Action plot etc.

I made the plot Walter asked for, but am hesitant to put it into the draft for two reasons 1) It's unclear what the "action" really means for times below around 40 or so, since when we compute it we assume the potential is stationary whereas in fact it is evolving, and 2) it doesn't tell us anything distinct from the energy vs. time, as far as I can tell.  Here is the plot:

I've made this plot before, but settled on energy instead of action for the draft, mainly because of reason 1).  It's your call though, we can throw this into the paper if you all agree it should go in.   Incidentally, I also took this opportunity to make a similar plot for one of the steep cases, with $\gamma=0.8$, for comparison. Here it is:

This demonstrates pretty clearly, that in comparison to the fiducial case, these actions are frozen almost exactly where they started.

I thought you might appreciate seeing the direct comparison for the energy plots, I know I've done this before but I couldn't remember if I'd shared with everyone.  These plots were the basis for my statements in the draft that the energy and actions for the steep cases evolve only minimally from the initial conditions.

Although the energies do redistribute a little more than the actions do, it's pretty clear that the density profile is not going to be able to change significantly, because the orbits do not have the opportunity to gain or shed energy from their original energies.  Another way to put this is that for the steep cases, the characteristic distance over which energy can be transported is much smaller than in the fiducial case.