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

## Statement of first principle[edit]

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

**Definition:**

Let be a macroscopic system relaxing in . Internal energy is the sum of kinetic energy of all the particle and their total interaction potential energy :

**Definition:**

Let a macroscopic system moving with respect to . It has a macroscopic kinetic energy . The total energy is the sum of the kinetic energy and the internal energy . \index{internal energy}

**Principle:**

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

**Principle:**

At each time, particulaire derivative (see example exmppartder) of the total energy is the sum of *external* strains power and of the heat \index{heat} received by the system.

This implies:

**Theorem:**

For a closed system,

**Theorem:**

If macroscopic kinetic energy is zero then:

**Remark:**

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

## Consequences of first principle[edit]

The fact that is a state function implies that:

- Variation of does not depend on the followed path, that is variation of depends only on the initial and final states.
- is a total differential that that Schwarz theorem can be applied. If is a function of two variables and then:

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

so that energy conservation can also be written:

eint

System modelization consists in evaluating , and . Power by relation eint is associated to the modelization.