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

这一建议的智力价值在于发展了理解抽象和具体的近可积哈密顿系统的不稳定性形成的技术。这包括天体力学中的经典三体问题，它特别描述了太阳-木星-小行星系统建模的三体问题。到目前为止，对这些和许多其他近可积系统的不稳定性的数学描述是有限的。相反，大尺度初始条件下的稳定性由著名的KAM理论描述。该项目的主要目标是分析长时间内复杂机械系统的运动稳定性，重点是行星的运动。这种复杂系统的行为可以被看作是规则的，如行星的运动，或混乱的，如飓风的运动。我们将分析规则行为和混沌行为之间的相互作用，以确定各种系统的稳定性长度，包括经典力学中复杂的三体问题，这涉及到确定除了相互引力之外没有任何影响的三个天体的运动。关于这个系统的不稳定性的知识相当有限。目的是发展技术来研究这些系统的稳定时间。该项目还将涉及研究生培训。学生将成为天体力学专家，并将有使用相对较新的数学工具处理经典问题的第一手经验。

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.