The thermodynamic principle of virtual dissipation is given a formulation with variations of the field components freed from any constraints. Application of the principle and the introduction of a new state variable called heat content leads to the integro-differential field equations of thermorheology. Rigorous Lagrangian equations are also obtained which avoid the use of the entropy produced as an auxiliary variable while taking into account the heat generated. Quasi-isothermal and quasi-adiabatic evolutions are considered. Attention is called to the completeness of the generalized coordinate description and to implications in finite-element and bifurcation analysis based on the Lagrangian equations.