# Coupled differential equations for charged particles

1. May 31, 2015

### SgrA*

Hello,

I wanted to study the behaviour of electrons in a spatially bounded system. I want to have a larger number of electrons, but I took 3 to start with and arrived at this system of coupled equations:
\begin{align}\begin{bmatrix} \mathbf{\ddot{x_{1}}}\\ \\ \mathbf{\ddot{x_{2}}}\\ \\ \mathbf{\ddot{x_{3}}} \end{bmatrix} = \frac{1}{4\pi\epsilon_0} \begin{bmatrix} \frac{q_1 q_2}{m_1} & \frac{q_1 q_3}{m_1} \\ \\ \frac{q_2 q_1}{m_2} & \frac{q_2 q_3}{m_2} \\ \\ \frac{q_3 q_1}{m_3} & \frac{q_3 q_2}{m_3} \\ \end{bmatrix} \begin{bmatrix} \frac{\mathbf{r_{12}}}{|r_{12}^{3}|} & \frac{\mathbf{r_{21}}}{|r_{21}^{3}|} & \frac{\mathbf{r_{31}}}{|r_{31}^{3}|}\\ \\ \frac{\mathbf{r_{13}}}{|r_{13}^{3}|} & \frac{\mathbf{r_{23}}}{|r_{23}^{3}|} & \frac{\mathbf{r_{32}}}{|r_{32}^{3}|} \end{bmatrix}\end{align}
I'm not sure how to solve it: I've only solved the coupled mass problem by diagonalization, but I had a 2x2 matrix there. What method can I use to solve this system?

Thanks!

2. Jun 1, 2015

### Orodruin

Staff Emeritus
Are you trying to solve the three-body problem? It does not have an analytical solution.