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Vol 227, No 4 (2017)
- Year: 2017
- Articles: 8
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14872
Article
Sessions of the Workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, “Urgent Problems of Geometry and Mechanics” Named After V. V. Trofimov
387-394
Chromatic Number with Several Forbidden Distances in the Space with the ℓq-Metric
Abstract
We study the chromatic number \( \overline{\chi}\left(X;\rho; k\right) \) of a metric space X with a metric ρ and k forbidden distances. We obtain an estimate of the form \( \overline{\chi}\left({\mathbb{R}}^n;\rho; k\right)\ge {(Bk)}^{Cn} \) for cases where the metric ρ on the set ℝn is generated by the ℓq-norm.
395-401
402-406
407-418
Integrable Motions of a Pendulum in a Two-Dimensional Plane
Abstract
In this paper, we examine new cases of integrability of dynamical systems on the tangent bundle to a low-dimensional sphere, including flat dynamical systems that describe a rigid body in a nonconservative force field. The problems studied are described by dynamical systems with variable dissipation with zero mean. We detect cases of integrability of equations of motion in transcendental functions (in terms of classification of singularity) that are expressed through finite combinations of elementary functions.
419-441
Transcendental First Integrals of Dynamical Systems on the Tangent Bundle to the Sphere
Abstract
In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.
442-460
Pendulum Systems with Dynamical Symmetry
Abstract
In this paper, we consider a class of oscillatory mechanical systems described by nonlinear second-order differential equations that contain parameters with variable dissipation and possess the property of dynamic symmetry. We study properties of mathematical models of such systems depending on parameters. Special attention is paid to bifurcations and reconstructions of phase portraits and the appearance and disappearance of cycles that envelope the phase cylinder or lie entirely on its covering. We present a complete parametric analysis of system with position-viscous friction and four parameters, where the force action linearly depends on the speed. The space of parameters is split into domains in which the topological behavior of the system is preserved. For some classes of pendulum systems whose right-hand sides depend on smooth functions, a qualitative analysis is performed, i.e., the space of systems considered is split into domains of different behavior of trajectories on the phase cylinder of quasi-velocities. We also perform a systematic analysis of the problem on an aerodynamic pendulum and detect periodic modes unknown earlier; these modes correspond to cycles in the mathematical model.
461-519
Asymptotics of Motions of Viscous Incompressible Fluids with Large Viscosity
Abstract
We examine a nonstationary initial-boundary-value problem on the motion of a viscous incompressible fluid with large viscosity. We obtain estimates of the convergence of solutions of this problem to solutions of the corresponding linearized problems as the viscosity tends to infinity. We also consider the case of the problem periodic in time.
520-530