Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields

TL;DR

This paper investigates whether the rotational motion of a rigid satellite under gravitational and magnetic influences can be integrated analytically, concluding non-integrability except in a specific symmetric case with a particular magnetic moment configuration.

Contribution

The study extends Ziglin theory to analyze the integrability of satellite dynamics, demonstrating non-integrability in general and identifying a special symmetric case with a specific magnetic moment relation.

Findings

Most configurations are non-integrable.

A unique symmetric case admits an additional first integral.

The analysis uses an extension of Morales-Ruiz and Ramis' Ziglin theory.

Abstract

In this paper we analyse the integrability of a dynamical system describing the rotational motion of a rigid satellite under the influence of gravitational and magnetic fields. In our investigations we apply an extension of the Ziglin theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric satellite the system does not admit an additional real meromorphic first integral except for one case when the value of the induced magnetic moment along the symmetry axis is related to the principal moments of inertia in a special way.