Mathematical Sciences: Geometry and Dynamics of Mechanical Systems with Symmetry

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9633161 Marsden The proposed research lies in the area of geometric mechanics. It has to do with laying a theoretical foundation for various aspects of classical mechanics. Nonholonomic systems with symmetry are to be investigated as an application of the ideas of Lagrangian reduction. Integrable systems and solitons will be pursued with a view towards certain applications in control dynamics. Bifurcations in Hamiltonian systems will be investigated in the context of singular reduction. Geometry of control systems will also be studied. Geometric mechanics aims to formulate theories and problems in classical mechanics in a mathematically rigorous setting. There have been several recent successes of geometric mechanics. The falling cat example, where the cat executes a complete re-orientation without exerting any angular momentum, has now been explained using the idea of a connection - a connection is a mathematical notion originally introduced to study so called principal bundles in differential geometry. The proposed research will further pursue these theoretical ideas as well as consider possible applications in such areas as control theory and robotics.