Mathematical Sciences: Reduction Theory in Mechanics and Classical Relativistic Fields

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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.