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In this activity the qualitative analysis of spatial problems of the real rigid body motions in a resistant medium is fulfilled. A nonlinear model that describes the interaction of a rigid body with a medium and takes into account (based on experimental data on the motion of circular cylinders in water) the dependence of the arm of the force on the normalized angular velocity of the body and the dependence of the moment of the force on the angle of attack is constructed. An analysis of plane and spatial models (in the presence or absence of an additional tracking force) leads to sufficient stability conditions for translational motion, as one of the key types of motions. Either stable or unstable self-oscillation can be observed under certain conditions.

The results of this work appeared in the process of studying a certain problem on the rigid body motion in a medium with resistance [Chaplygin, 1933], [Chaplygin, 1976], [Shamolin-1, 2007], [Shamolin-2, 2007], where we needed to deal with first integrals having nonstandard properties. Precisely, they are not analytic, not smooth, and on certain sets, they can be even discontinuous. Moreover, they are expressed through a finite combination of elementary functions. However, the latter circumstances allowed us to carry out a complete analysis of all phase trajectories and show those their properties which have a ”roughness” and are preserved for systems of a more general form having certain symmetries of latent type. Therefore, it is interesting to study sufficiently wide classes of dynamical systems that have analogous properties and moreover, coming from the dynamics of a rigid body interacting with a medium.

Institute of Mechanics, Lomonosov Moscow State University, Michurinskii Ave., 1, 119899 Moscow, Russian FederationThe results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium.Moreearlierthecomplete listsoftranscendentalfirstintegralsexpressed througha finitecombinationofelementary functionswere obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlightthose properties of them which exhibit the roughness and preserve for systems of a more general form.

The purpose of the activity is to elaborate the qualitative methods for studying the dynamics of rigid bodies interacting with a resisting medium under quasistationarity conditions. This material refers equally to the qualitative theory of ordinary differential equations and the dynamics of rigid bodies. We use the properties of body’s motion in a medium under conditions of the jet flow past this body. We study the plane model problems of the motion of a body with the cone form of its shape in a resisting medium. The new families of phase portraits of variable dissipation systems are obtained, their absolute or relative roughness is demonstrated. The integrable cases of equations of motion of rigid bodies are found. c � 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

In is well–known due to its complexity, the problem of the motion of a rigid body in an unbounded medium requires the introduction of certain simplifying restrictions. The main aim in this connection is to introduce hypotheses that would make it possible to study the motion of the rigid body separately from the motion of the medium in which the body is embedded. On the one hand, a similar approach was realized in the classical Kirchhoff problem on the motion of a body in an unbounded ideal incompressible fluid that undergoes an irrotational motion and is at rest at infinity. On the other hand, it is obvious that the above–mentioned Kirchhoff problem does not exhaust the possibilities of this kind of simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)