# Introduction to Mathematical Physics/Continuous approximation/Exercises

**Exercice:**

Give the equations governing the dynamics of a plate (negligible thickness) from powers taking into account the gradient of the speeds (first gradient theory). Compare with a approach starting form conservation laws.

**Exercice:**

Same question as previous problem, but with a rope clamped between two walls.

**Exercice:**

exoplasmapert

A plasma\index{plasma} is a set of charged particles, electrons and ions. A classical model of plasma is the "two fluid model": the system is described by two sets of functions density, speed, and pressure, one for each type of particles, electrons and ions: set characterizes the electrons and set characterizes the ions. The momentum conservation equation for the electrons is:

me

The momentum conservation equation for the ions is:

mi

Solve this non linear problem (find solution ) assuming:

- field is directed along direction and so defines parallel direction and a perpendicular direction (the plane perpendicular to the field).
- speeds can be written with and .
- field can be written: .
- densities can be written . Plasma satisfies the quasi--neutrality condition\index{quasi-neutrality} : .
- gases are considered perfect: .
- , have the values they have at equilibrium.