Fast rotations of the symmetric rigid body: a general study by Hamiltonian perturbation theory. Part I.

In this paper we study the dynamics of a fast rotating symmetric rigid body in an arbitrary analytic potential, by means of the techniques of Hamiltonian perturbation theory, specifically Nekhoroshev's theory. For a rigid body with a fixed point, this approach allows us to get an accurate description of the motion (with rigorous estimates) on time scales which increase very fast with the angular velocity Omega of the body, precisely as exp Omega(1/2). As we show, on such time scales, the body performs an approximately free motion around the instantaneous direction of the angular momentum vector m, which in turn moves slowly in space, with speed of order Omega(-1). The properties of such a slow drift of m strongly depend on the resonance properties of the initial angular velocity. For nonresonant initial conditions, the unit vector mu in the direction of m closely follows, on the unit sphere, the level curves of an averaged external potential, so the motion is essentially regular (chaotic effects, if any, are confined to small-amplitude oscillations around such motion). On the contrary, in case of resonance, mu can perform large-scale chaotic motions, which spread over genuinely two-dimensional regions of the unit sphere; numerical evidence of this phenomenon in an oversimplified model is also provided. Similar results are proven for prolate rigid bodies with no fixed point; an important difference is that, because of the presence of additional 'slow' degrees of freedom, a large-scale chaotic behaviour of m is now possible in nonresonant motions, too. Because of the presence of singularities of the action-angle coordinates, our description is not optimal for motions very close to gyroscopic rotations; the forthcoming part II will be devoted to these special motions.