Dual pairs in fluid dynamics

This article is a rigorous study of the dual pair structure of the ideal fluid (Phys D 7:305-323, 1983) and the dual pair structure for the n-dimensional Camassa-Holm (EPDiff) equation (The breadth of symplectic and poisson geometry: Festshrift in honor of Alan Weinstein, 2004), including the proofs of the necessary transitivity results. In the case of the ideal fluid, we show that a careful definition of the momentum maps leads naturally to central extensions of diffeomorphism groups such as the group of quantomorphisms and the Ismagilov central extension.