Existence of Liouvillian solutions in the Hess-Sretensky case of the problem of motion of a gyrostat with a fixed point

TL;DR

This paper investigates the conditions under which the equations of motion for a heavy gyrostat in the Hess-Sretensky case have Liouvillian solutions, extending classical integrability results using differential Galois theory.

Contribution

The paper derives the differential equation for the Hess-Sretensky case and applies the Kovacic algorithm to find conditions for Liouvillian solutions, advancing understanding of integrability in gyrostat dynamics.

Findings

Identified parameter conditions for Liouvillian solutions

Reduced the differential equation to rational function coefficients

Provided criteria for integrability in quadratures

Abstract

In 1890 W. Hess found the new special case of integrability of the Euler - Poisson equations of motion of a heavy rigid body with a fixed point. In 1963 L.N. Sretensky proved that the special case of integrability, similar to the Hess case, also exists in the problem of the motion of a heavy gyrostat - a heavy rigid body with a fixed point, which contains a rotating homogeneous rotor. Further numerous generalizations of the classical Hess case were proposed, which take place during the motion of a heavy rigid body and a gyrostat with a fixed point in various force fields. The first studies that provided a qualitative description of the motion of a heavy rigid body in the integrable Hess case were published almost immediately after this case was found. In 1892 P.A. Nekrasov proved, that the solution of the problem of motion of a heavy rigid body with a fixed point in the Hess case is reduced to the integration the second order linear homogeneous differential equation with variable coefficients. A similar result regarding the problem of the motion of a heavy gyrostat in the Hess - Sretensky case was presented by Sretensky. In this paper we present the derivation of the corresponding second order linear differential equation and reduce the coefficients of this equation to the form of rational functions. Then, using the Kovacic algorithm we study the problem of the existence of liouvillian solutions of the corresponding second order linear differential equation. We obtain the conditions for the parameters of the problem, under which the liouvillian solutions of the corresponding linear differential equation exist. Under these conditions equations of motion of a heavy gyrostat with a fixed point in the Hess - Sretensky case can be integrated in quadratures.