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2013
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Alexey Borisov

Alexey Borisov
1, Universitetskaya str., Izhevsk 426034, Russia
Institute of Computer Science, Russia

Director, Institute of Computer Science, Udmurt State University, Russia
Professor, Department of Computational Mechanics at UdSU
Director, Scientific and Publishing Center "Regular and Chaotic Dynamics"

Born: March 27, 1965 in Moscow, Russia
1984-1989: student of N.E. Bauman Moscow State Technical University (MSTU).
1992: Ph.D. (candidate of science). Thesis title: "Nonintegrability of Kirchhoff equations and related problems in rigid body dynamics", M.V. Lomonosov Moscow State University.
2001: Doctor in physics and mathematics. Thesis title: "Poisson structures and Lie Algebras in Hamiltonian Mechanics", M.V. Lomonosov Moscow State University.

Positions held:
1996-2001: Head of the Laboratory of Dynamical Chaos and Nonlinearity at the Udmurt State University, Izhevsk.
since 1998: Director of the Scientific and Publishing Center "Regular and Chaotic Dynamics".
since 2002: Head of the Laboratory of Nonlinear Dynamics at A.A. Blagonravov Mechanical Engineering Research Institute of Russian Academy of Sciences, Moscow.
since 2002: Head of the Department of the Mathematical Methods in Nonlinear Dynamics at the Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences.
since 2010: Vice-rector for information and computer technology of UdSU

Member of the Russian National Committee on Theoretical and Applied Mechanics (2001)
Corresponding Member of Russian Academy of Natural Sciences (2006)

Сo-founder and associate editor of the International Scientific Journal "Regular and Chaotic Dynamics"; co-founder and editor-in-chief of "Nelineinaya Dinamika" (Russian Journal of Nonlinear Dynamics).

In 2012 A.V.Borisov and I.S.Mamaev received the Sofia Kovalevskaya Award for a series of monographs devoted to the integrable systems of Hamiltonian mechanics.

Research supervision of 8 candidates of science and 3 doctors of science (I.S. Mamaev, A.A. Kilin, S.M. Ramodanov).

Publications:

Borisov A. V., Mamaev I. S.
An Inhomogeneous Chaplygin Sleigh
2017, vol. 22, no. 4, pp.  435-447
Abstract
In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics.
Keywords: Chaplygin sleigh, inhomogeneous nonholonomic constraints, conservation laws, qualitative analysis, resonance
Citation: Borisov A. V., Mamaev I. S.,  An Inhomogeneous Chaplygin Sleigh, Regular and Chaotic Dynamics, 2017, vol. 22, no. 4, pp. 435-447
DOI:10.1134/S1560354717040062
Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Hess–Appelrot Case and Quantization of the Rotation Number
2017, vol. 22, no. 2, pp.  180-196
Abstract
This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords: invariant submanifold, rotation number, Cantor ladder, limit cycles
Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The Hess–Appelrot Case and Quantization of the Rotation Number, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
DOI:10.1134/S156035471702006X
Borisov A. V., Kuznetsov S. P.
Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts
2016, vol. 21, no. 7-8, pp.  792-803
Abstract
For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.
Keywords: Chaplygin sleigh, nonholonomic mechanics, attractor, chaos, bifurcation
Citation: Borisov A. V., Kuznetsov S. P.,  Regular and Chaotic Motions of a Chaplygin Sleigh under Periodic Pulsed Torque Impacts, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 792-803
DOI:10.1134/S1560354716070029
Borisov A. V., Kazakov A. O., Pivovarova E. N.
Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
2016, vol. 21, no. 7-8, pp.  885-901
Abstract
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of period-doubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Keywords: Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact
Citation: Borisov A. V., Kazakov A. O., Pivovarova E. N.,  Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901
DOI:10.1134/S156035471607011X
Borisov A. V., Kazakov A. O., Sataev I. R.
Spiral Chaos in the Nonholonomic Model of a Chaplygin Top
2016, vol. 21, no. 7-8, pp.  939-954
Abstract
This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn about the influence of “strangeness” of the attractor on the motion pattern of the top.
Keywords: nonholonomic constraint, spiral chaos, discrete spiral attractor
Citation: Borisov A. V., Kazakov A. O., Sataev I. R.,  Spiral Chaos in the Nonholonomic Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954
DOI:10.1134/S1560354716070157
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
2016, vol. 21, no. 6, pp.  759-774
Abstract
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3, 6, 14)$, the other is defined by two generatrices and growth vector $(2, 3, 5, 8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords: sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
Citation: Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.,  Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
DOI:10.1134/S1560354716060125
Borisov A. V., Mamaev I. S., Bizyaev I. A.
The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
2016, vol. 21, no. 5, pp.  556-580
Abstract
In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.
Keywords: celestial mechanics, space of constant curvature, reduction, rigid body dynamics, Poincaré section
Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A.,  The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580
DOI:10.1134/S1560354716050075
Borisov A. V., Mamaev I. S., Bizyaev I. A.
Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period
2016, vol. 21, no. 4, pp.  455-476
Abstract
In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
Keywords: nonholonomic mechanics, nonholonomic constraint, d’Alembert–Lagrange principle, permutation relations
Citation: Borisov A. V., Mamaev I. S., Bizyaev I. A.,  Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
DOI:10.1134/S1560354716040055
Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Vortex Sources in a Deformation Flow
2016, vol. 21, no. 3, pp.  367-376
Abstract
This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse.
Keywords: integrability, vortex sources, reduction, deformation flow
Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The Dynamics of Vortex Sources in a Deformation Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376
DOI:10.1134/S1560354716030084
Borisov A. V., Mamaev I. S.
Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics
2016, vol. 21, no. 2, pp.  232-248
Abstract
The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré – Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).
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
DOI:10.1134/S1560354716020064
Bizyaev I. A., Borisov A. V., Mamaev I. S.
Dynamics of the Chaplygin Sleigh on a Cylinder
2016, vol. 21, no. 1, pp.  136-146
Abstract
This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found.

In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.

Keywords: Chaplygin sleigh, invariant measure, nonholonomic mechanics
Citation: 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
DOI:10.1134/S1560354716010081
Borisov A. V., Mamaev I. S., Kilin A. A., Bizyaev I. A.
Qualitative Analysis of the Dynamics of a Wheeled Vehicle
2015, vol. 20, no. 6, pp.  739-751
Abstract
This paper is concerned with the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined.
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
DOI:10.1134/S156035471506009X
Borisov A. V., Kilin A. A., Mamaev I. S.
On the Hadamard – Hamel Problem and the Dynamics of Wheeled Vehicles
2015, vol. 20, no. 6, pp.  752-766
Abstract
In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.
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
DOI:10.1134/S1560354715060106
Borisov A. V., Karavaev Y. L., Mamaev I. S., Erdakova N. N., Ivanova T. B., Tarasov V. V.
Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane
2015, vol. 20, no. 5, pp.  518-541
Abstract
In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
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
DOI:10.1134/S1560354715050020
Borisov A. V., Mamaev I. S.
Symmetries and Reduction in Nonholonomic Mechanics
2015, vol. 20, no. 5, pp.  553-604
Abstract
This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.
Keywords: reduction, symmetry, tensor invariant, first integral, symmetry group, symmetry field, nonholonomic constraint, Noether theorem
Citation: Borisov A. V., Mamaev I. S.,  Symmetries and Reduction in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 553-604
DOI:10.1134/S1560354715050044
Bizyaev I. A., Borisov A. V., Kazakov A. O.
Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors
2015, vol. 20, no. 5, pp.  605-626
Abstract
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.
Keywords: Suslov problem, nonholonomic constraint, reversal, strange attractor
Citation: 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
DOI:10.1134/S1560354715050056
Borisov A. V., Mamaev I. S., Bizyaev I. A.
The Jacobi Integral in Nonholonomic Mechanics
2015, vol. 20, no. 3, pp.  383-400
Abstract
In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.
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
DOI:10.1134/S1560354715030107
Borisov A. V., Kilin A. A., Mamaev I. S.
Dynamics and Control of an Omniwheel Vehicle
2015, vol. 20, no. 2, pp.  153-172
Abstract
A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained.
Keywords: omniwheel, roller-bearing wheel, nonholonomic constraint, dynamical system, invariant measure, integrability, controllability
Citation: 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
DOI:10.1134/S1560354715020045
Borisov A. V., Erdakova N. N., Ivanova T. B., Mamaev I. S.
The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane
2014, vol. 19, no. 6, pp.  607-634
Abstract
In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.
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
DOI:10.1134/S1560354714060021
Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Three Vortex Sources
2014, vol. 19, no. 6, pp.  694-701
Abstract
In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
Keywords: integrability, vortex sources, shape sphere, reduction, homothetic configurations
Citation: Bizyaev I. A., Borisov A. V., Mamaev I. S.,  The Dynamics of Three Vortex Sources, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701
DOI:10.1134/S1560354714060070
Borisov A. V., Kazakov A. O., Sataev I. R.
The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top
2014, vol. 19, no. 6, pp.  718-733
Abstract
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
DOI:10.1134/S1560354714060094
Bizyaev I. A., Borisov A. V., Mamaev I. S.
Superintegrable Generalizations of the Kepler and Hook Problems
2014, vol. 19, no. 3, pp.  415-434
Abstract
In this paper we consider superintegrable systems which are an immediate generalization of the Kepler and Hook problems, both in two-dimensional spaces — the plane $\mathbb{R}^2$ and the sphere $S^2$ — and in three-dimensional spaces $\mathbb{R}^3$ and $S^3$. Using the central projection and the reduction procedure proposed in [21], we show an interrelation between the superintegrable systems found previously and show new ones. In all cases the superintegrals are presented in explicit form.
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
DOI:10.1134/S1560354714030095
Bizyaev I. A., Borisov A. V., Mamaev I. S.
The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside
2014, vol. 19, no. 2, pp.  198-213
Abstract
In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.
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
DOI:10.1134/S156035471402004X
Borisov A. V., Kudryashov N. A.
Paul Painlevé and His Contribution to Science
2014, vol. 19, no. 1, pp.  1-19
Abstract
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.
Keywords: mathematician, politician, Painlevé equations; Painlevé transcendents; Painlevé paradox
Citation: Borisov A. V., Kudryashov N. A.,  Paul Painlevé and His Contribution to Science, Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 1-19
DOI:10.1134/S1560354714010018
Borisov A. V., Kilin A. A., Mamaev I. S.
The Problem of Drift and Recurrence for the Rolling Chaplygin Ball
2013, vol. 18, no. 6, pp.  832-859
Abstract
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.
Keywords: nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus
Citation: 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
DOI:10.1134/S1560354713060166
Borisov A. V., Mamaev I. S.
The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity
2013, vol. 18, no. 5, pp.  490-496
Abstract
We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.
Keywords: vortex motion, nonholonomic constraint, Chaplygin ball, invariant measure, integrability, rigid body, ideal fluid
Citation: 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
DOI:10.1134/S156035471305002X
Borisov A. V., Mamaev I. S.
Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere
2013, vol. 18, no. 4, pp.  356-371
Abstract
A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.
Keywords: integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical periodic solution
Citation: 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
DOI:10.1134/S1560354713040035
Borisov A. V., Mamaev I. S., Bizyaev I. A.
The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere
2013, vol. 18, no. 3, pp.  277-328
Abstract
In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.
Keywords: nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution
Citation: 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
DOI:10.1134/S1560354713030064
Borisov A. V., Kilin A. A., Mamaev I. S.
The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
2013, vol. 18, no. 1-2, pp.  33-62
Abstract
We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.
Keywords: ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics
Citation: 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
DOI:10.1134/S1560354713010036
Borisov A. V., Kilin A. A., Mamaev I. S.
How to Control the Chaplygin Ball Using Rotors. II
2013, vol. 18, no. 1-2, pp.  144-158
Abstract
In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.
Keywords: non-holonomic constraint, control, dry friction, viscous friction, stability, periodic solutions
Citation: 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
DOI:10.1134/S1560354713010103
Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V.
Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback
2012, vol. 17, no. 6, pp.  512-532
Abstract
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.
Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, strange attractor, Lyapunov exponents, bifurcation, fractal dimension
Citation: Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V.,  Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
DOI:10.1134/S1560354712060044
Bolsinov A. V., Borisov A. V., Mamaev I. S.
Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals
2012, vol. 17, no. 6, pp.  571-579
Abstract
In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
Keywords: non-holonomic constraint, Liouville foliation, invariant torus, invariant measure, integrability
Citation: 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
DOI:10.1134/S1560354712060081
Bolsinov A. V., Borisov A. V., Mamaev I. S.
The Bifurcation Analysis and the Conley Index in Mechanics
2012, vol. 17, no. 5, pp.  457-478
Abstract
The paper is devoted to the bifurcation analysis and the Conley index in Hamiltonian dynamical systems. We discuss the phenomenon of appearance (disappearance) of equilibrium points under the change of the Morse index of a critical point of a Hamiltonian. As an application of these techniques we find new relative equilibria in the problem of the motion of three point vortices of equal intensity in a circular domain.
Keywords: Morse index, Conley index, bifurcation analysis, bifurcation diagram, Hamiltonian dynamics, fixed point, relative equilibrium
Citation: 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
DOI:10.1134/S1560354712050073
Borisov A. V., Kilin A. A., Mamaev I. S.
How to Control Chaplygin’s Sphere Using Rotors
2012, vol. 17, no. 3-4, pp.  258-272
Abstract
In the paper we study the control of a balanced dynamically non-symmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.
Keywords: non-holonomic constraint, non-holonomic distribution, control, Chow–Rashevsky theorem, drift
Citation: 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
DOI:10.1134/S1560354712030045
Borisov A. V., Kilin A. A., Mamaev I. S.
Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support
2012, vol. 17, no. 2, pp.  170-190
Abstract
We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu.N.Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support – the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.
Keywords: nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, isomorphism, bifurcation analysis
Citation: 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
DOI:10.1134/S1560354712020062
Borisov A. V., Mamaev I. S.
Two Non-holonomic Integrable Problems Tracing Back to Chaplygin
2012, vol. 17, no. 2, pp.  191-198
Abstract
The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.
Keywords: non-holonomic constraint, integrability, invariant measure, gyroscope, quadrature, coupled rigid bodies
Citation: 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
DOI:10.1134/S1560354712020074
Borisov A. V., Kilin A. A., Mamaev I. S.
On the Model of Non-holonomic Billiard
2011, vol. 16, no. 6, pp.  653-662
Abstract
In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.
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
DOI:10.1134/S1560354711060062
Borisov A. V., Meleshko V. V., Stremler M. A., van Heijst G. J.
Hassan Aref (1950–2011)
2011, vol. 16, no. 6, pp.  671-684
Abstract
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
DOI:10.1134/S1560354711060086
Bolsinov A. V., Borisov A. V., Mamaev I. S.
Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds
2011, vol. 16, no. 5, pp.  443-464
Abstract
The problem of Hamiltonization of nonholonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in nonholonomic mechanics.
Keywords: conformally Hamiltonian system, nonholonomic system, invariant measure, periodic trajectory, invariant torus, integrable system
Citation: 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
DOI:10.1134/S1560354711050030
Borisov A. V., Kilin A. A., Mamaev I. S.
Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere
2011, vol. 16, no. 5, pp.  465-483
Abstract
We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of "clandestine" linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.
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
DOI:10.1134/S1560354711050042
Borisov A. V., Kilin A. A., Mamaev I. S.
Hamiltonicity and integrability of the Suslov problem
2011, vol. 16, no. 1-2, pp.  104-116
Abstract
The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed.
Keywords: Hamiltonian system, Poisson bracket, nonholonomic constraint, invariant measure, integrability
Citation: 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
DOI:10.1134/S1560354711010035
Borisov A. V., Mamaev I. S., Ramodanov S. M.
Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface
2010, vol. 15, no. 4-5, pp.  440-461
Abstract
The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.
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
DOI:10.1134/S1560354710040040
Borisov A. V., Kilin A. A., Mamaev I. S.
Superintegrable system on a sphere with the integral of higher degree
2009, vol. 14, no. 6, pp.  615-620
Abstract
We consider the motion of a material point on the surface of a sphere in the field of $2n + 1$ identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional $N$-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed.
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
DOI:10.1134/S156035470906001X
Borisov A. V., Mamaev I. S.
Isomorphisms of geodesic flows on quadrics
2009, vol. 14, no. 4-5, pp.  455-465
Abstract
We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability.
Keywords: quadric, geodesic flows, integrability, compactification, regularization, isomorphism
Citation: Borisov A. V., Mamaev I. S.,  Isomorphisms of geodesic flows on quadrics, Regular and Chaotic Dynamics, 2009, vol. 14, no. 4-5, pp. 455-465
DOI:10.1134/S1560354709040030
Borisov A. V., Mamaev I. S., Kilin A. A.
The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids
2009, vol. 14, no. 2, pp.  179-217
Abstract
The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).
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
DOI:10.1134/S1560354709020014
Borisov A. V., Kilin A. A., Mamaev I. S.
Multiparticle Systems. The Algebra of Integrals and Integrable Cases
2009, vol. 14, no. 1, pp.  18-41
Abstract
Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particle­interaction potential homogeneous of degree $\alpha = –2$ are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems.
Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle­ interaction potential homogeneous of degree $\alpha = –2$ are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.
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
DOI:10.1134/S1560354709010043
Borisov A. V., Fedorov Y. N., Mamaev I. S.
Chaplygin ball over a fixed sphere: an explicit integration
2008, vol. 13, no. 6, pp.  557-571
Abstract
We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel–Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems.
Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.
Keywords: Chaplygin ball, explicit integration, nonholonomic mechanics
Citation: 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
DOI:10.1134/S1560354708060063
Borisov A. V., Mamaev I. S.
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
2008, vol. 13, no. 5, pp.  443-490
Abstract
This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.
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
DOI:10.1134/S1560354708050079
Borisov A. V., Kilin A. A., Mamaev I. S.
Stability of Steady Rotations in the Nonholonomic Routh Problem
2008, vol. 13, no. 4, pp.  239-249
Abstract
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.
Keywords: nonholonomic constraint, stationary rotations, stability
Citation: 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
DOI:10.1134/S1560354708040011
Borisov A. V., Kilin A. A., Mamaev I. S.
Absolute and Relative Choreographies in Rigid Body Dynamics
2008, vol. 13, no. 3, pp.  204-220
Abstract
For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev–Chaplygin case, and the Steklov solution. The "genealogy" of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).
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
DOI:10.1134/S1560354708030064
Borisov A. V., Kilin A. A., Mamaev I. S.
Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point
2008, vol. 13, no. 3, pp.  221-233
Abstract
In this paper, we consider the transition to chaos in the phase portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotization are indicated: (1) the growth of the homoclinic structure and (2) the development of cascades of period doubling bifurcations. On the zero level of the area integral, an adiabatic behavior of the system (as the energy tends to zero) is noted. Meander tori induced by the break of the torsion property of the mapping are found.
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
DOI:10.1134/S1560354708030076
Borisov A. V., Kozlov V. V., Mamaev I. S.
Asymptotic stability and associated problems of dynamics of falling rigid body
2007, vol. 12, no. 5, pp.  531-565
Abstract
We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
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
DOI:10.1134/S1560354707050061
Borisov A. V., Mamaev I. S.
Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting
2007, vol. 12, no. 2, pp.  153-159
Abstract
Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.
Keywords: nonholonomic mechanics, reducing multiplier, hamiltonization, isomorphism
Citation: 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
DOI:10.1134/S1560354707010030
Borisov A. V., Mamaev I. S.
Interaction between Kirchhoff vortices and point vortices in an ideal fluid
2007, vol. 12, no. 1, pp.  68-80
Abstract
We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found by the variable separation method . A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.
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
DOI:10.1134/S1560354707010066
Borisov A. V., Mamaev I. S.
On the problem of motion of vortex sources on a plane
2006, vol. 11, no. 4, pp.  455-466
Abstract
Equations of motion of vortex sources (examined earlier by Fridman and Polubarinova) are studied, and the problems of their being Hamiltonian and integrable are discussed. A system of two vortex sources and three sources-sinks was examined. Their behavior was found to be regular. Qualitative analysis of this system was made, and the class of Liouville integrable systems is considered. Particular solutions analogous to the homothetic configurations in celestial mechanics are given.
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
DOI:10.1070/RD2006v011n04ABEH000363
Borisov A. V., Mamaev I. S.
Superintegrable systems on a sphere
2005, vol. 10, no. 3, pp.  257-266
Abstract
We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, (a compact space of constant curvature). In particular, these generalizations include addition of a spherical analogue of the magnetic monopole (the Poincaré–Appell system) and addition of a more complicated field which is a generalization of the MICZ-system. The mentioned systems are integrable superintegrable, and there exists the vector integral which is analogous to the Laplace–Runge–Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space $L^3$.
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
DOI: 10.1070/RD2005v010n03ABEH000314
Borisov A. V., Mamaev I. S., Kilin A. A.
Two-body problem on a sphere. Reduction, stochasticity, periodic orbits
2004, vol. 9, no. 3, pp.  265-279
Abstract
We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.
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
DOI:10.1070/RD2004v009n03ABEH000280
Borisov A. V., Mamaev I. S., Kilin A. A.
Absolute and relative choreographies in the problem of point vortices moving on a plane
2004, vol. 9, no. 2, pp.  101-111
Abstract
We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the $n$-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.
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
DOI:10.1070/RD2004v009n02ABEH000269
Borisov A. V., Mamaev I. S., Ramodanov S. M.
Motion of a circular cylinder and $n$ point vortices in a perfect fluid
2003, vol. 8, no. 4, pp.  449-462
Abstract
The paper studies the system of a rigid body interacting dynamically with point vortices in a perfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to be Hamiltonian (the corresponding Poisson bracket structure is rather complicated). We also reduced the number of degrees of freedom of the system by two using the reduction by symmetry technique and performed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.
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
DOI:10.1070/RD2003v008n04ABEH000257
Borisov A. V., Mamaev I. S.
An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid
2003, vol. 8, no. 2, pp.  163-166
Abstract
In this paper we present the nonlinear Poisson structure and two first integrals in the problem on plane motion of circular cylinder and $N$ point vortices in the ideal fluid. A priori this problem is not Hamiltonian. The particular case $N = 1$, i.e. the problem on interaction of cylinder and vortex, is integrable.
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
DOI:10.1070/RD2003v008n02ABEH000235
Borisov A. V., Mamaev I. S., Kilin A. A.
Dynamics of rolling disk
2003, vol. 8, no. 2, pp.  201-212
Abstract
In the paper we present the qualitative analysis of rolling motion without slipping of a homogeneous round disk on a horisontal plane. The problem was studied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its integrability. The behavior of the point of contact on a plane is investigated and conditions under which its trajectory is finit are obtained. The bifurcation diagrams are constructed.
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
DOI:10.1070/RD2003v008n02ABEH000237
Borisov A. V., Mamaev I. S.
The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics
2002, vol. 7, no. 2, pp.  177-200
Abstract
In this paper we study the cases of existence of an invariant measure, additional first integrals, and a Poisson structure in the problem of rigid body's rolling without sliding on a plane and a sphere. The problem of rigid body's motion on a plane was studied by S.A. Chaplygin, P. Appel, D. Korteweg. They showed that the equations of motion are reduced to a second-order linear differential equation in the case when the surface of the dynamically symmetrical body is a surface of revolution. These results were partially generalized by P. Woronetz, who studied the motion of a body of revolution and the motion of round disk with sharp edge on a sphere. In both cases the systems are Euler–Jacobi integrable and have additional integrals and invariant measure. It can be shown that by an appropriate change of time (determined by reducing multiplier), the reduced system is a Hamiltonian one. Here we consider some particular cases when the integrals and the invariant measure can be presented as finite algebraic expressions. We also consider a generalized problem of rolling of a dynamically nonsymmetric Chaplygin ball. The results of investigations are summarized in tables to illustrate the hierarchy of existence of various tensor invariants: invariant measure, integrals, and Poisson structure.
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
DOI:10.1070/RD2002v007n02ABEH000204
Borisov A. V., Mamaev I. S., Kilin A. A.
The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics
2002, vol. 7, no. 2, pp.  201-219
Abstract
The paper is concerned with the problem on rolling of a homogeneous ball on an arbitrary surface. New cases when the problem is solved by quadratures are presented. The paper also indicates a special case when an additional integral and invariant measure exist. Using this case, we obtain a nonholonomic generalization of the Jacobi problem for the inertial motion of a point on an ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder in the gravity field the ball's motion is bounded and, on the average, it does not move downwards. All the results of the paper considerably expand the results obtained by E. Routh in XIX century.
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
DOI:10.1070/RD2002v007n02ABEH000205
Borisov A. V., Mamaev I. S.
Generalization of the Goryachev–Chaplygin Case
2002, vol. 7, no. 1, pp.  21-30
Abstract
In this paper we present a generalization of the Goryachev–Chaplygin integrable case on a bundle of Poisson brackets, and on Sokolov terms in his new integrable case of Kirchhoff equations. We also present a new analogous integrable case for the quaternion form of rigid body dynamics equations. This form of equations is recently developed and we can use it for the description of rigid body motions in specific force fields, and for the study of different problems of quantum mechanics. In addition we present new invariant relations in the considered problems.
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
DOI:10.1070/RD2002v007n01ABEH000192
Borisov A. V., Mamaev I. S.
On the History of the Development of the Nonholonomic Dynamics
2002, vol. 7, no. 1, pp.  43-47
Abstract
The main directions in the development of the nonholonomic dynamics are briefly considered in this paper. The first direction is connected with the general formalizm of the equations of dynamics that differs from the Lagrangian and Hamiltonian methods of the equations of motion's construction. The second direction, substantially more important for dynamics, includes investigations concerning the analysis of the specific nonholonomic problems. We also point out rather promising direction in development of nonholonomic systems that is connected with intensive use of the modern computer-aided methods.
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
DOI:10.1070/RD2002v007n01ABEH000194
Borisov A. V., Mamaev I. S.
Euler–Poisson Equations and Integrable Cases
2001, vol. 6, no. 3, pp.  253-276
Abstract
In this paper we propose a new approach to the study of integrable cases based on intensive computer methods' application. We make a new investigation of Kovalevskaya and Goryachev–Chaplygin cases of Euler–Poisson equations and obtain many new results in rigid body dynamics in absolute space. Also we present the visualization of some special particular solutions.
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
DOI:10.1070/RD2001v006n03ABEH000176
Borisov A. V., Mamaev I. S., Kholmskaya A. G.
Kovalevskaya top and generalizations of integrable systems
2001, vol. 6, no. 1, pp.  1-16
Abstract
Generalizations of the Kovalevskaya, Chaplygin, Goryachev–Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and action-angle variables is described. Another integration method for the Kovalevskaya top on the bundle is found. This method uses a coordinate transformation that reduces the Kovalevskaya system to the Neumann system. The Kolosov analogy is considered. A generalization of a recent Gaffet system to the bundle of Poisson brackets is obtained at the end of the paper.
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
DOI:10.1070/RD2001v006n01ABEH000161
Borisov A. V., Kilin A. A.
Stability of Thomson's Configurations of Vortices on a Sphere
2000, vol. 5, no. 2, pp.  189-200
Abstract
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
DOI:10.1070/RD2000v005n02ABEH000141
Borisov A. V., Mamaev I. S.
Some comments to the paper by A.M.Perelomov "A note on geodesics on ellipsoid" RCD 2000 5(1) 89-91
2000, vol. 5, no. 1, pp.  92-94
Abstract
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
Borisov A. V., Dudoladov S. L.
Kovalevskaya Exponents and Poisson Structures
1999, vol. 4, no. 3, pp.  13-20
Abstract
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
DOI:10.1070/RD1999v004n03ABEH000111
Bolsinov A. V., Borisov A. V., Mamaev I. S.
Lie algebras in vortex dynamics and celestial mechanics — IV
1999, vol. 4, no. 1, pp.  23-50
Abstract
1.Classificaton of the algebra of $n$ vortices on a plane
2.Solvable problems of vortex dynamics
3.Algebraization and reduction in a three-body problem
The work [13] introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie–Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works [14,15]. In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem.
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
DOI:10.1070/RD1999v004n01ABEH000097
Borisov A. V., Lebedev V. G.
Dynamics of Three Vortices on a Plane and a Sphere — III. Noncompact Case. Problems of Collaps and Scattering
1998, vol. 3, no. 4, pp.  74-86
Abstract
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
DOI:10.1070/RD1998v003n04ABEH000094
Borisov A. V., Lebedev V. G.
Dynamics of three vortices on a plane and a sphere — II. General compact case
1998, vol. 3, no. 2, pp.  99-114
Abstract
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
DOI:10.1070/RD1998v003n02ABEH000075
Borisov A. V., Pavlov A. E.
Dynamics and statics of vortices on a plane and a sphere - I
1998, vol. 3, no. 1, pp.  28-38
Abstract
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
DOI:10.1070/RD1998v003n01ABEH000059
Borisov A. V., Mamaev I. S.
Non-linear Poisson brackets and isomorphisms in dynamics
1997, vol. 2, nos. 3-4, pp.  72-89
Abstract
In the paper the equations of motion of a rigid body in the Hamiltonian form on the subalgebra of algebra $e(4)$ are written. With the help of the algebraic methods a number of new isomorphisms in dynamics is established. We consider the lowering of the order as the process of decreasing rank of the Poisson structure with the algebraic point of view and indicate the possibility of arising the nonlinear Poisson brackets at this reduction as well.
Citation: Borisov A. V., Mamaev I. S.,  Non-linear Poisson brackets and isomorphisms in dynamics, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 72-89
DOI:10.1070/RD1997v002n03ABEH000049
Borisov A. V., Mamaev I. S.
Adiabatic Chaos in Rigid Body Dynamics
1997, vol. 2, no. 2, pp.  65-78
Abstract
We consider arising of adiabatic chaos in rigid body dynamics. The comparison of analytical diffusion coefficient describing probable effects in the chaos zone with numerical experiment is carried out. The analysis of split of asymptotic surfaces is carried out the curves of indfenition in the Poincare-Zhukovsky problem.
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
DOI:10.1070/RD1997v002n02ABEH000037
Borisov A. V., Simakov N. N.
Period Doubling Bifurcation in Rigid Body Dynamics
1997, vol. 2, no. 1, pp.  64-74
Abstract
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
DOI:10.1070/RD1997v002n01ABEH000027
Borisov A. V.
Necessary and Sufficient Conditions of Kirchhoff Equation Integrability
1996, vol. 1, no. 2, pp.  61-76
Abstract
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
DOI:10.1070/RD1996v001n02ABEH000016
Borisov A. V., Tsygvintsev A. V.
Kowalewski exponents and integrable systems of classic dynamics. I, II
1996, vol. 1, no. 1, pp.  15-37
Abstract
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
DOI:10.1070/RD1996v001n01ABEH000002

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