Bracket formulation as a source for the development of dynamic equations in continuum mechanics
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.
scientific article; zbMATH DE number 1567177