ИСТИНА

In 1890 German mathematician and physicist W. Hess found new special case of integrability of Euler - Poisson equations of motion of a heavy rigid body with a fixed point. In 1892 P.A. Nekrasov proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second order linear differential equation. To obtain this equation we firstly derive the Euler - Poisson equations in the special Kharlamov coordinate system. Using this form of the Euler - Poisson equations we derive the corresponding linear differential equation and transform its coefficients to the form of rational functions. Applying the Kovacic algorithm to the obtained differential equation, we proved that the liouvillian solutions of the corresponding second order linear differential equation exists only in the case, when the moving rigid body is the Lagrange top, or in the case, when the constant of the area integral is zero.