Borisov A. V., Mamaev I. S., Notes on new friction models and nonholonomic mechanics, Physics-Uspekhi, 2015, vol. 58, no. 12, pp. 1220-1222
Figures of equilibrium of an inhomogeneous self-gravitating fluid
Celestial Mechanics and Dynamical Astronomy, 2015, vol. 122, no. 1, pp. 1-26
Abstract
pdf (651.34 Kb)
This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
Keywords:
Self-gravitating fluid, Confocal stratification, Homothetic stratification, Chaplygin problem, Axisymmetric equilibrium figures, Space of constant curvature
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Celestial Mechanics and Dynamical Astronomy, 2015, vol. 122, no. 1, pp. 1-26
On the loss of contact of the Euler disk
Nonlinear Dynamics, 2015, vol. 79, no. 4, pp. 2287-2294
Abstract
pdf (829.12 Kb)
This paper is an experimental investigation of a round uniform disk rolling on a horizontal surface. Two methods for experimentally determining the loss of contact of the rolling disk from the horizontal surface before its stop are proposed. Results of experiments for disks having different masses and manufactured from different materials are presented. Causes of ?microlosses of contact? detected in the processes of motion are discussed.
Keywords:
Euler?s disk, Loss of contact, Experiment
Citation:
Borisov A. V., Mamaev I. S., Karavaev Y. L., On the loss of contact of the Euler disk, Nonlinear Dynamics, 2015, vol. 79, no. 4, pp. 2287-2294
Qualitative Analysis of the Dynamics of a Wheeled Vehicle
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751
Abstract
pdf (445.93 Kb)
This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined.
Keywords:
nonholonomic constraint, system dynamics, wheeled vehicle, Chaplygin system
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A., Qualitative Analysis of the Dynamics of a Wheeled Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 739-751
On the Hadamard ? Hamel Problem and the Dynamics of Wheeled Vehicles
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 752-766
Abstract
pdf (265.93 Kb)
In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.
Keywords:
nonholonomic constraint, wheeled vehicle, reduction, equations of motion
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., On the Hadamard ? Hamel Problem and the Dynamics of Wheeled Vehicles, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 752-766
Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
Abstract
pdf (640.12 Kb)
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626
Symmetries and Reduction in Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 553-604
Abstract
pdf (539.38 Kb)
This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.
Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane
Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 518-541
Abstract
pdf (516.92 Kb)
In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords:
dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation:
Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V., Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 518-541
The Jacobi Integral in Nonholonomic Mechanics
Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 383-400
Abstract
pdf (990.04 Kb)
In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
Keywords:
nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem
Citation:
Borisov A. V., Mamaev I. S., Bizyaev I. A., The Jacobi Integral in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 383-400
Dynamics and Control of an Omniwheel Vehicle
Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 153-172
Abstract
pdf (1.11 Mb)
A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained.
Borisov A. V., Kilin A. A., Mamaev I. S., Dynamics and Control of an Omniwheel Vehicle, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 153-172
Symmetries and Reduction in Nonholonomic Mechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 763?823
Abstract
pdf (909.32 Kb)
This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.
Borisov A. V., Mamaev I. S., Symmetries and Reduction in Nonholonomic Mechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 763?823
Topology and Bifurcations in Nonholonomic Mechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 735?762
Abstract
pdf (561.73 Kb)
This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie ?Poisson bracket of rank 2. This Lie ? Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.
Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and Bifurcations in Nonholonomic Mechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 735?762
On the dynamics of a body with an axisymmetric base sliding on a rough plane
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 547-577
Abstract
pdf (2.38 Mb)
In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords:
dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation:
Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 3, pp. 547-577
The Jacobi Integral in NonholonomicMechanics
Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 377-396
Abstract
pdf (1.9 Mb)
In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
Keywords:
nonholonomic constraint, Jacobi integral, Chaplygin sleigh, rotating table, Suslov problem
Citation:
Borisov A. V., Mamaev I. S., Bizyaev I. A., The Jacobi Integral in NonholonomicMechanics, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 2, pp. 377-396
Geometrisation of Chaplygin's reducing multiplier theorem
Nonlinearity, 2015, vol. 28, no. 7, pp. 2307?2318
Abstract
pdf (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
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
The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin?s Top
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
Abstract
pdf (1.29 Mb)
In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.
Keywords:
rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin?s Top, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733
The Dynamics of Three Vortex Sources
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701
Abstract
pdf (244.27 Kb)
In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system?s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane
Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 607-634
Abstract
pdf (965.59 Kb)
In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
Keywords:
dry friction, linear pressure distribution, planar motion, Coulomb law
Citation:
Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S., The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 607-634
Superintegrable Generalizations of the Kepler and Hook Problems
Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434
Abstract
pdf (300.95 Kb)
In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.
Keywords:
superintegrable systems, Kepler and Hook problems, isomorphism, central projection, reduction, highest degree polynomial superintegrals
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Superintegrable Generalizations of the Kepler and Hook Problems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 415-434
The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside
Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 198-213
Abstract
pdf (241.48 Kb)
In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.
Keywords:
nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 198-213
Paul 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.
On the dynamics of a body with an axisymmetric base sliding on a rough plane
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 483-495
Abstract
pdf (667.35 Kb)
In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system?s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.
Keywords:
dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation:
Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 4, pp. 483-495
Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Abstract
pdf (1.3 Mb)
We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.
Keywords:
Chaplygin?s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
Citation:
Borisov A. V., Kazakov A. O., Sataev I. R., Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 361-380
Invariant Measure and Hamiltonization of Nonholonomic Systems
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 355-359
Abstract
pdf (283.75 Kb)
This paper discusses new unresolved problems of nonholonomic mechanics. Hypotheses of the possibility of Hamiltonization and the existence of an invariant measure for such systems are advanced.
Borisov A. V., Mamaev I. S., Invariant Measure and Hamiltonization of Nonholonomic Systems, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 355-359
The dynamics of three vortex sources
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 319-327
Abstract
pdf (413.32 Kb)
In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system?s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
Bizyaev I. A., Borisov A. V., Mamaev I. S., The dynamics of three vortex sources, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 319-327
Figures of equilibrium of an inhomogeneous self-gravitating fluid
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 73-100
Abstract
pdf (492.78 Kb)
This paper 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
The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid
Fluid Dynamics Research, 2014, vol. 46, no. 3, 031415, 16 pp.
Abstract
pdf (746.69 Kb)
We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible and viscous fluid. Using the numerical simulation of the Navier?Stokes equations, we confirm the existence of leapfrogging of three equal vortex rings and suggest the possibility of detecting it experimentally. We also confirm the existence of leapfrogging of two vortex rings with opposite-signed vorticities in a viscous fluid.
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Tenenev V. A., The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid , Fluid Dynamics Research, 2014, vol. 46, no. 3, 031415, 16 pp.
The Problem of Drift and Recurrence for the Rolling Chaplygin Ball
Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 832-859
Abstract
pdf (702.93 Kb)
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
Borisov A. V., Kilin A. A., Mamaev I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 832-859
The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity
Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 490-496
Abstract
pdf (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 Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere
Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 277-328
Abstract
pdf (2.69 Mb)
In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body?s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.
Borisov A. V., Mamaev I. S., Bizyaev I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regular and Chaotic Dynamics, 2013, vol. 18, no. 3, pp. 277-328
How to Control the Chaplygin Ball Using Rotors. II
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 144-158
Abstract
pdf (1.73 Mb)
In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.
Borisov A. V., Kilin A. A., Mamaev I. S., How to Control the Chaplygin Ball Using Rotors. II, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 144-158
The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 33-62
Abstract
pdf (857.35 Kb)
We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
Borisov A. V., Kilin A. A., Mamaev I. S., The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 33-62
The problem of drift and recurrence for the rolling Chaplygin ball
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 721-754
Abstract
pdf (875.6 Kb)
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of a reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
Borisov A. V., Kilin A. A., Mamaev I. S., The problem of drift and recurrence for the rolling Chaplygin ball, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 721-754
Geometrization of the Chaplygin reducing-multiplier theorem
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 4, pp. 627-640
Abstract
pdf (373.67 Kb)
This paper develops the theory 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 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 loss of contact of the Euler disk
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 499-506
Abstract
pdf (362.06 Kb)
The paper presents experimental investigation of a homogeneous circular disk rolling on a horizontal plane. In this paper two methods of experimental determination of the loss of contact between the rolling disk and the horizontal surface before the abrupt halt are proposed. Experimental results for disks of different masses and different materials are presented. The reasons for ?micro losses? of contact with surface revealed during the rolling are discussed.
Keywords:
Euler disk, loss of contact, experiment
Citation:
Borisov A. V., Mamaev I. S., Karavaev Y. L., On the loss of contact of the Euler disk, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 499-506
The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 141-202
Abstract
pdf (7.91 Mb)
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
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
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
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
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
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
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
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
Hassan Aref (1950?2011)
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Abstract
pdf (777.79 Kb)
Citation:
Borisov A. V., Meleshko V. V., Stremler M. A., van Heijst G. J., Hassan Aref (1950?2011), Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
On the Model of Non-holonomic Billiard
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 653-662
Abstract
pdf (199.9 Kb)
In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.
Keywords:
billiard, impact, point map, nonintegrability, periodic solution, nonholonomic constraint, integral of motion
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., On the Model of Non-holonomic Billiard, Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 653-662
Hassan Aref (1950?2011)
Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Abstract
pdf (777.79 Kb)
Citation:
Borisov A. V., Meleshko V. V., Stremler M. A., van Heijst G. J., Hassan Aref (1950?2011), Regular and Chaotic Dynamics, 2011, vol. 16, no. 6, pp. 671-684
Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 465-483
Abstract
pdf (643.15 Kb)
We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.
Keywords:
nonholonomic constraint, rolling motion, Chaplygin ball, integral, invariant measure
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 465-483
Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds
Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 443-464
Abstract
pdf (425.83 Kb)
The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics.
Bolsinov A. V., Borisov A. V., Mamaev I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regular and Chaotic Dynamics, 2011, vol. 16, no. 5, pp. 443-464
Hamiltonicity and integrability of the Suslov problem
Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 104-116
Abstract
pdf (239.81 Kb)
The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed.
Borisov A. V., Kilin A. A., Mamaev I. S., Hamiltonicity and integrability of the Suslov problem, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 104-116
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
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
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
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
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
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
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
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
The dynamics of a Chaplygin sleigh
Journal of Applied Mathematics and Mechanics, 2009, vol. 73, no. 2, pp. 156-161
Abstract
pdf (263.77 Kb)
The problem of the motion of a Chaplygin sleigh on horizontal and inclined surfaces is considered. The possibility of representing the equations of motion in Hamiltonian form and of integration using Liouville?s theorem (with a redundant algebra of integrals) is investigated. The asymptotics for the rectilinear uniformly accelerated sliding of a sleigh along the line of steepest descent are determined in the case of an inclined plane. The zones in the plane of the initial conditions, corresponding to a different behaviour of the sleigh, are constructed using numerical calculations. The boundaries of these domains are of a complex fractal nature, which enables a conclusion to be drawn concerning the probable character from of the dynamic behaviour.
Citation:
Borisov A. V., Mamaev I. S., The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 2009, vol. 73, no. 2, pp. 156-161
Superintegrable system on a sphere with the integral of higher degree
Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 615-620
Abstract
pdf (125.27 Kb)
We consider the motion of a material point on the surface of a sphere in the field of $2n + 1$ identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional $N$-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed.
Keywords:
superintegrable systems, systems with a potential, Hooke center
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Superintegrable system on a sphere with the integral of higher degree, Regular and Chaotic Dynamics, 2009, vol. 14, no. 6, pp. 615-620
Isomorphisms of geodesic flows on quadrics
Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 455-465
Abstract
pdf (376.58 Kb)
We consider several well-known isomorphisms between Jacobi?s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.
The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids
Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 179-217
Abstract
pdf (885.59 Kb)
The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors? original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).
Keywords:
liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids, Regular and Chaotic Dynamics, 2009, vol. 14, no. 2, pp. 179-217
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 18-41
Abstract
pdf (472.45 Kb)
Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a 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
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
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.
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
Explicit integration of one problem in nonholonomic mechanics
Doklady Physics, 2008, vol. 53, no. 10, pp. 525-528
Abstract
pdf (229.05 Kb)
Citation:
Borisov A. V., Mamaev I. S., Marikhin V. G., Explicit integration of one problem in nonholonomic mechanics, Doklady Physics, 2008, vol. 53, no. 10, pp. 525-528
Chaplygin ball over a fixed sphere: an explicit integration
Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 557-571
Abstract
pdf (282.96 Kb)
We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel–Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.
Borisov A. V., Fedorov Y. N., Mamaev I. S., Chaplygin ball over a fixed sphere: an explicit integration, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 557-571
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 443-490
Abstract
pdf (508.23 Kb)
This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.
Keywords:
nonholonomic systems, implementation of constraints, conservation laws, hierarchy of dynamics, explicit integration
Citation:
Borisov A. V., Mamaev I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems , Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 443-490
Stability of Steady Rotations in the Nonholonomic Routh Problem
Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 239-249
Abstract
pdf (392.42 Kb)
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.
Borisov A. V., Kilin A. A., Mamaev I. S., Stability of Steady Rotations in the Nonholonomic Routh Problem, Regular and Chaotic Dynamics, 2008, vol. 13, no. 4, pp. 239-249
Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 221-233
Abstract
pdf (491.62 Kb)
In this paper, we consider the transition to chaos in the phase portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotization are indicated: (1) the growth of the homoclinic structure and (2) the development of cascades of period doubling bifurcations. On the zero level of the area integral, an adiabatic behavior of the system (as the energy tends to zero) is noted. Meander tori induced by the break of the torsion property of the mapping are found.
Keywords:
motion of a rigid body, phase portrait, mechanism of chaotization, bifurcations
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 221-233
Absolute and Relative Choreographies in Rigid Body Dynamics
Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 204-220
Abstract
pdf (447.19 Kb)
For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev–Chaplygin case, and the Steklov solution. The "genealogy" of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
Keywords:
rigid-body dynamics, periodic solutions, continuation by a parameter, bifurcation
Citation:
Borisov A. V., Kilin A. A., Mamaev I. S., Absolute and Relative Choreographies in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 204-220
E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 497-513
Abstract
pdf (286.99 Kb)
Citation:
Borisov A. V., Gazizullina L., Ramodanov S. M., E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere, Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 497-513
Algebraic reduction of systems on two- and three-dimensional spheres
Russian Journal of Nonlinear Dynamics, 2008, vol. 4, no. 4, pp. 407-416
Abstract
pdf (180.6 Kb)
The paper develops further the algebraic-reduction method for $SO(4)$-symmetric systems 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
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
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
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 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
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
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
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
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
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
Strange Attractors in Rattleback Dynamics
Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393-403
Abstract
pdf (484.78 Kb)
This review is dedicated to the dynamics of the rattleback, a phenomenon with curious physical properties that is
studied in nonholonomic mechanics. All known analytical results are collected here, and some results of our numerical
simulation are presented. In particular, three-dimensional Poincare maps associated with dynamical systems are systematically investigated for the first time. It is shown that the loss of
stability of periodic and quasiperiodic solutions, which gives rise
to strange attractors, is typical of the three-dimensional maps related to rattleback dynamics. This explains some newly discovered properties of the rattleback related to the transition from regular to chaotic solutions at certain values of the physical parameters.
Citation:
Borisov A. V., Mamaev I. S., Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393-403
The Hess case in the dynamics of a rigid body
Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 2, pp. 227-235
Abstract
pdf (605.38 Kb)
In the paper we consider modifications of the Hess integral suitable for various forms of the equations of motion for rigid body. This integral occurs due to some additional symmetry properties of the equations of motion. Moreover, we discuss the general conditions under which the integral exists. Assuming that these conditions are satisfied, we discuss the reduction of the order of the equations, their explicit integration and a qualitative analysis of motion. For the first time, the paper indicates new counterparts of the Hess case for the problem of a gyroscope fixed in a gimbal suspension and for Chaplygin's equations describing the motion of a heavy rigid body in an ideal fluid.
Citation:
Borisov A. V., Mamaev I. S., The Hess case in the dynamics of a rigid body, Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 2, pp. 227-235
Motion of a circular cylinder and $n$ point vortices in a perfect fluid
Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 449-462
Abstract
pdf (585.43 Kb)
The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.
Citation:
Borisov A. V., Mamaev I. S., Ramodanov S. M., Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 449-462
Dynamics of rolling disk
Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 201-212
Abstract
pdf (648.4 Kb)
In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., Dynamics of rolling disk, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 201-212
An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid
Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 163-166
Abstract
pdf (90.36 Kb)
In this paper we present the nonlinear Poisson structure and two first integrals in the problem on plane motion of circular cylinder and $N$ point vortices in the ideal fluid. A priori this problem is not Hamiltonian. The particular case $N = 1$, i.e. the problem on interaction of cylinder and vortex, is integrable.
Citation:
Borisov A. V., Mamaev I. S., An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 163-166
Obstacle to the Reduction of Nonholonomic Systems to the Hamiltonian Form
Doklady Physics, 2002, vol. 47, no. 12, pp. 892-894
Abstract
pdf (36.81 Kb)
In the paper, classical Chaplygin's problem on an unbalanced ball that is rolling without slipping on a plane is considered. Using numerical simulations, we have shown the possibility of mixing integrable non-holonomic systems on invariant tori. Therein lies the obstacle for this system to be Hamiltonian. It should be noted, that, nevertheless, in such systems there is an invariant measure and conservation of energy.
Citation:
Borisov A. V., Mamaev I. S., Obstacle to the Reduction of Nonholonomic Systems to the Hamiltonian Form, Doklady Physics, 2002, vol. 47, no. 12, pp. 892-894
A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid
Doklady Physics, 2002, vol. 47, no. 7, pp. 544-547
Abstract
pdf (85.97 Kb)
The problem of rolling motion without slipping of an unbalanced ball on 1) an arbitrary ellipsoid and 2) an ellipsoid of revolution is considered. In his famous treatise E. Routh showed that the problem of rolling motion of a body on a surface of revolution even in the presence of axisymmetrical potential fields is integrable. In case 1, we present a new integral of motion. New solutions expressed in elementary functions are found in case 2.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid, Doklady Physics, 2002, vol. 47, no. 7, pp. 544-547
Compatible Poisson Brackets on Lie Algebras
Mathematical Notes, 2002, vol. 72, no. 1, pp. 10-30
Abstract
pdf (244.55 Kb)
We discuss the relationship between the representation of an integrable system as an $L-A$-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.
The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 201-219
Abstract
pdf (628.29 Kb)
The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.
Citation:
Borisov A. V., Mamaev I. S., Kilin A. A., The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 201-219
The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 177-200
Abstract
pdf (784.15 Kb)
In this paper we study the cases of existence of an invariant measure, additional first integrals, and a Poisson structure in the problem of rigid body's rolling without sliding on a plane and a sphere. The problem of rigid body's motion on a plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of the dynamically symmetrical body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of a body of revolution and the motion of round disk with sharp edge on a sphere. In both cases the systems are Euler–Jacobi integrable and have additional integrals and invariant measure. It can be shown that by an appropriate change of time (determined by reducing multiplier), the reduced system is a Hamiltonian one. Here we consider some particular cases when the integrals and the invariant measure can be presented as finite algebraic expressions. We also consider a generalized problem of rolling of a dynamically nonsymmetric Chaplygin ball. The results of investigations are summarized in tables to illustrate the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure.
Citation:
Borisov A. V., Mamaev I. S., The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 2, pp. 177-200
On the History of the Development of the Nonholonomic Dynamics
Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 43-47
Abstract
pdf (183.63 Kb)
The main directions in the development of the nonholonomic dynamics are briefly considered in this paper. The first direction is connected with the general formalizm of the equations of dynamics that differs from the Lagrangian and Hamiltonian methods of the equations of motion's construction. The second direction, substantially more important for dynamics, includes investigations concerning the analysis of the specific nonholonomic problems. We also point out rather promising direction in development of nonholonomic systems that is connected with intensive use of the modern computer-aided methods.
Citation:
Borisov A. V., Mamaev I. S., On the History of the Development of the Nonholonomic Dynamics, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 43-47
Generalization of the Goryachev–Chaplygin Case
Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 21-30
Abstract
pdf (289.22 Kb)
In this paper we present a generalization of the Goryachev–Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.
Citation:
Borisov A. V., Mamaev I. S., Generalization of the Goryachev–Chaplygin Case, Regular and Chaotic Dynamics, 2002, vol. 7, no. 1, pp. 21-30
A New Integrable Case on $so(4)$
Doklady Physics, 2001, vol. 46, no. 12, pp. 888-889
Abstract
pdf (28.12 Kb)
In his paper "New integrable Case of Kirchoff's equations" (Theor. and Math. Physics. 2001) V.V. Sokolov proposed a new integrable case with additional integral of fourth degree. We have shown that the integral can be written in a more natural form and consider its generalization to a bundle of Poisson brackets.
Citation:
Borisov A. V., Mamaev I. S., Sokolov V. V., A New Integrable Case on $so(4)$, Doklady Physics, 2001, vol. 46, no. 12, pp. 888-889
Chaplygin's Ball Rolling Problem Is Hamiltonian
Mathematical Notes, 2001, vol. 70, no. 5, pp. 720-723
Abstract
pdf (339.65 Kb)
In this paper we introduce a new nonlinear Poisson bracket in the problem of rolling motion of a Chaplygin's ball. Thus, upon some change of time, the equations of motion become Hamiltonian. We have also established that the trajectories of this system are isomorphic to the trajectories of the Braden system which describes the motion of a point on a two-dimensional sphere in a potential field; using the isomorphism, we have shown that in the Chaplygin problem the variables are separable.
Keywords:
Chaplygin's ball rolling problem, potential force field, Poisson bracket, Euler\,--\,Jacobi theorem
Citation:
Borisov A. V., Mamaev I. S., Chaplygin's Ball Rolling Problem Is Hamiltonian, Mathematical Notes, 2001, vol. 70, no. 5, pp. 720-723
Euler?Poisson Equations and Integrable Cases
Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 253-276
Abstract
pdf (1.41 Mb)
In this paper we propose a new approach to the study of integrable cases based on intensive computer methods' application. We make a new investigation of Kovalevskaya and Goryachev–Chaplygin cases of Euler–Poisson equations and obtain many new results in rigid body dynamics in absolute space. Also we present the visualization of some special particular solutions.
Citation:
Borisov A. V., Mamaev I. S., Euler?Poisson Equations and Integrable Cases, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 253-276
Kovalevskaya top and generalizations of integrable systems
Regular and Chaotic Dynamics, 2001, vol. 6, no. 1, pp. 1-16
Abstract
pdf (286.68 Kb)
Generalizations of the Kovalevskaya, Chaplygin, Goryachev–Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and action-angle variables is described. Another integration method for the Kovalevskaya top on the bundle is found. This method uses a coordinate transformation that reduces the Kovalevskaya system to the Neumann system. The Kolosov analogy is considered. A generalization of a recent Gaffet system to the bundle of Poisson brackets is obtained at the end of the paper.
Citation:
Borisov A. V., Mamaev I. S., Kholmskaya A. G., Kovalevskaya top and generalizations of integrable systems, Regular and Chaotic Dynamics, 2001, vol. 6, no. 1, pp. 1-16
Stability of Thomson's Configurations of Vortices on a Sphere
Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200
Abstract
pdf (284.39 Kb)
In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated.
Citation:
Borisov A. V., Kilin A. A., Stability of Thomson's Configurations of Vortices on a Sphere, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200
Some comments to the paper by A.M.Perelomov "A note on geodesics on ellipsoid" RCD 2000 5(1) 89-91
Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 92-94
Abstract
pdf (178.43 Kb)
Citation:
Borisov A. V., Mamaev I. S., Some comments to the paper by A.M.Perelomov "A note on geodesics on ellipsoid" RCD 2000 5(1) 89-91, Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 92-94
The Kovalevskaya case and new integrable systems of dynamics
Vestnik molodyh uchenyh. "Prikladnaya matematika i mehanika", 2000, no. 4, pp. 13-25
Abstract
pdf (910.69 Kb)
Citation:
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
Lie algebras in vortex dynamics and celestial mechanics — IV
Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 23-50
Abstract
pdf (1.16 Mb)
1.Classificaton of the algebra of $n$ vortices on a plane
2.Solvable problems of vortex dynamics
3.Algebraization and reduction in a three-body problem
The work [13] introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie–Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works [14,15]. In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem.
Citation:
Bolsinov A. V., Borisov A. V., Mamaev I. S., Lie algebras in vortex dynamics and celestial mechanics — IV, Regular and Chaotic Dynamics, 1999, vol. 4, no. 1, pp. 23-50
Dynamics of Three Vortices on a Plane and a Sphere — III. Noncompact Case. Problems of Collaps and Scattering
Regular and Chaotic Dynamics, 1998, vol. 3, no. 4, pp. 74-86
Abstract
pdf (657.83 Kb)
In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and scattering of vortices and obtained the conditions of realization. We completed the bifurcation analysis and investigated the dependence of stability in linear approximation and frequency of rotation in relative coordinates for collinear and Thomson's configurations from value of a full moment and indicated the geometric interpretation for characteristic situations. We constructed a phase portrait and geometric projection for an integrable configuration of four vortices on a plane.
Citation:
Borisov A. V., Lebedev V. G., Dynamics of Three Vortices on a Plane and a Sphere — III. Noncompact Case. Problems of Collaps and Scattering, Regular and Chaotic Dynamics, 1998, vol. 3, no. 4, pp. 74-86
Dynamics of three vortices on a plane and a sphere — II. General compact case
Regular and Chaotic Dynamics, 1998, vol. 3, no. 2, pp. 99-114
Abstract
pdf (493.96 Kb)
Integrable problem of three vorteces on a plane and sphere are considered. The classification of Poisson structures is carried out. We accomplish the bifurcational analysis using the variables introduced in previous part of the work.
Citation:
Borisov A. V., Lebedev V. G., Dynamics of three vortices on a plane and a sphere — II. General compact case, Regular and Chaotic Dynamics, 1998, vol. 3, no. 2, pp. 99-114
Dynamics and statics of vortices on a plane and a sphere - I
Regular and Chaotic Dynamics, 1998, vol. 3, no. 1, pp. 28-38
Abstract
pdf (223.68 Kb)
In the present paper a description of a problem of point vortices on a plane and a sphere in the "internal" variables is discussed. The hamiltonian equations of motion of vortices on a plane are built on the Lie–Poisson algebras, and in the case of vortices on a sphere on the quadratic Jacobi algebras. The last ones are obtained by deformation of the corresponding linear algebras. Some partial solutions of the systems of three and four vortices are considered. Stationary and static vortex configurations are found.
Citation:
Borisov A. V., Pavlov A. E., Dynamics and statics of vortices on a plane and a sphere - I, Regular and Chaotic Dynamics, 1998, vol. 3, no. 1, pp. 28-38
A degenerate Poisson Structure and Lie algebras in the Two Problems of Hamiltonian Dynamics
Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, pp. 71-74
Abstract
pdf (3.81 Mb)
Citation:
Borisov A. V., Mamaev I. S., A degenerate Poisson Structure and Lie algebras in the Two Problems of Hamiltonian Dynamics, Proceedings of IX Seminar "Gravitational Energy and Gravity Waves", 1998, pp. 71-74
Kovalevskaya's method in rigid body dynamics
Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 1, pp. 27-32
Abstract
pdf (701.12 Kb)
An example from the field of rigid body dynamics, possessing a natural physical justification, is presented. The behaviour of the solutions of the equations of motion in the real domain, whatever the initial data, is regular; nevertheless, depending on the values of a certain control parameter, the solution of the system may branch in the complex time plane, and the system will have multi-valued first integrals. A denumerable sequence of single-valued polynomial integrals of arbitrarily high even degree is found (unlike Kovalevskaya's case, in which the degree of the first integral of the Euler–Poisson equations is four). As an extension, a system from non-holonomic mechanics is considered.
Citation:
Borisov A. V., Tsygvintsev A. V., Kovalevskaya's method in rigid body dynamics, Journal of Applied Mathematics and Mechanics, 1997, vol. 61, no. 1, pp. 27-32
Non-linear Poisson brackets and isomorphisms in dynamics
Regular and Chaotic Dynamics, 1997, vol. 2, no. 3-4, pp. 72-89
Abstract
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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
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
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
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
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.