Application of the Kovacic algorithm for the investigation of motion of a heavy rigid body with a fixed point in the Hess case

TL;DR

This paper applies the Kovacic algorithm to analyze the integrability of a heavy rigid body's motion in the Hess case, identifying conditions for Liouvillian solutions of the associated differential equation.

Contribution

It derives the linear differential equation for the Hess case and demonstrates, using Kovacic's algorithm, that Liouvillian solutions exist only for specific rigid body configurations.

Findings

Liouvillian solutions exist only for the Lagrange top or zero area integral case.

The differential equation coefficients are expressed in rational form.

The Kovacic algorithm confirms integrability conditions in the Hess case.

Abstract

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. In this paper the corresponding linear differential equation is derived and its coefficients are presented 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 the Lagrange top, or in the case when the constant of the area integral is zero.