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Symplectic reduction for semidirect products and central extensions. (English) Zbl 0973.53069

Summary: This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980’s by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle.
We shall start with the classical cases of commuting reduction and present a new proof and approach to semidirect product theory. We shall then give an idea of how the more general theory of group extensions proceeds. The case of central extensions is illustrated in this paper with the example of the Heisenberg group. The theory, however, applies to many other interesting examples such as the Bott Virasoro group and the KdV equation.

MSC:

53D20 Momentum maps; symplectic reduction
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