Bracket formulation as a source for the development of dynamic equations in continuum mechanics
type: Item
instance of: scholarly article
title: Bracket formulation as a source for the development of dynamic equations in continuum mechanics
zbMATH Open document ID: 0992.70015
DOI: 10.1016/S0377-0257(00)00131-2
author: Antony N. Beris
published in: Journal of Non-Newtonian Fluid Mechanics
publication date: 2002-09-03T00:00:00Z
review text: This paper is devoted to the bracket formulation of dynamic equations in nonequilibrium thermodynamics. This is an introduction to theoretical foundations of one-generator bracket formalism, where the one generator is Hamiltonian. The Hamiltonian represents the system's total (extended) internal energy but, through a Legendre transform similar to that used in equilibrium thermodynamics, its derivatives can also be calculated from the available expression for the system's extended free energy. First, the conservative component of the bracket is recognized as the Poisson bracket \(\{.,.\}\) of Hamiltonian mechanics. For instance, \(dF/dt=\{F,E\}\), where \(F\) is arbitrary functional, and \(E\) is total (extended) internal energy. The properties of the Poisson bracket are briefly reviewed, and the Poisson structure of the conservative bracket is manifested in a variety of other formalisms of dynamic equations of nonequilibrium thermodynamics. The nonconservative, dissipative component is presented in the one-generator formalism, a more extensive treatment of which can be found in a research monograph by the author and Edwards (1994). The authors also discuss the two-generator form of bracket equations that corresponds to the GENERIC framework of the description. The two generators are Hamiltonian which drives the conservative part of the dynamics, and the entropy which drives the dissipative part. Differences and similarities between the two- and one-generator formalisms are pointed out, and the author illustrates the advantage of the use of bracket formulation by a number of recent applications.
Mathematics Subject Classification ID: 70H99
Mathematics Subject Classification ID: 82C35
Mathematics Subject Classification ID: 70H25
Mathematics Subject Classification ID: 70H33
Mathematics Subject Classification ID: 70G45
zbMATH DE Number: 1567177
zbMATH Keywords: two-generator bracket formalism
zbMATH Keywords: nonequilibrium thermodynamics
zbMATH Keywords: GENERIC framework
zbMATH Keywords: one-generator bracket formalsm
zbMATH Keywords: Hamiltonian
zbMATH Keywords: extended internal energy
zbMATH Keywords: Legendre transform
zbMATH Keywords: extended free energy
zbMATH Keywords: Poisson bracket
zbMATH Keywords: Poissons structure
zbMATH Keywords: entropy
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: s0377-0257(00)00131-2
OpenAlex ID: W2049092740
description: scientific article; zbMATH DE number 1567177