Monodromy of Hamiltonian Systems with Complexity 1 Torus Actions

K. Efstathiou and N. Martynchuk
Journal of Geometry and Physics, 2016

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We consider the monodromy of $n$-torus bundles in $n$ degree of freedom integrable Hamiltonian systems with a complexity 1 torus action, that is, a Hamiltonian $\mathbb T^{n-1}$ action. We show that orbits with $\mathbb T^1$ isotropy are associated to non-trivial monodromy and we give a simple formula for computing the monodromy matrix in this case. In the case of $2$ degree of freedom systems such orbits correspond to fixed points of the $\mathbb T^1$ action. Thus we demonstrate that, given a $\mathbb T^{n-1}$ invariant Hamiltonian $H$, it is the $\mathbb T^{n-1}$ action, rather than $H$, that determines monodromy.