Geodesic
An analytical mechanics approach to the first law of thermodynamics and construction of a variational hierarchy
Theoretical and applied mechanics, Tome 48 (2021) no. 1

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A simple procedure is presented for the study of the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second-order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian--Hamiltonian theory, integrals of motion, bracket formalism, and Noether's theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a \emph{variational hierarchy}, where at each level or iteration the full implication of the least action principle can be shown again.
Keywords: continuum mechanics, first law of thermodynamics, least action principle, dissipation, variational hierarchy 
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Hamid Said. An analytical mechanics approach to the first law of thermodynamics and construction of a variational hierarchy. Theoretical and applied mechanics, Tome 48 (2021) no. 1. http://geodesic.mathdoc.fr/item/TAM_2021_48_1_a0/