Mathematical Sciences: Reduction Theory in Mechanics and Classical Relativistic Fields

将探讨具有对称性的机械系统的约简。所要研究的问题是由稳定性理论和经典相对论场理论的最新进展所激发的。将使用的块对角化技术是基于最近解决旋转和内部模式分离问题的方法。我们将研究这种技术背后的一般几何理论。利用基于约简思想的经典场论的最新进展，发展相应的协变约简理论和相关主题。这个项目将扩展我们对相对论性磁场、电场和引力场稳定性的理解。这些可以描述运动的场，如果在各种扰动下系统以某种方式保持其基本性质，则被称为稳定场。几十年来，这些问题一直受到严密的数学研究，这个项目将扩大我们对它们的认识。

Reduction of mechanical systems with symmetry will be pursued. The problems to be studied were motivated by recent advances in stability theory and the theory of classical relativistic fields. The block diagonalization technique to be used is based on a recent solution to the problem of separation of rotational and internal modes. The general geometric theory underlying this technique will be investigated. Recent advances in classical field theory based on reduction ideas will be exploited to develop corresponding covariant reduction theory and related topics. This project will extend our understanding of the stability of relativistic magnetic, electrical, and gravitational fields. These fields, which may describe motion, are said to be stable if under various sorts of perturbations the system maintains its basic nature in some way. These have come under intense mathematical scrutiny for several decades and this project will extend our knowledge of them.