# Introduction to Mathematical Physics/Continuous approximation/Energy conservation and first principle of thermodynamics

## Statement of first principle

Energy conservation law corresponds to the first principle of thermodynamics ([#References|references]). \index{first principle of thermodynamics}

Definition:

Let ${\displaystyle S}$ be a macroscopic system relaxing in ${\displaystyle R_{0}}$. Internal energy ${\displaystyle U}$ is the sum of kinetic energy of all the particle ${\displaystyle E_{cm}}$ and their total interaction potential energy ${\displaystyle E_{p}}$:

${\displaystyle U=E_{cm}+E_{p}}$

Definition:

Let a macroscopic system moving with respect to ${\displaystyle R}$. It has a macroscopic kinetic energy ${\displaystyle E_{c}}$. The total energy ${\displaystyle E_{tot}}$ is the sum of the kinetic energy ${\displaystyle E_{c}}$ and the internal energy ${\displaystyle U}$. \index{internal energy}

${\displaystyle E_{tot}=E_{c}+U}$

Principle:

Internal energy ${\displaystyle U}$ is a state function\footnote{ That means that an elementary variation ${\displaystyle dU}$ is a total differential. } . Total energy ${\displaystyle E_{tot}}$ can vary only by exchanges with the exterior.

Principle:

At each time, particulaire derivative (see example exmppartder) of the total energy ${\displaystyle E_{tot}}$ is the sum of external strains power ${\displaystyle P_{e}}$ and of the heat ${\displaystyle {\dot {Q}}}$ \index{heat} received by the system.

${\displaystyle {\frac {dE_{tot}}{dt}}=P_{e}+{\dot {Q}}}$

This implies:

Theorem:

For a closed system, ${\displaystyle dE_{tot}=\delta W_{e}+\delta Q}$

Theorem:

If macroscopic kinetic energy is zero then:

${\displaystyle dU=\delta W+\delta Q}$

Remark:

Energy conservation can also be obtained taking the third moment of Vlasov equation (see equation eqvlasov).

## Consequences of first principle

The fact that ${\displaystyle U}$ is a state function implies that:

• Variation of ${\displaystyle U}$ does not depend on the followed path, that is variation of ${\displaystyle U}$ depends only on the initial and final states.
• ${\displaystyle dU}$ is a total differential that that Schwarz theorem can be applied. If ${\displaystyle U}$ is a function of two variables ${\displaystyle x}$ and ${\displaystyle y}$ then:

${\displaystyle {\frac {\partial ^{2}U}{\partial x\partial y}}={\frac {\partial ^{2}U}{\partial y\partial x}}}$

Let us precise the relation between dynamics and first principle of thermodynamics. From the kinetic energy theorem:

${\displaystyle {\frac {dE_{c}}{dt}}=P_{e}+P_{i}}$

so that energy conservation can also be written:

eint

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

System modelization consists in evaluating ${\displaystyle E_{c}}$, ${\displaystyle P_{e}}$ and ${\displaystyle P_{i}}$. Power ${\displaystyle P_{i}}$ by relation eint is associated to the ${\displaystyle U}$ modelization.