Geometric Analysis of Mechanical Systems with Symmetry

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AbstractProposal: DMS-9802106Principal Investigator: Jerrold E. MarsdenThis proposal intends to advance funamental research in reductiontheory for mechanical systems with symmetry in the following ways.First, the theory of reduction by stages for group extensions will bedeveloped in both the Hamiltonian (cotangent, symplectic and Poissoncontexts) and the Lagrangian (variational context) points of view.Second, the geometry and dynamics of Lagrangian reduction, especiallythe Lagrange-Poincare equations and the Lagrange-d'Alembert-Poincareequations will be investigated. In addition, the theory of geometricphases for systems reduced using invariants and Kahler structures aswell as the use of blowing up techniques in singular reduction, andmultisymplectic geometry will be advanced.The engineering analysis of mechanical systems such as roboticlocomotion and control devices makes use of the proposers mathematicaltechniques in geometric mechanics. This mathematical foundation willbe of increased importance as the sophistication of such engineeringsystems get more complex and exhibit more exotic dynamical behaviorsand require more delicate control strategies. The topics in thisproposal are critical enabling tools for such research and have provento be of great utility in a variety of systems such as satellitestabilization and underwater vehicle dynamics. They are also being putto fundamental use in numerical simulation packages for nonlineardynamical systems in mechanics, including collision, fluid, andelastic systems as well as in satellite mission planning.