Arnold Diffusion, Quasi-ergodic Hypothesis, Instabilities for the Planar 3 Body Problem, and Central Configurations

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The intellectual merit of this proposal lies in developing techniques to understand formation of instabilities for abstract and concrete nearly integrable Hamiltonian systems. This includes the classical 3 body problem from celestial mechanics, which in particular describes the 3 body problem modeling the Sun-Jupiter-Asteroid system. So far mathematical description of instabilities for these and many other nearly integrable systems is limited. On the contrary stability for large measure of initial conditions is described by the famous KAM theory.The main goal of the project is analysis of the stability of motion of complex mechanical systems over long periods of time, focusing on the motion of planets. The behavior of such complex systems can be seen either as regular, as in the motion of planets, or chaotic, as in the motion of a hurricane. We shall analyze the interplay between regular and chaotic behavior to determine the length of stability of various systems, including the complicated three-body problem in classical mechanics, which involves determining the motion of three celestial bodies moving under no influence other than that of their mutual gravitation. Knowledge of instabilities of this system is fairly limited. The object is to develop techniques to investigate the stability time of these systems. The project will also involve graduate student training. Students will become expert celestial mechanics and will have first-hand experience in working on classical problems using relatively new mathematical tools.