New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium (Q1407374)

The objective of the paper is to introduce new cases of integrability in the plane and spacial dynamics of rigid bodies interacting with a resisting medium. The study focuses first on a rigid body executing a plane-parallel motion in a medium with quadratic resistance law. One assumes that the action of the medium on the outer surface of the body, which is a flat plate, is reduced to the force acting along the line that is orthogonal to the plate. The paper begins by introducing the equations of a free-moving body together with additional assumptions on the classes of dynamical functions corresponding to experimental data jet flows without damping. It discusses equations of motion when the velocity of the center of mass is constant, the constraint attained by adding a suitable control driving force. The further study focuses on study of transcendental first integrals of the above systems written in quasi-velocity variables and broadens the discussion to include the analysis of the topological structure of the phase portraits. The partition of the quasi-velocity phase plane into regions with distinct trajectories behavior is analyzed and classification of singular points is given in many cases. Having introduced certain simplifying assumptions, the author gives a qualitative treatment to many cases of the plane and space motions of a body in a resisting medium including the physical pendulum systems and in certain cases writes explicitly the closed algebraic expressions for transcendental first integrals.