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Computing the Jacobian matrix for a solar system simulation

  1. Aug 6, 2010 #1
    Hello physicists!

    I'm a comp sci student and I am trying to graphically model a simplified version of the solar system as part of a programming exercise. In order to apply the gravitational forces to the planets, I need to compute the Jacobian matrix as it relates to two particles (planetary objects). My (limited) understanding is that the resulting matrix will be 9x9, but I am unsure how it is constructed exactly. This is what I have come up with so far:

    [tex]J = \begin{pmatrix}
    \frac{\partial F_1}{\partial r_1} & \frac{\partial F_1}{\partial r_2} & \frac{\partial F_1}{\partial r_3}\\
    \frac{\partial F_2}{\partial r_1} & \frac{\partial F_2}{\partial r_2} & \frac{\partial F_2}{\partial r_3}\\
    \frac{\partial F_3}{\partial r_1} & \frac{\partial F_3}{\partial r_2} & \frac{\partial F_3}{\partial r_3}

    where [tex]\frac{\partial F_i}{\partial r_i}[/tex] is 3x3, [tex]F[/tex] is the gravitational force [tex]F = \frac{Gm_1m_2}{r^2}[/tex] and [tex]r[/tex] is the respective dimension component [tex](x, y, z)[/tex].

    Some clarification would be much appreciated :-)

  2. jcsd
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