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Volume 20, Number 3

Volume 20, Number 3, 2015
On the 65th birthday of professor V.V. Kozlov

Kozlov V. V.
The Dynamics of Systems with Servoconstraints. I
Abstract
The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.
Keywords: servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides
Citation: Kozlov V. V., The Dynamics of Systems with Servoconstraints. I, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 205-224
DOI:10.1134/S1560354715030016
Burov A. A.,  Shalimova E. S.
On the Motion of a Heavy Material Point on a Rotating Sphere (Dry Friction Case)
Abstract
The problem of motion of a heavy particle on a sphere uniformly rotating about a fixed axis is considered in the case of dry friction. It is assumed that the angle of inclination of the rotation axis is constant. The existence of equilibria in an absolute coordinate system and their linear stability are discussed. The equilibria in a relative coordinate system rotating with the sphere are also studied. These equilibria are generally nonisolated. The dependence of the equilibrium sets of this kind on the system parameters is also considered.
Keywords: dry friction, motion of a particle on a sphere, absolute and relative equilibria, bifurcations of equilibria
Citation: Burov A. A.,  Shalimova E. S., On the Motion of a Heavy Material Point on a Rotating Sphere (Dry Friction Case), Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 225-233
DOI:10.1134/S1560354715030028
Carvalho A.,  Cabral H. E.
Lyapunov Orbits in the $n$-Vortex Problem on the Sphere
Abstract
In the phase space reduced by rotation, we prove the existence of periodic orbits of the $(n + 1)$-vortex problem emanating from a relative equilibrium formed by $n$ unit vortices at the vertices of a regular polygon at a fixed latitude and an additional vortex of intensity κ at the north pole when the ideal fluid moves on the surface of a sphere.
Keywords: point vortex problem, relative equilibria, periodic orbits, Lyapunov center theorem
Citation: Carvalho A.,  Cabral H. E., Lyapunov Orbits in the $n$-Vortex Problem on the Sphere, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 234-246
DOI:10.1134/S156035471503003X
Albouy A.
Projective Dynamics and First Integrals
Abstract
We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami’s theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.
Keywords: bi-hamiltonian, Beltrami’s theorem, Young tableau symmetry, free motion, force field, decomposability preserving
Citation: Albouy A., Projective Dynamics and First Integrals, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 247-276
DOI:10.1134/S1560354715030041
Damianou P. A.,  Sabourin H.,  Vanhaecke P.
Intermediate Toda Systems
Abstract
We construct a large family of Hamiltonian systems which interpolate between the classical Kostant–Toda lattice and the full Kostant–Toda lattice and we discuss their integrability. There is one such system for every nilpotent ideal $\mathcal{I}$ in a Borel subalgebra $\mathfrak{b}_+$ of an arbitrary simple Lie algebra $\mathfrak{g}$. The classical Kostant–Toda lattice corresponds to the case of $\mathcal{I}=[\mathfrak{n}_+, \mathfrak{n}_+]$, where $\mathfrak{n}_+$ is the unipotent ideal of $\mathfrak{b}_+$, while the full Kostant–Toda lattice corresponds to $\mathcal{I}=\{0\}$. We mainly focus on the case $\mathcal{I}=[[\mathfrak{n}_+, \mathfrak{n}_+], \mathfrak{n}_+]$. In this case, using the theory of root systems of simple Lie algebras, we compute the rank of the underlying Poisson manifolds and construct a set of (rational) functions in involution, large enough to ensure Liouville integrability. These functions are restrictions of well-chosen integrals of the full Kostant–Toda lattice, except for the case of the Lie algebras of type $C$ and $D$ where a different function (Casimir) is needed. The latter fact, and other ones listed in the paper, point at the Liouville integrability of all the systems we construct, but also at the nontriviality of obtaining the result in full generality.
Keywords: Toda lattices, integrable systems
Citation: Damianou P. A.,  Sabourin H.,  Vanhaecke P., Intermediate Toda Systems, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 277-292
DOI:10.1134/S1560354715030053
Dragovic V.,  Gajic B.,  Jovanovic B.
Note on Free Symmetric Rigid Body Motion
Abstract
We consider the Euler equations of motion of a free symmetric rigid body around a fixed point, restricted to the invariant subspace given by the zero values of the corresponding linear Noether integrals. In the case of the $SO(n − 2)$-symmetry, we show that almost all trajectories are periodic and that the motion can be expressed in terms of elliptic functions. In the case of the $SO(n − 3)$-symmetry, we prove the solvability of the problem by using a recent Kozlov’s result on the Euler–Jacobi–Lie theorem.
Keywords: Euler equations, Manakov integrals, spectral curve, reduced Poisson space
Citation: Dragovic V.,  Gajic B.,  Jovanovic B., Note on Free Symmetric Rigid Body Motion, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 293-308
DOI:10.1134/S1560354715030065
Markeev A. P.
On the Birkhoff Transformation in the Case of Complete Degeneracy of the Quadratic Part of the Hamiltonian
Abstract
A time-periodic one-degree-of-freedom system is investigated. The system is assumed to have an equilibrium point in the neighborhood of which the Hamiltonian is represented as a convergent series. This series does not contain any second-degree terms, while the terms up to some finite degree $l$ do not depend explicitly on time. An algorithm for constructing a canonical transformation is proposed that simplifies the structure of the Hamiltonian to terms of degree $l$ inclusive.
As an application, a special case is considered when the expansion of the Hamiltonian begins with third-degree terms. For this case, sufficient conditions for instability of the equilibrium are obtained depending on the forms of the fourth and fifth degrees.
Keywords: Hamiltonian system, canonical transformations, stability
Citation: Markeev A. P., On the Birkhoff Transformation in the Case of Complete Degeneracy of the Quadratic Part of the Hamiltonian, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 309-316
DOI:10.1134/S1560354715030077
Akbarzadeh R.,  Haghighatdoost G.
The Topology of Liouville Foliation for the Borisov–Mamaev–Sokolov Integrable Case on the Lie Algebra $so(4)$
Abstract
In 2001, A.V. Borisov, I.S. Mamaev, and V.V. Sokolov discovered a new integrable case on the Lie algebra $so(4)$. This system coincides with the Poincaré equations on the Lie algebra $so(4)$, which describe the motion of a body with cavities filled with an incompressible vortex fluid. Moreover, the Poincaré equations describe the motion of a four-dimensional gyroscope. In this paper topological properties of this system are studied. In particular, for the system under consideration the bifurcation diagrams of the momentum mapping are constructed and all Fomenko invariants are calculated. Thereby, a classification of isoenergy surfaces for this system up to the rough Liouville equivalence is obtained.
Keywords: integrable Hamiltonian systems, isoenergy surfaces, Kirchhoff equations, Liouville foliation, bifurcation diagram, Borisov–Mamaev–Sokolov case, topological invariant
Citation: Akbarzadeh R.,  Haghighatdoost G., The Topology of Liouville Foliation for the Borisov–Mamaev–Sokolov Integrable Case on the Lie Algebra $so(4)$, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 317-344
DOI:10.1134/S1560354715030089
Kuznetsov S. P.
Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models
Abstract
Results are reviewed concerning the planar problem of a plate falling in a resisting medium studied with models based on ordinary differential equations for a small number of dynamical variables. A unified model is introduced to conduct a comparative analysis of the dynamical behaviors of models of Kozlov, Tanabe–Kaneko, Belmonte–Eisenberg–Moses and Andersen–Pesavento–Wang using common dimensionless variables and parameters. It is shown that the overall structure of the parameter spaces for the different models manifests certain similarities caused by the same inherent symmetry and by the universal nature of the phenomena involved in nonlinear dynamics (fixed points, limit cycles, attractors, and bifurcations).
Keywords: body motion in a fluid, oscillations, autorotation, flutter, attractor, bifurcation, chaos, Lyapunov exponent
Citation: Kuznetsov S. P., Plate Falling in a Fluid: Regular and Chaotic Dynamics of Finite-dimensional Models, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 345-382
DOI:10.1134/S1560354715030090
Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A.
The Jacobi Integral in Nonholonomic Mechanics
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

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