CHAOS! and some non-computer analysis...

I would never have guessed that I would still be working on the repulsive charge trapping charge problem. It is so simple yet it has a surprising amount of depth. So, this week although I continued analyzing the problem using the computer (I've mostly fixed the programming issues :)), we (Dr. Manne/I) finally figured out (without a computer!) some bounding criteria for the problem for the 1/r^n force. This was done by looking at the problem from a rotating frame, writing out the forces (the Coriolis, centrifugal, and 1/r^2 force) and the energies associated with the problem, and then playing with the resulting equations.
So, switching reference frames is pretty helpful/interesting...

Besides that, I had Mathematica solve about a 100,000 differential equations and plot a few other quantities associated with the problem that I hadn't explored. I am trying to find patterns among these graphs, but most of them are so complicated (some are chaotic) it is useless (although under certain conditions sometimes these crazy graphs will start to morph into nice and simple cosine/sine/circular graphs which is cool).

That said, here are a few fun graphs:
Velocity is the velocity of the orbiting particle and the escape distance is the distance between the particles when the inner particle is 1 unit away from the origin.
These first 3 are for a repulsive 1/r^2 force between the two particles:


If you view the graph on the left from a rotating frame, it turns into the graph on the right.



Just a different orbiting velocity and you get the "Taco Bell" graph:And this graph is if the force between the particle is 1/r^2:

And that is about it for this week.