  We study certain types of piecewise smooth Lagrangian fibrations of smooth
symplectic manifolds, which we call stitched Lagrangian fibrations. We extend
the classical theory of action-angle coordinates to these fibrations by
encoding the information on the non-smoothness into certain invariants
consisting, roughly, of a sequence of closed 1-forms on a torus. The main
motivation for this work is given by the piecewise smooth Lagrangian fibrations
previously constructed by the authors (in Some piece-wise smooth Lagrangian
fibrations, http://seminariomatematico.dm.unito.it/rendiconti/63-3.html), which
topologically coincide with the local models used by M. Gross in his paper
"Topological Mirror Symmetry", http://arxiv.org/abs/math.AG/9909015..
