# From Lagrangian mechanics to nonequilibrium thermodynamics: a variational perspective

## TL;DR

This paper reviews recent advances in variational formulations of nonequilibrium thermodynamics, extending classical mechanics principles to include irreversible processes in both finite and infinite dimensional systems.

## Contribution

It systematically extends Hamilton's principle to encompass irreversible thermodynamic processes in discrete and continuum systems.

## Key findings

- Variational principles can describe irreversible thermodynamics.
- Application to Navier-Stokes-Fourier systems demonstrates the approach.
- Framework unifies mechanics and thermodynamics through variational methods.

## Abstract

In this paper, we survey our recent results on the variational formulation of nonequilibrium thermodynamics for the finite dimensional case of discrete systems as well as for the infinite dimensional case of continuum systems. Starting with the fundamental variational principle of classical mechanics, namely, Hamilton's principle, we show, with the help of thermodynamic systems with gradually increasing level complexity, how to systematically extend it to include irreversible processes. In the finite dimensional cases, we treat systems experiencing the irreversible processes of mechanical friction, heat and mass transfer, both in the adiabatically closed and in the open cases. On the continuum side, we illustrate our theory with the example of multicomponent Navier-Stokes-Fourier systems.