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The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function.

I am wondering if someone can give me an interesting, or useful, example of a Hamiltonian system for which $X$ is not the cotangent bundle of a manifold.

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marked as duplicate by ACuriousMind, honeste_vivere, Qmechanic Jun 27 '16 at 22:00

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Often $X$ is a coadjoint orbit of a Lie group. These have a natural symplectic structure; see https://en.wikipedia.org/wiki/Symplectic_reduction

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