Liouvillian Solutions in the Problem of Motion of a Heavy Gyrostat with a Fixed Point under the Action of Gyroscopic Forces in the Hess Case

The problem of motion of a heavy gyrostat with a fixed point under the action of gyroscopic forces, corresponding to the classical Hess case in the problem of motion of a heavy rigid body with a fixed point, is considered. We derive that the problem of motion of a gyrostat is reduced to solving the second-order linear differential equation with rational coefficients. Using the Kovacic algorithm, we obtain the conditions under which the general solution of the corresponding second-order linear differential equation is expressed in terms of Liouvillian functions and, therefore, it can be presented in explicit form. We prove that under the obtained conditions the equations of motion can be integrated by quadratures.