On a Class of Precessions of a Rigid Body with a Fixed Point under the Action of Forces of Three Homogeneous Force Fields

This paper is concerned with a special class of precessions of a rigid body having a fixed point in a force field which is a superposition of three homogeneous force fields. It is assumed that the velocity of proper rotation of the body is twice as large as its velocity of precession. The conditions for the existence of the precessions under study are written in the form of a system of algebraic equations for the parameters of the problem. Its solvability is proved for a dynamically symmetric body. It is proved that, if the ellipsoid of inertia of the body is a sphere, then the nutation angle is equal to $\arccos \frac{1}{3}$. The resulting solution of the equations of motion of the body is represented as elliptic Jacobi functions.