### Bouncy Ball Problem

(Classical Mechanics with Philip Morrison)

[Mathematica Notebook, PDF, MathML]

Here the problem was to analyze a seemingly straightforward problem: describe the 1-D motion of a perfectly elastic
ball bouncing on a table moving up-and-down at a constant (sinusoidal) rate. Enrico Fermi proposed this as a model
for the origin of cosmic radiation: here, the ball is
a charged particle and the table is a moving magnetic field.

The assigned problem was simply to find a map from the ball's state at one collision to the state on its next collision.
However, I was curious to find out what a computer simulation of the ball's motion could reveal about the global structure
of the problem. (Specifically, I wanted to construct a picture of the 4-D phase space {*h*_{n}, *φ*_{n}; *h*_{n+1},
*φ*_{n+1}} for apex height *h* and table phase *φ*).

Since it's not immediately clear how the collision map can be solved in closed-form, one might like to use
a computer approximate-root-finding algorithm. However, two problems present themselves: this is very, very slow,
and is vulnerable to being trapped by false roots (near-misses):

To understand why this is such a problem, you have to know a bit about the mathematics driving this technology. Computer
root-finding algorithms generally find a solution that matches within a given
tolerance. You can turn that tolerance smaller, but the search time grows faster than the accuracy improves -- and
you cannot set the tolerance to zero.

Faced with this limitation, I turned back to mathematics. By non-dimensionalizing the system and making an observation
(once the ball falls below *h*=*A* the maximum height of the table, it will collide within that period),
I was able to reduce the parameter space and actually find a closed-form solution.

Then, in the regime where the table isn't pumping infinite energy into the ball, I used computer simulation to find
the following highly interesting map of the phase space:

There are some initial conditions for the ball which give stable orbits: if the ball hits a little early, it will
gain energy and travel farther (its next arrival will have a later phase); if it hits late it will lose a bit of velocity
and thus phase. The phase and velocity remain stably constrained:

There
are other conditions for which the ball hops periodically among a few stable phase/velocity regions:

and
finally, there are starting situations for which the ball's motion becomes classically chaotic: after enough time,
any two nearby starting positions will end up arbitrarily far apart:

The phase space map above displays 500 iterations each of 50 initial conditions; the chaotic
orbit I've highlighted shows 10,000 iterations of a single orbit. Each point involves computing the return map

It's safe to say that I would never have been able to generate the above graphs without the technological aid of
a computer graphing program. However, it is only by understanding the limitations of the solutions this technology
provides -- and the mathematical origins of those limitations -- that I was able to sensibly navigate the intricacies
of understanding the mechanics (or in other words, to follow the bouncing ball).