Variational Methods in Hamiltonian Mechanics

PI：John N.本项目的目的是用变分方法构造哈密顿系统的轨道。 重点是与阿诺德扩散有关的问题。 对于2中严格凸可积系统的小扰动， 或3自由度，马瑟最近成功地证明了阿诺德扩散的强形式。 对于尖点剩余集中的扰动，存在其作用以相当任意的规定方式变化的轨道。该项目的一个主要目标是将这些结果推广到3个自由度的严格凸系统的小扰动。该项目涉及各种物理应用中出现的基本数学问题，例如托卡马克中等离子体的包容（用于通过核聚变产生能量）和行星系统中轨道的稳定性问题。 在每种情况下，它都与动力系统的某些轨道是否漂移有关。 就托卡马克而言，数学问题与等离子体中的氢原子是否与容器壁碰撞的物理问题有关;在太阳系的情况下，它涉及到行星是否一直围绕太阳旋转的问题，或者行星之间的相互引力是否会导致其中一颗行星在很长一段时间内远离太阳。 然而，需要指出的是，这个项目涉及的是一个起源于这类物理问题研究的基本数学问题;它并不直接解决这些问题。

PI: John N. Mather, Princeton UniversityDMS-0245336AbstractThe aim of this project is to construct orbits of Hamiltonian systems by variational methods. The focus is on questions related to Arnold diffusion. For small perturbations of strictly convex integrable systems in 2 or 3 degrees of freedom, Mather has recently succeeded in proving a strong form of Arnold diffusion. For perturbations in a cusp residual set, there are orbits whose action varies in a fairly arbitrary prescribed way. A major goal of this project will be to generalize these results to small perturbations of strictly convex systems in 3 degrees of freedom.This project deals with fundamental mathematical questions that arise in various physical application, e.g. the containment of a plasma in a tokomak (for the production of energy through nuclear fusion) and the question of the stability of orbits in planetary systems. In each case, it has to do with whether certain orbits of dynamical systems wander. In the case of the tokomak, the mathematical questions is related to the physical question of whether the hydrogen atoms in the plasma collide with the walls of the container; in the case of the solar system it is related to the question of whether the planets remain orbiting the sun for all time, or whether the mutual gravitational attraction of the planets could cause one of the planets to wander away from the sun over a very long period of time. It needs to be pointed out, however, that this project deals with a fundamental mathematical problem that originated in the study of physical problems of this sort; it does not address these problems directly.