On the Stability of the Regular Grioli Precession in a Particular Case

Abstract

In a homogeneous gravitational field, the movement of a rigid body around a fixed point is considered. No dynamic symmetry of the body is assumed, and its centroid lies on the perpendicular to a circular section of the moment ellipsoid, restored from the fixed point. Within this mass geometry a regular axial precession of the body is possible, such that the precession axis does not coincide with the vertical axis (the Grioli precession). We investigate the stability of that precession for the special case where the ellipsoid is strongly stretched along an axis close to the axis containing the centroid. The regular precession of the body over an axis inclined with respect to vertical axis has been discovered by Grioli in 1947 (see [1]). This investigation is extended in [2–5]: in particular, it is shown that no heavy rigid body has regular precessions different from the classical precession of a dynamically symmetric body (in the Lagrange case) and the regular precession described by Grioli. The history of the discovery and investigation of the precession motions of rigid bodies is presented in [6–8]. In [9], the investigation of the stability problem for the Grioli precession is originated. Further, various ways that the problem has been posed (both numerical and analytical) are considered in [10–15]. However, no complete or strict solution for all admissible values of parameters of the problem has yet been obtained. Below, we present new results on the orbital stability of the Grioli precession.