  A simple procedure is presented to study 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 variational hierarchy, where at each level or iteration the full
implication of the least action principle can be shown again.
