Mathematical Sciences: Dynamical Systems

这两位调查员正在从事不同的项目。然而，它们都与Kolmogorov - Arnold - Moser理论有关。首席研究员马瑟正在进行一个项目，以证明“阿诺德扩散”存在于正法向扭转的可积哈密顿系统的小扰动中。希望是通过变分方法来解决这个问题，或者至少是它的一个版本。Forni希望将Kolmogorov、Arnold和Moser的基本摄动定理推广到高属曲面上的线性流动。Kolmogorov, Arnold， Moser理论和关于Arnold扩散的已知结果都对太阳系的稳定性问题有重要的应用。这些研究涉及艾萨克·牛顿的太阳系数学模型，并试图回答这个问题，即数学模型是否对所有时间都是稳定的，而不仅仅是宇宙的年龄。哈密顿系统是牛顿太阳系模型的数学抽象，它适用于许多其他的机械系统。这两个项目都处理与太阳系稳定性的已知结果有关的更一般情况下的问题。

The two investigators are working on different projects. Each, however, is related to Kolmogorov - Arnold - Moser theory. The principal investigator, Mather, is working on a project to prove the existence of "Arnold diffusion" in small perturbations of integrable Hamiltonian systems of positive normal torsion. The hope is to solve this problem, or at least a version of it, by variational methods. Forni hopes to extend the fundamental perturbation theorem of Kolmogorov, Arnold, and Moser for linear flows on a surface of higher genus. Kolmogorov, Arnold, Moser theory and known results about Arnold diffusion both have significant applications to the problem of the stability of the solar system. These studies deal with Isaac Newton's mathematical model of the solar system and attempt to answer the question as to whether the mathematical model is stable for all time, not just the age of the universe. Hamiltonian systems are a mathematical abstraction of Newton's model of the solar system which applies to many other mechanical systems. Both projects deal with questions in this more general situation related to known results on the stability of the solar systems.