- 1990k: 70007 70E99 (58F40, 76D25) Samsonov V.A., Shamolin M.V. On the problem of the motion of a body in a resisting medium (Russian). Vestnik Moskov. Univ. Ser.I Mat. Mekh. 1989, no 3, 51-54, 105.
- 1993k: 70028 70H05 (34C05, 34C25, 58F40, 70E15) Shamolin, M.V. On the problem of the motion of a body in a resistant medium. (Russian. Russian summary) Vestnik Moskov. Univ. Ser.I Mat. Mekh. 1992, no.1, 52-58, 112.
- 1994b: 34060 34C99 (34C05, 34C25, 76D99) Shamolin, M.V. Application of the methods of Poincare topographical systems and comparison systems in some concrete systems of differential equations. (Russian. Russian summary) Vestnik Moskov. Univ. Ser.I Mat. Mekh. 1993, no.2, 66-70, 113.
- 1994i: 70027 70K99 (34C99, 70K20, 76B05) Shamolin, M.V. Phase portrait classification in a problem on the motion of a body in a resisting medium in the presence of a linear damping moment. (Russian. Russian summary) Prikl. Mat. Mekh. 57 (1993), no.4, 40-49; translation in J. Appl. Math. Mech. 57 (1993), no.4, 623-632.
- 1995d: 34060 34C23 (34C05, 34C25, 70E15) Shamolin, M.V. Closed trajectories of varios topological types in the problem of the motion of a body in a resisting medium. (Russian. Russian summary) Vestnik Moskov. Univ. Ser.I Mat. Mekh. (1992), no.2, 52-56, 112.
- 1995e: 34036 34C35 (34C99) Shamolin, M.V. Existence and uniqueness of trajectories that have points of infinity as limit sets for dynamical systems on the plane. (Russian. Russian summary) Vestnik Moskov. Univ. Ser.I Mat. Mekh. (1993), no.1, 68-71, 112.
- 1995g: 70006 70E15 (34C99, 70K05) Shamolin, M.V. (RS-MOSC; Moscow) A new two-parameter family of phase portraits in the problem of the motion of a body in a medium. (Russian) Dokl. Akad. Nauk 337 (1994), no.5, 611-614; translated in Phys. Dokl. 39 (1994), no.8, 587-590.
- 1997f: 70010 70E15 Shamolin, M.V. (RS-MOSC; Moscow) Determination of relative robustness and a two-parameter family of phase portraits in the dynamics of a rigid body. (Russian) Uspekhi Mat. Nauk 51 (1996), no.1 (307), 175-176; translation in Russian Math. Surveys 51 (1996), no.1, 165-166.
- 1998b: 70009 70E15 (34C05, 70K05, 76B10) Shamolin, M.V. (RS-Mosc; Moscow) Manifold of phase portraits types in the dynamics of a rigid body interacting with a resisting medium. (Russian) Dokl. Akad. Nauk 349 (1996), no.2, 193-197; translation in Phys. Docl. 41 (1996), no.7, 320-324.
- 1999a: 34089 34C05 34C11 Shamolin, M.V. (RS-MOSC; Moscow) Spartial Poincare topographical systems and comparison systems. (Russian) Uspekhi Mat. Nauk 52 (1997), no.3 (315), 177-178; translation in Russian Math. Surveys 52 (1997), no.3, 621-622.
- 1999e: 70027 70E99 34C99 Shamolin, M.V. An introduction to the problem of the braking of a body in a resisting medium, and a new two-parameter family of phase portraits. (Russian. Russian summary) Vestnik Moskov. Univ. Ser.I Mat. Mekh. 1996, no.4, 57-69,112.

*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.

*******

*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.

*******

*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.

*******

*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.

*******

*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.

*******

*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.

*******

*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.