A Lagrangian approach to classical thermodynamics

The specification of microstates of interacting dynamical systems is different in Lagrangian and Hamiltonian approaches whenever the interaction Lagrangian depends on generalised velocities. In almost all cases of physical interest however, velocity-dependent interaction Lagrangians do not couple velocities belonging to different subsystems. For these cases we define reduced system and bath Lagrangian macrostates, which like the underlying microstates differ from their Hamiltonian counterparts. We then derive exact first and second laws of thermodynamics without any modification of the original system and bath quantities. This approach yields manifestly gauge-invariant definitions of work and free energy, and a gauge-invariant Jarzynski equality is derived. The formalism is applied in deriving the thermodynamic laws for a material system within the radiation reservoir. The Lagrangian partition of the total energy is manifestly gauge-invariant and is in accordance with Poynting's theorem.