Mathematical Sciences: Mathematical Aspects of Classical Field Theory

本研究探讨了经典场论中初值约束与规范变换之间的联系。标准算例的初值分析表明：(1)欧拉-拉格朗日方程同时存在欠定和过定；（ii）（第一类）约束由与规范群相关的能量动量图的消失给出；(3)演化方程可以写成伴随式。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。分析的关键工具是“能量动量图”。这项研究的总体目标是发展对物理实体（如引力、电磁力和核力）背后的物理理论的数学结构的理解。最近的研究表明，所有的基本物理过程都可以用一种非常相似的数学方式来描述，这一观察导致了几个引人注目的特征和现象的发现，这些特征和现象对这些过程产生了新的见解。这些见解反过来又在机器人、最优控制和激光光学等远离其根源的领域产生了应用。提出的研究将努力统一和扩展基于几何和对称考虑的现有方法，从而获得对物理理论结构的根本新理解。

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.