EFFECTIVE STABILITY OF HAMILTONIAN PLANETARY SYSTEMS

The stability of the Solar System is a classical, long standing and challenging problem, already pointed out by Newton. In this thesis we revisit the problem in the light of Kolmogorov and Nekhoroshev theorems, with the aim of proving that they apply to realistic approximations of the Sun-Jupiter-Saturn and Sun-Jupiter-Saturn-Uranus systems. The present thesis is devoted to the study of three main problems, namely: (i) the applicability of Kolmogorov and Nekhoroshev theories to the problem of three bodies; (ii) the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system; (iii) the explicit construction of the normal form for elliptic tori in planetary systems. This work gives an original contribution on the methods for studying the stability of planetary systems. It contains an analytical contribution, namely the proof of the existence of low dimensional elliptic tori for the planetary system obtained via an algorithmic constructive method, and the explicit calculation of the stability time for the Sun-Jupiter-Saturn system and for the planar secular Sun-Jupiter-Saturn-Uranus system. We emphasize that we obtain the first realistic estimates for these problems based on a well established theoretical framework.