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# Volume 6, Number 3, 2001

 Kozlov V. V. Kinetics of Collisionless Continuous Medium Abstract In this article we develop Poincaré ideas about a heat balance of ideal gas considered as a collisionless continuous medium. We obtain the theorems on diffusion in nondegenerate completely integrable systems. As a corollary we show that for any initial distribution the gas will be eventually irreversibly and uniformly distributed over all volume, although every particle during this process approaches arbitrarily close to the initial position indefinitely many times. However, such individual returnability is not uniform, which results in diffusion in a reversible and conservative system. Balancing of pressure and internal energy of ideal gas is proved, the formulas for limit values of these quantities are given and the classical law for ideal gas in a heat balance is deduced. It is shown that the increase of entropy of gas under the adiabatic extension follows from the law of motion of a collisionless continuous medium. Citation: Kozlov V. V., Kinetics of Collisionless Continuous Medium, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 235-251 DOI:10.1070/RD2001v006n03ABEH000175
 Borisov A. V.,  Mamaev I. S. Euler–Poisson Equations and Integrable Cases 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
 Morales-Ruiz J. J.,  Peris J. M. On the Dynamical Meaning of the Picard–Vessiot Theory Abstract In this paper we present a dynamical interpretation of the Differential Galois Theory of Linear Differential Equations (also called the Picard$ndash;Vessiot Theory). The key point is that when a linear differential equation is not solvable in closed form then by a theorem of Tits the monodromy group for fuchsian equations (or a generalization of it for irregular singularities: the Ramis monodromy group) contains a free non-abelian group. Roughly this free group gives us a very complicated dynamics on some suitable spaces. Citation: Morales-Ruiz J. J., Peris J. M., On the Dynamical Meaning of the Picard–Vessiot Theory, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 277-290 DOI:10.1070/RD2001v006n03ABEH000177  Kilin A. A. The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis Abstract The motion of Chaplygin ball with and without gyroscope in the absolute space is analyzed. In particular, the trajectories of the point of contact are studied in detail. We discuss the motions in the absolute space, that correspond to the different types of motion in the moving frame of reference related to the body. The existence of the bounded trajectories of the ball's motion is shown by means of numerical methods in the case when the problem is reduced to a certain Hamiltonian system. Citation: Kilin A. A., The Dynamics of Chaplygin Ball: the Qualitative and Computer Analysis, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291-306 DOI:10.1070/RD2001v006n03ABEH000178  Tsiganov A. V. On the Invariant Separated Variables Abstract An integrable Hamiltonian system on a Poisson manifold consists of a Lagrangian foliation$\mathscr{F}$and a Hamilton function$H$. The invariant separated variables are independent on values of integrals of motion and Casimir functions. It means that they are invariant with respect to abelian group of symplectic diffeomorphisms of$\mathscr{F}$and belong to the invariant intersection of all the subfoliations of$\mathscr{F}$. In this paper we show that for many known integrable systems this invariance property allows us to calculate their separated variables explicitly. Citation: Tsiganov A. V., On the Invariant Separated Variables, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 307-326 DOI:10.1070/RD2001v006n03ABEH000179  Kistovich A. V., Chashechkin Y. D. Regular Method for Searching of Differential Equations Discrete Symmetries Abstract A new constructive regular method to search discrete symmetries of a set of differential equations is proposed. The method is based on automorphism properties of a basic$1\$-forms for initial and transformed equations. In difference with well-known methods of discrete symmetry searching the proposed method does not need in a priori knowledge and as sequence in a preliminary searching of infinitesimal symmetries for studied equations. Citation: Kistovich A. V.,  Chashechkin Y. D., Regular Method for Searching of Differential Equations Discrete Symmetries, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 327-336 DOI:10.1070/RD2001v006n03ABEH000180
 Moser J. Remark on the Paper "On Invariant Curves of Area-Preserving Mapping of an Annulus" Abstract Citation: Moser J., Remark on the Paper "On Invariant Curves of Area-Preserving Mapping of an Annulus", Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 337-338 DOI:10.1070/RD2001v006n03ABEH000181

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