THE NEKHOROSHEV THEOREM AND THE ASTEROID BELT DYNAMICAL SYSTEM

The present paper reviews the Nekhoroshev theorem from the point of vi ew of physicists and astronomers. We point out that Nekhoroshev result is strictly connected with the existence of a specific structure of t he phase space, the existence of which can be checked with several num erical tools. This is true also for a degenerate system such as the on e describing the motion of an asteroid in the so called main belt. The main difference is that in some parts of the belt, the Nekhoroshev re sult cannot apply a priori. Mean motion resonances of order smaller th an the logarithm of the mass of Jupiter and first order secular resona nces must be excluded. In the remaining parts, conversely, the Nekhoro shev theorem can be proved, provided some parameters, such as the mass es, the eccentricities and the inclinations of the planets are small e nough. At the light of this result a massive campaign of numerical int egrations of real and fictitious asteroids should allow to understand which is the real dynamical structure of the asteroid belt.