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Long term stability of proper rotations of the perturbed Euler rigid body. (English) Zbl 1087.70003

Summary: We study the long term stability of the proper rotations of the perturbed Euler 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 sufficiently 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 affect in an essential way the motion of the angular momentum in space. Preliminary numerical results indicate that the theory is, in many aspects, optimal, although in some points it can still be improved.

MSC:

70E20 Perturbation methods for rigid body dynamics
70E17 Motion of a rigid body with a fixed point
70E50 Stability problems in rigid body dynamics
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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