Application of the Kovacic Algorithm to the Problem of Motion of a Heavy Rigid Body with a Fixed Point in a Hess Case

Abstract

In 1890 W. Hess found new partial case of integrability of Euler – Poisson equations describing the motion of a heavy rigid body about 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 in a Hess case is reduced to integration the second order linear differential equation. In this paper the derive the corresponding linear differential equation and present its coefficients in the rational form. Using the Kovacic algorithm 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 a Lagrange top, or in the case when the constant of the area integral is zero.