## 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.

Thanks for your attention.

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

Good bye.