Lagrange equations coupled to a thermal equation: mechanics as   consequence of thermodynamics
  Following the analytic approach to thermodynamics developed by Stueckelberg, we study the evolution equations of a closed thermodynamic system consisting of point particles in a fluid. We obtain a system of coupled differential equations describing the mechanical and the thermal evolution of the system. The coupling between these evolution equations is due to the action of a viscous friction term. Finally, we apply our coupled evolution equations to study the thermodynamics of an isolated system consisting of identical point particles interacting through a harmonic potential. 
C. Gruber and S. D. Brechet
physics.flu-dyn cond-mat.stat-mech