Novel integrable cases in the dynamics of a body interacting with a medium taking into account dependence of the moment of the resistance force on the angular velocity

Planar and spatial non-linear models of the action of a medium on a rigid body which take into account the dependence of the arm of the force on the reduced angular velocity of the body are constructed, when the moment of the force is also a function of the angle of attack. New cases of complete integrability in terms of elementary functions are found, enabling of qualitative similarities to be discovered between the motions of free bodies in a resistive medium and the oscillations of bodies that are partially fixed and are immersed in a flow of the medium. It is shown that if the additional damping action of the medium on the body that appears in the system is significant, the attainment of stability of the rectilinear translational deceleration of the body is possible when it moves with finite angles of attack. The question of the roughness of the description of this phenomenon is of current interest: a finer property of relative roughness is discovered when reduced dynamical systems are investigated. ©2008.