# Lagrange Equations Coupled to a Thermal Equation: Mechanics as Consequence of Thermodynamics

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## Abstract

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## 1. Introduction

## 2. Axiomatic Formulation of the First and Second Laws

#### 2.1. First Law

#### 2.2. Second Law

- (a)
- Evolution Part :If the system is adiabatically closed, the entropy S is a non-decreasing function with respect to time, i.e.,$$\frac{dS}{dt}=I\left(t\right)\u2a7e0\phantom{\rule{4pt}{0ex}},$$
- (b)
- Equilibrium Part :If the system is isolated, as time tends to infinity (i.e., $t\to +\infty $) the entropy tends towards a finite local maximum [5], compatible with the constraints (internal walls and isolation conditions), i.e.,$$\underset{t\to +\infty}{lim}S\left(t\right)=\underset{\rho \phantom{\rule{4pt}{0ex}}\text{compatible}}{max}S\left[\rho \right]\phantom{\rule{4pt}{0ex}},$$

## 3. System of Point Particles: First Law

## 4. System of Point Particles: Second Law

#### 4.1. Thermodynamics of an Isolated System

#### 4.2. Thermodynamics of an Adiabatically Closed System

#### 4.3. Thermodynamics of a Closed System

#### 4.4. Equilibrium Part of the Second Law

- (i)
- The extremum condition ${\delta}^{\left(1\right)}E{|}_{S=\overline{S}}=0$ implies that,
- $\frac{\partial E}{\partial {v}^{i}}{|}_{S=\overline{S}}=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{i.e.,}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\sum _{j}{g}_{ij}\left(q\right){v}^{j}=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{thus}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{v}^{j}=0\phantom{\rule{4pt}{0ex}}\text{since}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{g}_{ij}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{positive}\phantom{\rule{4.pt}{0ex}}\text{definite.}$
- $\frac{\partial E}{\partial {q}^{i}}{|}_{S=\overline{S}}=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{which}\phantom{\rule{4.pt}{0ex}}\text{gives}\phantom{\rule{4.pt}{0ex}}\text{(with}\phantom{\rule{4.pt}{0ex}}{v}^{j}=0\text{)}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{Q}^{\phantom{\rule{0.166667em}{0ex}}\text{int}}=-\frac{\partial U}{\partial {q}^{i}}=0\phantom{\rule{4pt}{0ex}}.$In other words, the extremum condition implies that the system evolves to an equilibrium point of the time evolution equation (26), characterised by $q=\overline{q}$ and $v=\overline{v}=0$.

- (ii)
- The condition $\frac{1}{T}{\delta}^{\left(2\right)}E{|}_{S=\overline{S},\phantom{\rule{0.166667em}{0ex}}q=\overline{q},\phantom{\rule{0.166667em}{0ex}}v=\overline{v}=0}\u2a7e0$ implies first that the matrix $\frac{1}{T}{g}_{ij}\left(\overline{q}\right)$ is non-negative and thus, for our mechanical system with positive kinetic energy, the temperature is necessarily positive. This in turn implies that the friction coefficients matrix is non-negative. Moreover, this same condition also implies that the matrix $\frac{1}{T}\phantom{\rule{0.166667em}{0ex}}\frac{{\partial}^{2}U}{\partial {q}^{i}\partial {q}^{j}}\left(\overline{S},\overline{q}\right)$ must be non-negative. In conclusion, the equilibrium part of the second law implies that the system evolves towards a stable equilibrium point. This is the zeroth law of thermodynamics [3].

## 5. From Thermodynamics to Mechanics

## 6. Thermodynamics of an Isolated System of Point Particles Interacting through a Harmonic Potential

## 7. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Gruber, C.; Brechet, S.D.
Lagrange Equations Coupled to a Thermal Equation: Mechanics as Consequence of Thermodynamics. *Entropy* **2011**, *13*, 367-378.
https://doi.org/10.3390/e13020367

**AMA Style**

Gruber C, Brechet SD.
Lagrange Equations Coupled to a Thermal Equation: Mechanics as Consequence of Thermodynamics. *Entropy*. 2011; 13(2):367-378.
https://doi.org/10.3390/e13020367

**Chicago/Turabian Style**

Gruber, Christian, and Sylvain D. Brechet.
2011. "Lagrange Equations Coupled to a Thermal Equation: Mechanics as Consequence of Thermodynamics" *Entropy* 13, no. 2: 367-378.
https://doi.org/10.3390/e13020367