Mathematical Sciences: Mathematical Aspects of Classical Field Theory

这项研究探索了经典场论中初值约束和规范变换之间的联系。标准算例的初值分析表明：(I)欧拉-拉格朗日方程同时是欠定和超定的；(Ii)(第一类)约束是由与规范群有关的能量-动量映射的消失给出的；(Iii)演化方程可以写成伴随形式。Noether定理和Dirac-Bergmann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”。这项研究的总体目标是了解作为物理实体基础的物理理论的数学结构，如引力、电磁和核力。最近的工作表明，所有基本的物理过程都可以用非常相似的方式在数学上描述，这一观察导致了几个显著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。这些洞察力反过来又在机器人、最优控制和激光光学等领域产生了应用，而不是消除它们的根源。拟议的研究将努力统一和扩展现有的基于几何和对称性考虑的方法，从而从根本上对物理理论的结构有一个新的理解。

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.