This is a very large commit.  Here are some of the things I did:
  1. I reorganized the code so that it is easier to read.
  2. I fixed some issues with Patrick's energy code
  3. I added analytic velocity
  4. I separated velocity and position in graphs and added titles
  5. I changed a lot of variable names so that they made sense
  6. I fixed a one-off error in Nathan's code so that it there are
  actually 'resolution' numver of outputs

When you run this, I am not sure if the tex will compile if you don't
have the libraries on your computer, but see...