Summary (translated from the Russian): "We consider a variant of a problem of the motion of a body in a resisting medium under the assumption that the interaction of the medium with the body is confined to a part of the surface of a body, which has the form of a flat plate. For plane-parallel motion we completely analyse the case of constant velocity of a center of the plate. We prove the nonexistence of auto-oscillations and prove transcendental integrability."
70: Mechanics of particles and systems.
70E: Dynamics of rigid body.
58: Global analysis, analysis on manifolds.
58F: Ordinary differential equations on manifolds; dynamical systems.
76: Fluid mechanics.
76D: Incompressible viscous fluids.
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Summary (translated from the Russian): "We continue a qualitative analysis of a model variant of the interaction of a body with a resistant medium. Under the assumption that the motion is plane-parallel we completely analyze the case of constant velocity of the center of mass. We prove the presence of nonisolated periodic solutions, the absence of limit cycles and transcendental integrability, and present necessary and sufficient conditions for expressing the integral in terms of elementary functions."
70: Mechanics of particles and systems.
70H: Hamiltonian and Lagrangian mechanics.
34: Ordinary differential equations.
34C: Qualitative theory.
58: Global analysis, analysis on manifolds.
58F: Ordinary differential equations on manifolds; dynamical systems.
70E: Dynamics of rigid body.
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Summary (translated from the Russian): "We consider autonomous systems on the plane or a two-dimensional cylinder and study questions of the existence for various classes of systems of Poincare topographical systems or more general comparative systems. As applications we consider dynamical systems that describe the plane-parallel motion of a body in a resisting medium as well as various model variants of it."
34: Ordinary differential equations.
34C: Qualitative theory.
76: Fluid mechanics.
76D: Incompressible viscous fluids.
Summary (translated from the Russian): "We present a qualitative analysis of a dynamical system that describes a model version of the problem of the plane-parallel motion of a body in a medium with jet or separated flow when the entire interaction of the medium with the body is concentrated on a part of the surface of the body having the form of a flat plate. The force of the interaction is directed along the normal to the plate, and the point of application of this force depends only on the angle of attack. A thrust force acts along the mean perpendicular to the plate, which ensures that the value of the velocity of the center of the plate remains constant throughout the motion. In addition, a damping moment, linear with respect to the angular velocity, is imposed on the body. We carry out the phase portrait classification of the system depending on the coefficient of the moment. We note the mechanical and topological analogies with a pendulum fixed in a flowing medium."
70: Mechanics of particles and systems.
70K: Hamiltonian and Lagrangian mechanics.
34: Ordinary differential equations.
34C: Qualitative theory.
76: Fluid mechanics.
76B: Incompressible inviscid fluids, potential theory.
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Summary (translated from the Russian): "We consider dynamical systems on a two-dimensional cylinder. We sharpen the theorems of Hopf, Bendixon and Dulac, after which it becomes possible to study closed trajectories of various topological types in connection with the problem of the motion of a body in a resisting medium. We give an example of a class of systems in the phase space of which there exists a continuum of closed trajectories of different types."
34: Ordinary differential equations.
34C: Qualitative theory.
70: Mechanics of particles and systems.
70E: Dynamics of rigid body.
Summary (translated from the Russian): "We consider dynamical systems on the plane, cylinder and sphere. For some classes of systems we prove the existence and uniqueness of trajectories going out to infinity in the plane. For one-parameter systems of equations having monotonicity properties on two-dimensional oriented surfaces, we examine the problem of the existence and uniqueness of limit sets and their monotone dependence on the parameters."
34: Ordinary differential equations.
34C: Qualitative theory.
The paper deals with the Kirchhoff problem on the motion of a rigid body in an infinite ideal incompressible fluid medium. The author considers a sixth order dynamic system from which a second order subsystem splits off. The complete topological classification of phase portraits is carried out and a two-parameter family of phase portraits consisting of an uncountable set of topologically distinct phase portraits is isolated.
V.A. Sobolev (Samara)
70: Mechanics of particles and systems.
70E: Dynamics of rigid body.
34: Ordinary differential equations.
34C: Qualitative theory.
70K: Hamiltonian and Lagrangian mechanics.
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The author gives a defenition of the relative robustness of a system of differential equations that differs from previously used definitions. It contains two main points: sufficient smallness of the homeomorphism that produces the equivalence, and C(1)-topology in the space of vector fields. As an example, the author considers a problem that describes the dynamics of a rigid body interacting with a medium. He proves a theorem on absolute robustness, from which it follows that there exists a two-parameter family of phase portraits in which a degenerate transition occurs in the passage from one topological portrait type to another. It should be noted that the space in which the system is absolutely robust has finite mesure, while the space in which the system is a system of the first degree of robustness has mesure zero in the original space.
Gennady Victorovich Gorr (UKR-AOS-A1; Donetsk)
70: Mechanics of particles and systems.
70E: Dynamics of rigid body.
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The author considers a version of the plane-parallel motion of a rigid body in a resisting medium. He assumes that a part of the body's surface has a shape of a flat plate and that the interaction of the medium with the body is concentrated at precisely this part. A similar model proves useful in the investigation of bodies moving in a jet flow.
By eliminating the cyclic coordinates, the equations of motion are reduced to a second order autonomous system. The author uses qualitative methods to study and classify the phase trajectories of the reduced dynamical system. In particilar, he studies limit cycles and singular points of the vector field, and the behavior of stable and unstable separatrices. The presence of free parameters adds additional complexity to the system. For example, the author presents a two-parameter family of dynamical systems with a countable set of topologically different phase portraits.
Igor Gashenenko (UKR-AOS-A1; Donetsk)
70: Mechanics of particles and systems.
70E: Dynamics of rigid body.
34: Ordinary differential equations.
34C: Qualitative theory.
70K: Hamiltonian and Lagrangian mechanics.
76: Fluid mechanics.
76B: Incompressible inviscid fluids, potential theory.
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The notions of the Poincare topographical
system, the characteristic function and the comparison system
are generalized for the higher-dimensional case. Theorem.
Assume that in the 1-connected domain
containing a unique
singular point 
of
the smooth vector field v, there exists the hypersurface
,
such that there
exists a Poincare topographical system, having a center at
and defined
by a smooth function V, extended along 
up to 
, filling the
domain 
and such
that 
in K. Then in the
domain D there is no closed curve consisting of the trajectories of
the vector field v and intersecting
. Applications to
the center-focus problem are discussed.
Natalia Borisovna Medvedeva (RS-CHEL; Chelyabinsk)
34: Ordinary differential equations.
34C: Qualitative theory.
Summary (translated from the Russian): We actually begin with a consideration of a model version of the problem of free plane-parallel braking of a rigid body in a resisting medium under conditions of jet-type or detached flow. We carry out a qualitative analysis of the system of differential equations that describe a given class of motions and, based on it, obtain a new two-parameter family of phase portraits consisting of an uncountable set of nonequivalent portraits without limit cycles. See also 58069, 73068.
70: Mechanics of particles and systems.
70E: Dynamics of rigid body.
34: Ordinary differential equations.
34C: Qualitative theory.