- #1
SgrA*
- 16
- 0
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:
[itex]\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}
[/itex]
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!
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:
[itex]\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}
[/itex]
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!