Gennady Gorr

The problem of the motion of a rigid body with a fixed point in a potential force field is considered. A new case of three nonlinear invariant relations of the equations of motion is presented. The properties of Euler angles, Rodrigues – Hamilton parameters, and angular velocity hodographs in the Poinsot method are investigated using an integrated approach in the interpretation of body motion.

Two particular cases of the Kovalevskaya solution are studied. A modified Poinsot method is applied for the kinematic interpretation of the body motion. According to this method, the body motion is represented by rolling without sliding of the mobile hodograph of the vector collinear to the angular velocity vector along the stationary hodograph of this vector. Two variants are considered: the first variant is characterized by a plane hodograph of the auxiliary vector; the second variant corresponds to the case where the hodograph of this vector is located on the inertia ellipsoid of the body.

The Bobylev–Steklov solution belongs to one of the most well-known particular solutions of the Euler–Poisson equation of the problem of motion of a heavy rigid body with a fixed point. It is characterized by two linear invariant relations and can be expressed as elliptic functions of time. The interpretation of the motion of the Bobylev–Steklov gyroscope was carried out by P.V. Kharlamov using the Poinsot method. Analysis of the neighborhood of the Bobylev–Steklov solution in the integral manifold of the Euler–Poisson equations was presented by B.S. Bardin for the case where this solution describes pendulum motions. It is therefore of interest to study the general case of the above-mentioned manifold. Using the first Lyapunov method, a new class of asymptotic motions is obtained for a heavy rigid body whose limit motions are described by the Bobylev–Steklov solution.