Hamiltonian monodromy and Morse theory

N. Martynchuk, H. W. Broer, K. Efstathiou

Communications in Mathematical Physics, 375(2), 1373-1392 (2020)
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Preprint on Arxiv
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We show that Hamiltonian monodromy of an integrable two degrees of freedom system with a global circle action can be computed by applying Morse theory to the Hamiltonian of the system. Our proof is based on Takens's index theorem, which specifies how the energy-$h$ Chern number changes when $h$ passes a non-degenerate critical value, and a choice of admissible cycles in Fomenko-Zieschang theory. Connections of our result to some of the existing approaches to monodromy are discussed.