Comparison of jacobi-integrable cases of the plane and three-dimensional motion of a body in a medium in the case of jet flow

The complete integrability of the plane problem of the motion of a rigid body in a resisting medium under jet flow conditions is shown, when one first integral, which is transcendental function of the quasi-velocities (in the sense of the theory of functions of a complex variable having essentially singular points), exists in the system of dynamic equations. It is assumed that all the interaction of the medium with the body is concentrated in that part of the surface of the body which has the form of a (one-dimensional) plate. The plane problem is extended to the three-dimensional case; then a complete set of first integrals exists in the system of dynamic equations: one analytic, one meromorphic and one transcendental. It is assumed here that all the interaction of the medium with the body is concentrated in that part of the surface of the body which has the form of a plane (two-dimensional) disc. An attempt is also made to construct an extension of the “low-dimensional” cases to the dynamics of a so-called four-dimensional rigid body, interacting with a medium which is concentrated in that part of the (three-dimensional) surface of the body which has the form of a (three-dimensional) sphere. The vector of the angular velocity of the motion of such a body in this case is six-dimensional, while the velocity of the centre of mass is four-dimensional.