Monodromy in two degrees of freedom integrable systems
type: Item
instance of: scholarly article
title: Monodromy in two degrees of freedom integrable systems
zbMATH Open document ID: 0776.58017
DOI: 10.1016/0393-0440(92)90006-M
author: Maorong Zou
published in: Journal of Geometry and Physics
publication date: 1993-06-29T00:00:00Z
review text: Let \(f:M^ 4\to\mathbb{R}^ 2\) be an integrable Hamiltonian system. Assume three conditions: 1.) The set of critical values of \(f\) contains an isolated point \(p\). \(f\) is singular only at one point \(x_ 0\in f^{-1}(p)\). 2.) All regular fibers \(f^{-1}(r)\) for \(r\) near \(p\) are diffeomorphic to the 2-torus. 3.) Near \(x_ 0\), \(f\) has a certain normal form. Then for any fixed regular value \(r_ 0\) near \(p\) and any simple positively oriented loop around \(p\) connecting \(r_ 0\) to itself, the monodromy operator \(H_ 1(f^{-1}(r_ 0);\mathbb{Z})\to H_ 1(f^{- 1}(r_ 0);\mathbb{Z})\) is shown to be represented by the matrix \({1\;0\choose 1\;1}\) in some appropriately chosen basis for \(H_ 1(f^{-1}(r_ 0);\mathbb{Z})\).
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 219766
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: monodromy
reviewed by: Christian Bär
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 219766