Triangulated riemann surfaces with boundary and the Weil-Petersson Poisson structure

Given a hyperbolic surface with geodesic boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmiiller space T(S). We express the Weil-Petersson Poisson structure of T(S) in this system of coordinates, and we prove that it limits pointwise to the piecewise-linear Poisson structure defined by Kontsevich on the arc complex of S. At the same time, we obtain a formula for the first-order variation of the distance between two closed geodesics under Fenchel-Nielsen deformation.