Stability of a four body relative equilibrium

In the present work we study the lineal stability of a relative equilibrium for the problem of the gyrostat in newtonian interaction with three spherical rigid bodies or punctual masses. Geometrically the relative equilibrium is characterized by a particular symmetry, i.e., the rigid bodies have all the same mass m and form an equilateral triangle. On the other hand, the gyrostat of mass m0 with revolution symmetry around the third axis of inertia is located in the center of this triangle rotating with an angular velocity ωe, that will be determined, perpendicular to the plane formed by the previous spherical masses.This research was partially supported by the Spanish Ministerio de Ciencia y Tecnología (Project BFM2003-02137) and by the Consejería de Educación y Cultura de la Comunidad Autónoma de la Región de Murcia (Project Séneca 2002: PC-MC/3/00074/FS/02)