Symplectic structure, Lagrangian, and involutiveness of first integrals of the principal chiral field equation
type: Item
instance of: scholarly article
title: Symplectic structure, Lagrangian, and involutiveness of first integrals of the principal chiral field equation
zbMATH Open document ID: 0543.35090
DOI: 10.1007/BF01208263
author: L. A. Dickey
published in: Communications in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: The matrix equation (1) \(\bar U_ t-\bar V_ x=[\bar U,\bar V]\) is considered where \(\bar U=\zeta^{-1}U_ 0+U_ 1+\zeta A\), \(\bar V=- \zeta^{-1}U_ 0+U_ 1+\zeta A;\quad U_ 1,A,U_ 0\) are \(N\times N\) matrices and A is a constant diagonal matrix with distinct diagonal elements. (1) must be satisfied identically with respect to the parameter \(\zeta\). These equations could be transformed into the form \(M_{\eta}+N_{\xi}=0\), \(M_{\eta}-N_{\xi}=[N,M]\), \(\eta =x-t\), \(\xi =x+t\), which is referred to as chiral equations and M, N are called chiral fields. The main objective is to established the connection between the symplectic structure of chiral fields and the first integrals. Two symplectic structures are defined. The Lagrangian and Hamiltonian are constructed, corresponding to one of the symplectic structures. Next, the first integrals are constructed and shown to be in involution.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 58E99
Mathematics Subject Classification ID: 81V99
zbMATH DE Number: 3863749
zbMATH Keywords: symplectic structure
zbMATH Keywords: chiral fields
zbMATH Keywords: first integrals
zbMATH Keywords: symplectic structures
zbMATH Keywords: Lagrangian
zbMATH Keywords: Hamiltonian
reviewed by: Edward Malec
MaRDI profile type: Publication
cites work: Instanton-like solutions in chiral models
cites work: Hamiltonian structures and Lax equations generated by matrix differential operators with polynomial dependence on a parameter
cites work: Q3889459
full work available at URL: https://doi.org/10.1007/bf01208263
OpenAlex ID: W2093560939
description: scientific article; zbMATH DE number 3863749