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Coupled differential equations for charged particles

  1. May 31, 2015 #1
    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!
     
  2. jcsd
  3. Jun 1, 2015 #2

    Orodruin

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