Symplectic reduction (http://en.wikipedia.org/wiki/Moment_map#Symplectic_quotients and http://ncatlab.org/nlab/show/moment+map) is probably the most important application of the moment map. Also, cf. the classical paper by Atiyah and Bott "The moment map and equivariant cohomology" (https://www.researchgate.net/publication/243785209_The_moment_map_and_equivariant_cohomology) for the applications of somewhat different kind.

You can check e.g. the lecture notes at http://www.math.polytechnique.fr/~berline/cours-Fudan.pdf for learning more about the moment map.

Article Bott R.: The moment map and equivariant cohomology. Topology. 23, 1-28

@Artur

Thanks for your concise references. A question that has not been clarified for me is "clear examples" of "Polytops" that identified after a moment map of a Torus action on a symplectic manifold, in other words due to a paper by Atiyah:

"ANGULAR MOMENTUM,CONVEX POLYHEDRA AND ALGEBRAIC GEOMETRY "

The image of a moment map attached to the Toral group action on a symplectic manifold is a convex polyhedra; But i have not found enough examples of this theorem. Please introduce me the references containing these examples.

Regards; Amiri