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2013
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Volume 19, Number 2

Volume 19, Number 2, 2014
Proceedings of GDIS 2013, Izhevsk

The Fourth International Conference "Geometry, Dynamics, Integrable Systems" GDIS 2013 Held in Izhevsk, Russia, June 10–14, 2013
Kozlov V. V.
Remarks on Integrable Systems
Abstract
The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension $\geqslant n$. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.
Keywords: integrability by quadratures, adjoint system, Hamiltonian equations, Euler–Jacobi theorem, Lie theorem, symmetries
Citation: Kozlov V. V., Remarks on Integrable Systems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 145-161
DOI:10.1134/S1560354714020014
Dragovic V.,  Kukić K.
Systems of Kowalevski Type and Discriminantly Separable Polynomials
Abstract
Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two theta-functions in a procedure which is similar to the classical one for the Kowalevski top. The discriminantly separable polynomials play the role of the Kowalevski fundamental equation. Natural examples include the Sokolov systems and the Jurdjevic elasticae.
Keywords: integrable systems, Kowalevski top, discriminantly separable polynomials, systems of Kowalevski type
Citation: Dragovic V.,  Kukić K., Systems of Kowalevski Type and Discriminantly Separable Polynomials, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 162-184
DOI:10.1134/S1560354714020026
Tsiganov A. V.
On the Lie Integrability Theorem for the Chaplygin Ball
Abstract
The necessary number of commuting vector fields for the Chaplygin ball in the absolute space is constructed. We propose to get these vector fields in the framework of the Poisson geometry similar to Hamiltonian mechanics.
Keywords: nonholonomic dynamical system, Poisson bracket, Lie theorem, Chaplygin ball
Citation: Tsiganov A. V., On the Lie Integrability Theorem for the Chaplygin Ball, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 185-197
DOI:10.1134/S1560354714020038
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
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
Ivanov A. P.
On the Impulsive Dynamics of M-blocks
Abstract
This paper is concerned with the motion of a cubic rigid body (cube) with a rotor, caused by a sudden brake of the rotor, which imparts its angular momentum to the body. This produces an impulsive reaction of the support, leading to a jump or rolling from one face to another. Such dynamics was demonstrated by researchers from Massachusetts Institute of Technology at the IEEE/RSJ International Conference on Intelligent Robots and Systems in Tokio in November 2013. The robot, called by them M-block, is 4 cm in size and uses an internal flywheel mechanism rotating at 20 000 rev/min. Initially the cube rests on a horizontal plane. When the brake is set, the relative rotation slows down, and its energy is imparted to the case. The subsequent motion is illustrated in a clip [13]. Here the general approach to the analysis of dynamics of M-cube is proposed, including equations of impulsive motion and methods of their solution. Some particular cases are studied in details.
Keywords: gyrostat, impulsive dynamics, friction, robotic dynamics
Citation: Ivanov A. P., On the Impulsive Dynamics of M-blocks, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 214-225
DOI:10.1134/S1560354714020051
Kharlamov M. P.
Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field
Abstract
For the integrable system on $e(3,2)$ found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.
Keywords: generalized two-field gyrostat, critical subsystems, Appelrot classes, invariant relations, types of critical points
Citation: Kharlamov M. P., Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 226-244
DOI:10.1134/S1560354714020063
Jovanovic B.
Heisenberg Model in Pseudo-Euclidean Spaces
Abstract
We construct analogues of the classical Heisenberg spin chain model (or the discrete Neumann system), on pseudo-spheres and light-like cones in the pseudo-Euclidean spaces and show their complete Hamiltonian integrability. Further, we prove that the Heisenberg model on a light-like cone leads to a new example of the integrable discrete contact system.
Keywords: discrete Hamiltonian and contact systems, the Lax representation, complete integrability
Citation: Jovanovic B., Heisenberg Model in Pseudo-Euclidean Spaces, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 245-250
DOI:10.1134/S1560354714020075
Bounemoura A.,  Fischler S.
The Classical KAM Theorem for Hamiltonian Systems via Rational Approximations
Abstract
In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno–Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in [4] for perturbations of constant vector fields on the torus.
Keywords: perturbation of integrable Hamiltonian systems, KAM theory, Diophantine duality, periodic approximations
Citation: Bounemoura A.,  Fischler S., The Classical KAM Theorem for Hamiltonian Systems via Rational Approximations, Regular and Chaotic Dynamics, 2014, vol. 19, no. 2, pp. 251-265
DOI:10.1134/S1560354714020087

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