Long--term stability of proper rotations of the Euler perturbed rigid body

We study the long term stability of the proper rotations of the Euler perturbed rigid body, in the framework of Nekhoroshev theory. For simplicity we treat here in detail only the kinetically symmetric case (the potential needs not to be symmetric), but we indicate how to extend the results to the triaxial case. We show that the proper rotations around the symmetry axis are Nekhoroshev stable: more precisely, if the initial datum is suÆciently close to a proper rotation, then for a very long time it remains such, and the tip of the unit vector \mu parallel to the angular momentum precesses, up to a small noise, along the level curves of a regular function on the unit sphere. If the proper rotations are resonant, chaotic motions with positive Lyapunov exponents are possible, but chaos (unlike the case of ordinary motions, that is motions not close to proper rotations) is always localized, and does not aect in an essential way the motion of the angular momentum in space. Preliminary numerical result indicate that the theory is, in many aspects, optimal, although in some points it can still be improved.