Stability of Free Rotation of a Rigid Body

Abstract

The paper shows that a greater excursion of initial conditions, than that previously derived by the author, permits stability of free rotation of a rigid body about either of its stable principal axes. The results are exhibited by means of contour plots, in a new diagram that represents the dynamical properties of any rigid body. For time intervals short compared with geological eras, the paper also shows that, if the earth were truly rigid, its pole of rotation would move about its pole of figure in an ellipse of small eccentricity, dependent only on the principal moments of inertia. Because of non-rigidity the observed path of the pole of rotation is very different from this. The question is raised whether the data on polar motion could ever be taken so accurately and so frequently that a sort of osculating polar ellipse, correspond­ ing to an instantaneously rigid earth, could be fitted reliably at closely spaced intervals. If so, the path of the center of the ellipse would give the path of the pole of figure, which may have a seasonal period of one year. A recent paper by the author, henceforth referred to as V1969, derived some new quantitative results concerning an old problem, the stability of free rotation of a rigid body. The method used only the integrals of motion and the physical assumption that the body's angular velocity 00 is a continuous function oftime. The present paper deals with the same topic, but yields as its main results suf­ ficient conditions for stability of rotation that are much less restrictive. It then depicts these conditions by means of contour lines in a new diagram that classifies all rigid bodies dynamically. Finally, it improves the earlier discussion of polar wandering in the earth. For a rigid body rotating freely or in a uniform gravitational field, if T is its kinetic energy of rotation and L its angular momentum, both taken relative to the center of mass as origin, their constancy results in the equations 2T = Aooi + Boo~ + Cw~ = Awio + Bw~o + Cw~o L2 = A 2 wi + B2W~ + C2W~ = A 2 wio + B2W~0 + C2W~0'