## 20.4 Planetary Motion

Here we have a second order equation, and now there are two dependent variables. All this means that we need add more columns to handle them.

Here are the equations: The dependent variables will be x and y, and the independent variable t.

We will, for simplicity assume that at the start y = 0 and x' = 0, the planet having coordinates x(t) and y(t), while we assume the sun is fixed at (0, 0).

The general equations of motion have the form

We can devote a column to t, one to x, one to x', one to y, one to y', one to r, and one to x" and finally, one to y".

The t column, as usual now, will increase by d from row to row. The r, x" and y" columns will express the equations just above for these things in terms of x and y.

x and y will start at their initial values and increase from row to row by where the values of the x' and x" are taken from the previous row, with a similar expression for y.

x' and y' will start at their initial values and increase by where

This may look awful, but it isn't. Only one formula need be entered per column, and then copied down, and all these column entries these are quite straightforward.

Exercise 20.3 Set this up, and Chart x vs y in the motion. See if you can arrange to get elliptical orbits.

I apologize for the skimpiness of this description, but this is the end, so you should have the ability to figure out what should be done here.

If you want to learn about multivariable calculus, you could do worse than consult 18.013A.

Good bye.