An overview on the Nekhoroshev theorem

The Nekhoroshev theorem has provided in the last decades an important framework to study the long-term stability of quasi-integrable dynamical systems. While KAM theorem predicts the stability of invariant tori for non–resonant motions, the Nekhoroshev theorem provides exponential stability estimates for resonant motions. The first part of this chapter reviews the mechanisms which are at the basis of the proof of Nekhoroshev theorem. The application of Nekhoroshev theorem to systems of interest for Celestial Mechanics encounters difficulties due to the presence of the so–called proper degeneracy, or super–integrability, of the Kepler problem. For this reason, for concrete systems such as planetary systems or asteroids of the main belt, the application of the Nekhoroshev theorem requires modifications of the theory. At variance with the non-degenerate case, these systems can have chaotic diffusion in relatively short times. The second part of this chapter reviews the problems related to degenerate systems, and describes the mechanism of production of chaotic motions in short times.