Path-Dependent Energy Lagrangian for Irreversible Thermomechanical Systems

We present a minimal Path-Dependent Energy Lagrangian (PDEL) that generates, from a single action, the balance equations of mechanics and the entropy/heat equation for irreversible thermomechanical systems. The reversible part is the Helmholtz free energy, while irreversible effects enter through a history integral of channel powers. A single upper-limit/tangential variation rule makes the same instantaneous power appear as a dissipative force in the mechanical/internal-variable equations and as a positive source in the entropy/heat equation, closing the first law without double counting and guaranteeing nonnegative entropy production under mild monotonicity assumptions. PDEL preserves the classical Lagrangian mechanics while subsuming standard dissipative models (Kelvin–Voigt viscosity, diffusion) and their viscous heating, and clarifies the reversible character of thermo-mechanical cross terms. The formulation offers a compact alternative to Rayleigh/Onsager appendices and GENERIC/metriplectic brackets, with limited algebraic complexity and straightforward extension to multiphysics.