Today I heard the claim that in practice, all symplectic manifolds that people care about arise as the Hamiltonian reduction of a cotangent bundle $T^{\ast}(M)$ under the action of a Lie group $G$ ($M$ and $G$ may both be infinite-dimensional in general, I think). For example many moduli spaces of interest arise in this way. Is it literally true that every symplectic manifold arises in this way? Can we moreover arrange for $G$ and $M$ to be finite-dimensional? 



Is every symplectic manifold a Hamiltonian reduction of a cotangent bundle?
Actually if you allow infinite dimension, every symplectic manifold is a coadjoint orbit of its group of symplectomorphisms. That is even more... how to say? Symplectic :-) If you want a reference there is a diffeological version of this theorem [here](http://math.huji.ac.il/~piz/Site/The%20Articles/6B9FFBDA-EB3F-4222-8CDD-8A5359C87D34.html) It is also in the Memoir [here](http://math.huji.ac.il/~piz/documents/TMMID.pdf) §10


