Implicit Hamiltonian systems with symmetry
type: Item
instance of: scholarly article
title: Implicit Hamiltonian systems with symmetry
zbMATH Open document ID: 0921.70014
DOI: 10.1016/S0034-4877(98)80176-6
published in: Reports on Mathematical Physics
publication date: 1999-10-06T00:00:00Z
review text: The classical topic is treated here by exploiting some new and rather promising technical tools. Let \(X\) be a manifold, \(D\subset TX+T^*X\) a vector subbundle, \(D^\perp\subset TX+T^*X\) the subbundle of all \((X,\alpha)\) satisfying \(\langle\alpha\mid\widehat X\rangle+\langle \widehat\alpha\mid X\rangle=0\) for all \((\widehat X,\widehat\alpha)\in D\). Then \(D\) is called a Dirac structure if \(D=D^\perp\) and \(\langle L_{X_1}\alpha_2\mid X_3\rangle+ \langle L_{X_2} \alpha_3\mid X_1\rangle+ \langle L_{X_3}\alpha_1\mid X_2\rangle=0\) for all \((X_i,\alpha_i)\in D\). The author defines an implicit Hamiltonian system by the inclusion \((\dot x,dH)\in D\) where \(H\in C^\infty(X)\) is a given function (the Hamiltonian). The inclusion can be represented by a mixed set of differential and algebraic equations \(F(x,\dot x)=0\). The infinitesimal symmetries \(f\in TX\) are defined by \(L_fD\subset D\). The author shows that the symmetries permit to reduce \(X\) to a quotient manifold \(\widetilde X\) equipped with a reduced Dirac structure \(\widetilde D\).
Mathematics Subject Classification ID: 70H33
Mathematics Subject Classification ID: 70F25
Mathematics Subject Classification ID: 37C80
zbMATH DE Number: 1233896
zbMATH Keywords: vector subbundle
zbMATH Keywords: Dirac structure
zbMATH Keywords: inclusion
zbMATH Keywords: infinitesimal symmetries
zbMATH Keywords: reduced Dirac structure
author: Arjan van der Schaft
reviewed by: Jan Chrastina
MaRDI profile type: Publication
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full work available at URL: s0034-4877(98)80176-6
OpenAlex ID: W2105970071
description: scientific article; zbMATH DE number 1233896