Index theory for symplectic paths with applications
type: Item
instance of: scholarly article
title: Index theory for symplectic paths with applications
zbMATH Open document ID: 1012.37012
published in: Progress in Mathematics
publication date: 2003-01-14T00:00:00Z
review text: In order to briefly describe the subject of this monograph recall that a real \(2n\times 2n\)-matrix \(M\) is called symplectic if \(M^TJM=J\), where \(J\) is equal to \(\left[\begin{smallmatrix} 0&-I\\ I&0 \end{smallmatrix}\right]\). The space of symplectic matrices is denoted by \(\text{Sp}(2n)\) and its codimension-one subspace consisting of matrices \(M\) such that \(\omega\) is its eignenvalue is denoted by \(\text{Sp}(2n)_\omega^0\). For \(\tau>0\), the space of symplectic matrix paths is defined as \[ P_\tau(2n):=\{\gamma\in C([0,\tau],\text{Sp}(2n))\colon \gamma(0)=I\}. \] A symplectic matrix path arises naturally as the fundamental matrix of a linear Hamiltonian system \(\dot x=JB(t)x\) for some matrix-valued function \(B\), and the theory of Hamiltonian systems motivated introduction of invariants related to symplectic paths. Let \(\gamma\in P_\tau(2n)\). For every element \(\omega\) of the unit circle in \(\mathbb{C}\), the index with respect to \(\omega\) of \(\gamma\) is the pair \[ (i_\omega(\gamma),\nu_\omega(\gamma))\in {\mathbb{Z}}\times \{0,\ldots,2n\}, \] where \(\nu_\omega(\gamma)\), called the nullity, is equal to the dimension (over \(\mathbb{C}\)) of the kernel of \(\gamma(\tau)-\omega I\), and \(i_\omega(\gamma)\), called the rotation, can be described in the nondegenerate case \(\nu_\omega(\gamma)=0\) as the intersection number of the joint path \(\gamma\ast\xi\) and \(\text{Sp}(2n)_\omega^0\), where \(\xi\) connects the diagonal symplectic matrix having a sequence of \(2\)-s and \(1/2\)-s as the diagonal entries, with \(I\). The index was introduced in the paper of \textit{C. Conley} and \textit{E. Zehnder} [Commun. Pure Appl. Math. 37, 207-253 (1984; Zbl 0559.58019)] in the case of nondegenerate paths, \(\omega=1\), and \(n\geq 2\), and it was further extended in papers of Y. Long, E. Zehnder, C. Viterbo, J. Robbin, and D. Salamon. The presented in the monograph final version together with axiomatic characterization comes from the paper by…
reviewed by: Roman Srzednicki
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 37J05
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1853002
zbMATH Keywords: symplectic matrix
zbMATH Keywords: symplectic group
zbMATH Keywords: Conley-Zehnder index
zbMATH Keywords: Maslov index
zbMATH Keywords: Morse index
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: Lagrangian system
zbMATH Keywords: geodesic
author: Yiming Long
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1853002