Nonequilibrium thermodynamics with constraints: constrained functional derivatives and their transformations

Abstract A general methodology of incorporating physical constraints into applications of bracket theory in nonequilibrium thermodynamics is introduced at the macroscopic level expressed in terms of a set of local field variables. Constraints on the state variables are incorporated into the system description by suitably modifying the functional derivatives due to the presence of constraints through an extension of Lagrange’s method of undetermined multipliers to local field variables, whereby the dynamics of conserved quantities are restricted to submanifolds that are compatible with the imposed constraints. This article presents the mathematical prerequisites of the general methodology and a classification of constraints, along with a discussion of characteristic applications to microstructured materials.