Rotation Forms and Local Hamiltonian Monodromy

TL;DR

This paper introduces a residue-based method for computing monodromy in integrable systems, applicable to both classical and non-compact cases, and demonstrates its effectiveness on well-known examples.

Contribution

It develops a local residue formula approach for monodromy computation, unifying geometric and analytical methods, and extends applicability to non-compact fibers.

Findings

Successfully applied to champagne bottle, spherical pendulum, hydrogen atom

Reproduces classical monodromy results for focus-focus singularities

Shows equivalence of scattering monodromy with classical monodromy in compact cases

Abstract

The monodromy of torus bundles associated to completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach is that the residue-like formula can be shown to be local in a neighborhood of a singularity, hence allowing the definition of monodromy also in the case of non-compact fibers. This idea has been introduced in the literature under the name of scattering monodromy. We prove the coincidence of the two definitions with the monodromy of an appropriately chosen compactification.