# Electron equations of motion for plasma

I'm reading through an Introduction to Plasma Physics by Francis F. Chen, and in the simplified derivation for plasma oscillation in 1D, the book quotes the electron equations of motion as:

$$mn_e\left[\frac{\partial\mathbf{v}_e}{\partial t}+(\mathbf{v}_e\cdot\mathbf{\nabla})\mathbf{v}_e\right]=-en_e\mathbf{E}$$

This looks like the usual form of $$m\frac{d\mathbf{v}_e}{dt}=\mathbf{F}=-e\mathbf{E}$$, but with an additional $$(\mathbf{v}_e\cdot\mathbf{\nabla})\mathbf{v}_e$$ term. In this case the term is just $$v_e^{(x)}\frac{\partial\mathbf{v}_e}{\partial x}$$. It looks familiar, but I can't recall where this originates from. Other deviations I've read do not include it, and it is later ignored when considering the case when $$\left|\mathbf{v}\right|<<1$$. From this assumption, I would think special relativity but I haven't found a mention of it yet.

Does anyone know where this extra term comes from?

• Isn't that just the material derivative? – Kyle Kanos May 25 at 1:15
• Also known as the convective derivative: $\frac{d}{dt}=\frac{\partial}{\partial t} + v \cdot \nabla$ - the total time rate of change of a quantity moving instantaneously with the velocity $v$. – Cinaed Simson May 27 at 7:42
• Thank you! I forgot to account for the fact that the force would change with position. It's been a while. – Levi Hall May 29 at 18:23

## 1 Answer

In one dimension, we have: $$\frac{dv_e}{dt}=\frac{\partial v_e}{\partial t}+\frac{\partial v_e}{\partial x}\frac{dx}{dt}=\frac{\partial v_e}{\partial t}+v_e\frac{\partial v_e}{\partial x}$$

In 3 dimensions, the x derivative "becomes" a gradient. Thus we get: $$\frac{dv_e}{dt}=\frac{\partial v_e}{\partial t}+(v_e\nabla)v_e$$