Mathematical Sciences: Mathematical Aspects of Classical Field Theory

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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.