# Introduction to Mathematical Physics/Energy in continuous media/Introduction

The first principle of thermodynamics (see section secpremierprinci) allows to bind the internal energy variation to the internal strains power \index{strains}:

${\displaystyle {\frac {dU}{dt}}={\dot {Q}}-P_{i}}$

if the heat flow is assumed to be zero, the internal energy variation is:

${\displaystyle dU=-P_{i}dt}$

This relation allows to bind mechanical strains (${\displaystyle P_{i}}$ term) to system's thermodynamical properties (${\displaystyle dU}$ term). When modelizing a system some "thermodynamical" variables ${\displaystyle X}$ are chosen. They can be scalars ${\displaystyle x}$, vectors ${\displaystyle x_{i}}$, tensors ${\displaystyle x_{ij}}$, \dots Differential ${\displaystyle dU}$ can be naturally expressed using those thermodynamical variables ${\displaystyle X}$ by using a relation that can be symbolically written:

${\displaystyle dU=FdX}$

where ${\displaystyle F}$ is the conjugated\footnote{ This is the same duality relation noticed between strains and speeds and their gradients when dealing with powers.}

thermodynamical variable of variable


${\displaystyle X}$. In general it is looked for expressing ${\displaystyle F}$ as a function of ${\displaystyle X}$ .

Remark:

If ${\displaystyle X}$ is a scalar ${\displaystyle x}$, the energy differential is:

${\displaystyle dU=f.dx}$

If ${\displaystyle X}$ is a vector ${\displaystyle x_{i}}$, the energy differential is:

${\displaystyle dU=f_{i}dx_{i}}$

Si ${\displaystyle X}$ is a tensor ${\displaystyle x_{ij}}$, the energy differential is:

${\displaystyle dU=f_{ij}dx_{ij}}$

Remark:

If a displacement ${\displaystyle x}$, is considered as thermodynamical variable, then the conjugated variable ${\displaystyle f}$ has the dimension of a force.

Remark:

One can go from a description using variable ${\displaystyle X}$ as thermodynamical variable to a description using the conjugated variable ${\displaystyle F}$ of ${\displaystyle X}$ as thermodynamical variables by using a Legendre transformation (see section secmaxient and ([#References

The next step is, using physical arguments, to find an {\bf expression of the internal energy ${\displaystyle U(X)}$} \index{internal energy} as a function of thermodynamical variables ${\displaystyle X}$. Relation ${\displaystyle F(X)}$ is obtained by differentiating ${\displaystyle U}$ with respect to ${\displaystyle X}$, symbolically:

${\displaystyle F(X)={\frac {\partial U}{\partial X}}}$

In this chapter several examples of this modelization approach are presented.