Path-Dependent Energy Lagrangian for Irreversible Thermomechanical Systems

TL;DR

This paper introduces a minimal Path-Dependent Energy Lagrangian (PDEL) framework that unifies reversible and irreversible thermomechanical processes, ensuring energy conservation and nonnegative entropy production.

Contribution

It develops a unified variational principle for thermomechanical systems that captures irreversible effects through a history integral, simplifying existing formalisms.

Findings

PDEL reproduces classical mechanics and dissipative models.

Ensures nonnegative entropy production under mild conditions.

Provides a compact, extendable formulation for multiphysics systems.

Abstract

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.