  This article is the second of a pair of articles about the Goldman symplectic
form on the PGL(V)-Hitchin component of a closed, connected, oriented,
hyperbolic surface S. We show that any ideal triangulation on S and any
compatible bridge system determine a symplectic trivialization of the tangent
bundle to the PGL(V)-Hitchin component of S. Using this, we prove that a large
class of vector fields defined in the companion paper [SWZ20] are Hamiltonian.
This is then used to prove that the explicit global coordinate system defined
in the companion paper [SWZ20] is a global Darboux coordinate system for the
PGL(V)-Hitchin component.
