Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics
Russian Mathematical Surveys, 2017, vol. 72, no. 5, pp. 783-840
Abstract
pdf (1.09 Mb)
This is a survey of the main forms of equations of dynamical
systems with non-integrable constraints, divided into two large groups.
The first group contains systems arising in vakonomic mechanics and optimal
control theory, with the equations of motion obtained from the variational
principle, and the second contains systems in classical non-holonomic
mechanics, when the constraints are ideal and therefore the D’Alembert–Lagrange principle holds.
Borisov A. V., Mamaev I. S., Bizyaev I. A., Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics, Russian Mathematical Surveys, 2017, vol. 72, no. 5, pp. 783-840
A Chaplygin sleigh with a moving point mass
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2017, vol. 27, no. 4, pp. 583-589
Abstract
pdf (314.6 Kb)
Nonholonomic mechanical systems arise in the context of many problems of practical significance. A famous model in nonholonomic mechanics is the Chaplygin sleigh. The Chaplygin sleigh is a rigid body with a sharp weightless wheel in contact with the (supporting) surface. The sharp edge of the wheel prevents the wheel from sliding in the direction perpendicular to its plane. This paper is concerned with a Chaplygin sleigh with time-varying mass distribution, which arises due to the motion of a point in the direction transverse to the plane of the knife edge. Equations of motion are obtained from which a closed system of equations with time-periodic coefficients decouples. This system governs the evolution of the translational and angular velocities of the sleigh. It is shown that if the projection of the center of mass of the whole system onto the axis along the knife edge is zero, the translational velocity of the sleigh increases. The trajectory of the point of contact is, as a rule, unbounded.
Keywords:
nonholonomic mechanics, Chaplygin sleigh, acceleration, first integrals
Citation:
Bizyaev I. A., A Chaplygin sleigh with a moving point mass, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2017, vol. 27, no. 4, pp. 583-589
Invariant measure in the problem of a disk rolling on a plane
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2017, vol. 27, no. 4, pp. 576-582
Abstract
pdf (312.38 Kb)
This paper addresses the dynamics of a disk rolling on an absolutely rough plane. It is proved that the equations of motion have an invariant measure with continuous density only in two cases: a dynamically symmetric disk and a disk with a special mass distribution. In the former case, the equations of motion possess two additional integrals and are integrable by quadratures by the Euler-Jacobi theorem. In the latter case, the absence of additional integrals is shown using a Poincaré map. In both cases, the volume of any domain in phase space (calculated with the help of the density) is preserved by the phase flow. Nonholonomic mechanics is populated with systems both with and without an invariant measure.
Keywords:
nonholonomic mechanics, Schwarzschild-Littlewood theorem, manifold of falls, chaotic dynamics
Citation:
Bizyaev I. A., Invariant measure in the problem of a disk rolling on a plane, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2017, vol. 27, no. 4, pp. 576-582
The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration
Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 955–975
Abstract
pdf (1.91 Mb)
This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 955–975
The Inertial Motion of a Roller Racer
Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 239-247
Abstract
pdf (3.26 Mb)
This paper addresses the problem of the inertial motion of a roller racer, which reduces to investigating a dynamical system on a (two-dimensional) torus and to classifying singular points on it. It is shown that the motion of the roller racer in absolute space is
asymptotic. A restriction on the system parameters in which this motion is bounded (compact) is presented.
The Hess–Appelrot Case and Quantization of the Rotation Number
Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
Abstract
pdf (991.1 Kb)
This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.
Keywords:
invariant submanifold, rotation number, Cantor ladder, limit cycles
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot Case and Quantization of the Rotation Number, Regular and Chaotic Dynamics, 2017, vol. 22, no. 2, pp. 180-196
The Hess–Appelrot case and quantization of the rotation number
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 433-452
Abstract
pdf (556.32 Kb)
This paper is concerned with the Hess case in the Euler –Poisson equations and with its
generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces
to investigating the vector field on a torus and that the graph showing the dependence of the
rotation number on parameters has horizontal segments (limit cycles) only for integer values of
the rotation number. In addition, an example of a Hamiltonian system is given which possesses
an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation
number on parameters is a Cantor ladder.
Keywords:
invariant submanifold, rotation number, Cantor ladder, limit cycles
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot case and quantization of the rotation number, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 3, pp. 433-452
Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 1, pp. 129-146
Abstract
pdf (2.82 Mb)
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords:
sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
Citation:
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S., Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 1, pp. 129-146
Chaplygin sleigh with periodically oscillating internal mass
EPL, 2017, vol. 119, no. 6, 60008, 7 pp.
Abstract
pdf (854.88 Kb)
We consider the movement of Chaplygin sleigh on a plane that is a solid body with
imposed nonholonomic constraint, which excludes the possibility of motions transversal to the constraint element (“knife-edge”), and complement the model with an attached mass, periodically oscillating relatively to the main platform of the sleigh. Numerical simulations indicate the occurrence of either unrestricted acceleration of the sleigh, or motions with bounded velocities and
momenta, depending on parameters. We note the presence of phenomena characteristic to nonholonomic systems with complex dynamics; in particular, attractors occur responsible for chaotic
motions. In addition, quasiperiodic regimes take place similar to those observed in conservative nonlinear dynamics.
Citation:
Bizyaev I. A., Borisov A. V., Kuznetsov S. P., Chaplygin sleigh with periodically oscillating internal mass, EPL, 2017, vol. 119, no. 6, 60008, 7 pp.
Generalizations of the Kovalevskaya Case and Quaternions
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 33-44
Abstract
pdf (190.82 Kb)
This paper provides a detailed description of various reduction schemes in rigid
body dynamics. The analysis of one of such nontrivial reductions makes it possible to put the
cases already found in order and to obtain new generalizations of the Kovalevskaya case to $e(3)$.
Note that the indicated reduction allows one to obtain in a natural way some singular additive
terms that were proposed earlier by D.N. Goryachev.
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Generalizations of the Kovalevskaya Case and Quaternions, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 33-44
The Hess–Appelrot system and its nonholonomic analogs
Proceedings of the Steklov Institute of Mathematics, 2016, vol. 294, pp. 268-292
Abstract
pdf (1.23 Mb)
This paper is concerned with the nonholonomic Suslov problem
and its generalization proposed by Chaplygin. The issue of the existence
of an invariant measure with singular density (having singularities at some
points of phase space) is discussed.
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Hess–Appelrot system and its nonholonomic analogs, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 294, pp. 268-292
Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
Abstract
pdf (3.48 Mb)
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3, 6, 14)$, the other is defined by two generatrices and growth vector $(2, 3, 5, 8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords:
sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
Citation:
Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S., Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity
Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580
Abstract
pdf (1.45 Mb)
In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.
Keywords:
celestial mechanics, space of constant curvature, reduction, rigid body dynamics, Poincaré section
Citation:
Borisov A. V., Mamaev I. S., Bizyaev I. A., The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 556-580
Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period
Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
Abstract
pdf (1.87 Mb)
In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
Borisov A. V., Mamaev I. S., Bizyaev I. A., Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
The Dynamics of Vortex Sources in a Deformation Flow
Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376
Abstract
pdf (1.37 Mb)
This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse.
Bizyaev I. A., Borisov A. V., Mamaev I. S., The Dynamics of Vortex Sources in a Deformation Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376
Dynamics of the Chaplygin Sleigh on a Cylinder
Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146
Abstract
pdf (268.54 Kb)
This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found.
In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.
Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin Sleigh on a Cylinder, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146
Dynamics of the Chaplygin sleigh on a cylinder
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 675–687
Abstract
pdf (331.42 Kb)
This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.
Bizyaev I. A., Borisov A. V., Mamaev I. S., Dynamics of the Chaplygin sleigh on a cylinder, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 675–687
Historical and critical review of the development of nonholonomic mechanics: the classical period
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 3, pp. 385-411
Abstract
pdf (1.9 Mb)
In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
Borisov A. V., Mamaev I. S., Bizyaev I. A., Historical and critical review of the development of nonholonomic mechanics: the classical period, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 3, pp. 385-411
Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors
Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 263-287
Abstract
pdf (3.54 Mb)
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
Bizyaev I. A., Borisov A. V., Kazakov A. O., Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 2, pp. 263-287
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.
Homogeneous systems with quadratic integrals, Lie–Poisson quasi-brackets, and the Kovalevskaya method
Sbornik: Mathematics, 2015, vol. 206, no. 12, pp. 29–54
Abstract
pdf (481.65 Kb)
We consider differential equations with quadratic right-hand sides which admit two quadratic first integrals, one of which is a positive definite quadratic form. We present general conditions under which a linear change of variables reduces this system to some "canonical" form. Under these conditions the system turns out to be nondivergent and is reduced to Hamiltonian form, however, the corresponding linear Lie–Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case the equations are reduced to the classical equations of the Euler top, and in the four-dimensional space the system turns out to be superintegrable and coincides with the Euler–Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplication with which the Poisson bracket satisfies the Jacobi identity. In the general case, we prove that there is no reducing multiplier for $n>5$. As an example, we consider a system of Lotka–Volterra type with quadratic right-hand sides, which was studied already by Kovalevskaya from the viewpoint of the conditions for uniqueness of its solutions as functions of complex time.
Keywords:
first integrals, conformally Hamiltonian system, Poisson bracket, Kovalevskaya system, dynamical systems with quadratic right-hand sides
Citation:
Bizyaev I. A., Kozlov V. V., Homogeneous systems with quadratic integrals, Lie–Poisson quasi-brackets, and the Kovalevskaya method, Sbornik: Mathematics, 2015, vol. 206, no. 12, pp. 29–54
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
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
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
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
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
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
Topology and Bifurcations in Nonholonomic Mechanics
International Journal of Bifurcation and Chaos, 2015, vol. 25, no. 10, 1530028, 21 pp.
Abstract
pdf (645.53 Kb)
This paper develops topological methods for qualitative analysis of the behavior of nonholonomic
dynamical systems. Their application is illustrated by considering a new integrable system of
nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic,
it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible
types of integral manifolds are found and a classification of trajectories on them is presented.
Bizyaev I. A., Bolsinov A. V., Borisov A. V., Mamaev I. S., Topology and Bifurcations in Nonholonomic Mechanics, International Journal of Bifurcation and Chaos, 2015, vol. 25, no. 10, 1530028, 21 pp.
Hamiltonization of Elementary Nonholonomic Systems
Russian Journal of Mathematical Physics, 2015, vol. 22, no. 4, pp. 444-453
Abstract
pdf (115.49 Kb)
In this paper, we develop the method of Chaplygin’s reducing multiplier; using this method, we obtain a conformally Hamiltonian representation for three nonholonomic systems, namely, for the nonholonomic oscillator, for the Heisenberg system, and for the Chaplygin sleigh. Furthermore, in the case of oscillator and nonholonomic Chaplygin sleigh, we show that the problem reduces to the study of motion of a mass point (in a potential field) on a plane and, in the case of Heisenberg system, on the sphere. Moreover, we consider an example of a nonholonomic system (suggested by Blackall) to which one cannot apply the method of reducing multiplier.
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Hamiltonization of Elementary Nonholonomic Systems, Russian Journal of Mathematical Physics, 2015, vol. 22, no. 4, pp. 444-453
Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top
Doklady Mathematics, 2014, vol. 90, no. 2, pp. 631–634
Abstract
pdf (203.47 Kb)
Citation:
Bizyaev I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Doklady Mathematics, 2014, vol. 90, no. 2, pp. 631–634
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.
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
The dynamics of three vortex sources
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 319-327
Abstract
pdf (413.32 Kb)
In this paper, the integrability of the equations of a system of three vortex sources is shown. A reduced system describing, up to similarity, the evolution of the system’s configurations is obtained. Possible phase portraits and various relative equilibria of the system are presented.
Bizyaev I. A., Borisov A. V., Mamaev I. S., The dynamics of three vortex sources, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 319-327
On a generalization of systems of Calogero type
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, pp. 209-212
Abstract
pdf (237.84 Kb)
This paper is concerned with a three-body system on a straight line in a potential field proposed by Tsiganov. The Liouville integrability of this system is shown. Reduction and separation of variables are performed.
Keywords:
Calogero systems, reduction, integrable systems, Jacobi problem
Citation:
Bizyaev I. A., On a generalization of systems of Calogero type, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 2, pp. 209-212
Figures of equilibrium of an inhomogeneous self-gravitating fluid
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 73-100
Abstract
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This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with density stratification and a steady-state velocity field. As in the classical setting, it is assumed that the figure or its layers uniformly rotate about an axis fixed in space. As is well known, when there is no rotation, only a ball can be a figure of equilibrium.
It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with inherent constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification.
We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.
Keywords:
self-gravitating fluid, confocal stratification, homothetic stratification, space of constant curvature
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 73-100
On the Routh sphere problem
Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, no. 8, 085202, 11 pp.
Abstract
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We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on six-dimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra $e^*$(3). It allows us to relate the nonholonomic Routh system with the Hamiltonian system on a cotangent bundle to the sphere with a canonical Poisson structure.
Citation:
Bizyaev I. A., Tsiganov A. V., On the Routh sphere problem , Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, no. 8, 085202, 11 pp.
The 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
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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
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
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In this paper we investigate two systems consisting of a spherical shell rolling on a plane without slipping and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is fixed at the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of the nonholonomic hinge. The equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics.
Keywords:
nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge
Citation:
Bizyaev I. A., Borisov A. V., Mamaev I. S., The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 547-566
Integrability and stochastic behavior in some nonholonomic dynamics problems
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 257-265
Abstract
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In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of an ellipsoid on a plane and a sphere. We research these problems using Poincare maps, which investigation helps to discover a new integrable case.
Keywords:
nonholonomic constraint, invariant measure, first integral, Poincare map, integrability and chaos
Citation:
Bizyaev I. A., Kazakov A. O., Integrability and stochastic behavior in some nonholonomic dynamics problems, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 257-265
The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 141-202
Abstract
pdf (7.91 Mb)
In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.
Borisov A. V., Mamaev I. S., Bizyaev I. A., The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 2, pp. 141-202
On the Routh sphere
Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 3, pp. 569-583
Abstract
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We discuss an embedding of the vector field associated with the nonholonomic Routh sphere in subgroup of the commuting Hamiltonian vector fields associated with this system. We prove that the corresponding Poisson brackets are reduced to canonical ones in the region without of homoclinic trajectories.
Figures of equilibrium of liquid self-gravitating inhomogeneous mass
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2011, no. 3, pp. 142-153
Abstract
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We consider the inhomogeneous self-gravitating liquid spheroid with confocal stratification which rotates around the minor semiaxis and is in equilibrium. General relationships for pressure, angular velocity and gravitational potential of the spheroid with any density function are obtained. Special cases of piecewise constant and continuous density functions are analyzed.
Bizyaev I. A., Ivanova T. B., Figures of equilibrium of liquid self-gravitating inhomogeneous mass, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2011, no. 3, pp. 142-153