# An Analytical Mechanics Approach to the First Law of Thermodynamics and Construction of a Variational Hierarchy

## TL;DR

This paper develops a variational hierarchy for thermomechanical systems by transforming energy conservation equations with dissipation into hyperbolic PDEs, enabling a Lagrangian-Hamiltonian framework and iterative variational principles.

## Contribution

It introduces a novel procedure to convert nonlinear energy equations into hyperbolic PDEs, establishing a comprehensive analytic mechanics framework for dissipative thermomechanical systems.

## Key findings

- Transformation of energy equations into hyperbolic PDEs
- Development of a variational hierarchy with iterative principles
- Establishment of a Lagrangian-Hamiltonian theory for dissipative systems

## Abstract

A simple procedure is presented to study the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian-Hamiltonian theory, integrals of motion, bracket formalism, and Noether's theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a variational hierarchy, where at each level or iteration the full implication of the least action principle can be shown again.