Hamiltonian monodromy and Morse theory

N. Martynchuk, H. W. Broer, K. Efstathiou (2020)
Communications in Mathematical Physics

Abstract

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.

@article{2019_MBE_Monodromy_Morse,
 abstract = {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.},
 archiveprefix = {arxiv},
 author = {Martynchuk, N. and Broer, H. W. and Efstathiou, K.},
 doi = {10.1007/s00220-019-03578-2},
 eprint = {1901.00705},
 journal = {Communications in Mathematical Physics},
 journalurl = {https://link.springer.com/journal/220},
 number = {2},
 pages = {1373-1392},
 primaryclass = {math-ph},
 title = {Hamiltonian monodromy and Morse theory},
 volume = {375},
 year = {2020}
}