How to use numerical analysis to calculate the motion of any planet?
I have just started reading the <PERSON> lectures and in chapter 9, he explains how one might use numerical analysis to calculate the motion of a spring which can be extended to calculate the motion of earth around the sun ( assuming the sun is fixed ).
<PERSON> states that one may calculate the motion of Neptune or Jupiter or whatever even if the sun is not taken to be stationary. These are the equations he states:
Now let us see how we can calculate the motion of Neptune, Jupiter, Uranus, or any other planet. If we have a great many planets, and let the sun move too, can we do the same thing? Of course we can. We calculate the force on a particular planet, let us say planet number i, which has a position $x_i,y_i,z_i$ (i=1 may represent the sun, i=2 Mercury, i=3 Venus, and so on). 319
We must know the positions of all the planets. The force acting on one is due to all the other bodies which are located, let us say, at positions $x_j,y_j,z_j$. Therefore the equations are: \begin{align} m_i\,\frac{dv_{ix}}{dt}&= \sum_{j=1}^N-\frac{Gm_im_j(x_i-x_j)}{r_{ij}^3},\notag\ \label{Eq:I:9:18} m_i\,\frac{dv_{iy}}{dt}&= \sum_{j=1}^N-\frac{Gm_im_j( y_i- y_j )}{r_{ij}^3},\ m_i\,\frac{dv_{iz}}{dt}&= \sum_{j=1}^N-\frac{Gm_im_j( z_i- z_j )}{r_{ij}^3}.\notag \end{align} Where \begin{equation} \label{Eq:I:9:19} r_{ij}=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2}. \end{equation}
Then he states that,"Then when we have all initial positions and velocities we can calculate all the accelerations by first calculating all the distances". Alright, if we do have all initial velocities and distances of all the planets, how do we use that to calculate all future positions and velocities of planet $i$ as the positions and velocities of planet $j$ will be affected by planet $i$ as well! ?
Here is the link to the chapter:http://www.feynmanlectures.caltech.edu/I_09.html 766