Examples
In the case of a Hamiltonian action of the circle G = U(1), the Lie algebra dual is naturally identified with R, and the moment map is simply the Hamiltonian function that generates the circle action.
Another classical case occurs when M is the cotangent bundle of R3 and G is the Euclidean group generated by rotations and translations. That is, G is a six-dimensional group, the semidirect product of SO(3) and R3. The six components of the moment map are then the three angular momenta and the three linear momenta.
Read more about this topic: Moment Map
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