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}

This implies:

Consequences of first principle
The fact that $$U$$ is a state function implies that: Let us precise the relation between dynamics and first principle of thermodynamics. From the kinetic energy theorem:
 * Variation of $$U$$ does not depend on the followed path, that is variation   of $$U$$ depends only on the initial and final states.
 * $$dU$$ is a total differential that that Schwarz theorem can be   applied. If $$U$$ is a function of two variables $$x$$ and $$y$$ then:

so that energy conservation can also be written:

System modelization consists in evaluating $$E_c$$, $$P_e$$ and $$P_i$$. Power $$P_i$$ by relation eint is associated to the $$U$$ modelization.