  We introduce a completely integrable system on the Grassmannian of 2-planes
in an n-space associated with any triangulation of a polygon with n sides, and
compute the potential function for its Lagrangian torus fiber. The moment
polytopes of this system for different triangulations are related by an
integral piecewise-linear transformation, and the corresponding potential
functions are related by its geometric lift in the sense of Berenstein and
Zelevinsky.
