I first learned how to compute monodromy of torus bundles in integrable Hamiltonian systems from Cushman’s and Bates’s “Global Aspects of Classical Integrable Systems”, now at its second edition. The computation of monodromy there effectively boils down to the computation of the variation of the rotation number along properly chosen closed paths in the system.

Recently, together with Andrea Giacobbe, Pavao Mardešić, and Dominique Sugny we needed to better understand this computation. Studying the problem led us to a much better appreciation of the subtleties involved and revealed the deep connection between this analytical computation and the geometry of the fibration.

The main novelty in our work is the introduction and detailed study of rotation 1-forms. These have similarities with connection 1-forms, well known from the theory of principal bundles. We show that the information about monodromy is encoded in the poles of the rotation 1-form, points where the form is not defined. In particular, the variation of the rotation number can be expressed as the sum of residue-like quantities around such poles. We use this fact to re-prove that monodromy for systems with isolated focus-focus points is locally determined.

Furthermore, we consider the problem of monodromy for systems with cylinder fibers and we define non-compact monodromy by properly identifying the ends of the cylinders. Such identification turns cylinders into tori in such a way that it fixes the monodromy of the resulting torus bundle. We compare non-compact monodromy to scattering monodromy and we show that they coincide for systems with $A_1$ singularities, that is, focus-focus points.

Details can be found in our paper which is now available from the arXiv.