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A simple procedure is presented to study the conservation of energy equationwith dissipation in continuum mechanics in 1D. This procedure is used totransform this nonlinear evolution-diffusion equation into a hyperbolic PDE;specifically, a second order quasi-linear wave equation. An immediateimplication of this procedure is the formation of a least action principle forthe balance of energy with dissipation. The corresponding action functionalenables us to establish a complete analytic mechanics for thermomechanicalsystems: a Lagrangian-Hamiltonian theory, integrals of motion, bracketformalism, and Noether's theorem. Furthermore, we apply our procedureiteratively and produce an infinite sequence of interlocked variationalprinciples, a variational hierarchy, where at each level or iteration the fullimplication of the least action principle can be shown again.

An Analytical Mechanics Approach to the First Law of Thermodynamics and Construction of a Variational Hierarchy