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Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know the set of regular values is not simply connected.

Question Short of computing local action-angle coordinates or the period lattice, how can we compute the monodromy of this system? Or simply detect whether it is non-trivial?

The only way I know to do this is to compute the transition functions of the period lattice bundle explicitly (a la Cushman and Bates), which seems to be roughly equivalent to several other approaches (such as computing local action angle coordinates). Are there other known tricks for doing this, or famous examples I should be aware of? For example, if I can realize my system as a Lax pair, are there `algebraic' ways to detect monodromy?

Edit: cf. http://arxiv.org/pdf/1401.3630.pdf it appears that (at least in 2 degrees of freedom) if you have an isolated focus singular value in the base, then (in the Hamiltonian case) the monodromy around that point can be computed from the number of singular points in the fiber. So one strategy is to study the topology of the singular fibers. Apparently this approach breaks down if the system is non-Hamiltonian. Does this approach generalize to Hamiltonian systems with arbitrary degrees of freedom? For example, in degree 3 we would be looking at monodromy around a critical line, so we might need to study the topology of a bundle of singular tori over this line.

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For example, are there interesting ways to show that the rotation numbers are multivalued functions? –  Jeremy Lane Feb 20 at 5:25

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After reading a bit into the problem, it seems that 'Cushman's principle' is the answer I am looking for. In the case of the spherical pendulum, one can observe with Morse theory that the energy level sets change topology as one passes through the isolated critical value. This indicates that a pull-back of the torus bundle to a circle around the critical value should be non-trivial, which is precisely the situation of non-trivial topological monodromy. This is a nice observation because doing Morse theory in this situation is much more simple than computing action-angle coordinates. On the other hand, this does not compute the monodromy.

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