Monodromy of Hamiltonian systems with complexity 1 torus actions

K. Efstathiou, N. Martynchuk (2017)
Journal of Geometry and Physics

Abstract

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 $T^{n} - 1$ action. We show that orbits with $T^{1}$ 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 $T^{1}$ action. Thus we demonstrate that, given a $T^{n} - 1$ invariant Hamiltonian $H$ , it is the $T^{n} - 1$ action, rather than $H$ , that determines monodromy.

@article{2017_EM_Monodromy_Complexity_1,
 abstract = {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 $T^n−1$ action. We show that orbits with $T^1$ 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 $T^1$ action. Thus we demonstrate that, given a $T^n−1$ invariant Hamiltonian $H$, it is the $T^n−1$ action, rather than $H$, that determines monodromy.},
 author = {Efstathiou, K. and Martynchuk, N.},
 doi = {10.1016/j.geomphys.2016.05.014},
 journal = {Journal of Geometry and Physics},
 journalurl = {https://www.sciencedirect.com/journal/journal-of-geometry-and-physics},
 pages = {104-115},
 title = {Monodromy of Hamiltonian systems with complexity 1 torus actions},
 volume = {115},
 year = {2017}
}