Mathematical Sciences: Mathematical Aspects of Classical Field Theory

这项研究探讨了初始值与 经典场的约束与规范变换 理论 标准算例的初值分析表明， （1）Euler-Lagrange方程同时 （二）第一次和第二次的分别。 约束是由能量-动量映射的消失给出的 与规范群有关的;（iii）演化方程 可以写成伴随形式。 诺特定理和狄拉克- 伯格曼约束分析有助于预测和解释 这些特征。 分析中的关键工具是 “能量-动量图。" 本研究的总体目标是开发一种 理解物理理论的数学结构 潜在的物理实体，如引力， 电磁力和核力 最近的研究工作表明 所有基本的物理过程都可以用 数学上类似的方式，这一观察导致了 发现了几个显著的特征和现象， 对这些过程产生新的见解。 这些见解，在 反过来，已经在一些领域产生了应用， 他们的来源，如机器人，最佳控制和激光光学。 拟议的研究将奋进统一和扩大目前的 方法，基于几何和对称的考虑， 从而获得了一个全新的理解， 物理理论的结构。

This research explores connections between initial value constraints and gauge transformations in classical field theories. Initial value analyses of standard examples indicate that (i) the Euler-Lagrange equations are simultaneously underdetermined and overdetermined; (ii) the (first class) constraints are given by the vanishing of the energy-momentum map associated to the gauge group; and (iii) the evolution equations can be written in adjoint form. Noether's theorem and the Dirac- Bergmann analysis of constraints help to predict and explain these features. The key tool in the analysis is the "energy- momentum map." The general goal of this research is to develop an understanding of the mathematical structure of physical theories underlying physical entities such as gravitation, electromagnetism and the nuclear forces. Recent work has shown that all basic physical processes can be described in a very similar manner mathematically, and this observation has led to the discovery of several striking features and phenomena which yield new insights into these processes. These insights, in turn, have generated applications in fields far from removed their source, such as robotics, optimal control and laser optics. The proposed research will endeavor to unify and extend current approaches, based on geometric and symmetry considerations, thereby obtaining a fundamentally new understanding of the structure of physical theories.