Re: reflections on a neglected problem

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

Leigh,

I am glad you brought up this problem (posed by Chip Sample) again.  I
worked on it and tried myself to get the list interested once before.  It
is a fascinating problem indeed and I continue to be amazed that I
hadn't stumbled across it before.

It appears that we are at virtually identical points of understanding. 
I've made no progress since my first couple of days of thinking about it. 
I have simulated the system using Interactive Physics and solved it
numerically with Maple and, reassuringly, they give essentially identical
results for all regimes of mass and charge ratio.  The bound cases *are*
the most interesting and it appears to me that the oscillations that occur
are hinting at some other conserved integral of the motion.  There are two
obvious periods of motion that both pairs of particles engage in--a basic
"bounce period" and a longer period envelope during which the bounce
amplitudes increase and decrease in a fairly regular way. I've tried
looking for canonical transformations of the coordinates which might yield
some other non-obvious conserved momentum to no avail. 

It's interesting to note that in the M/m = 1, Q/q = -1 case to which you
refer, the particles reach *infinite* kinetic energy *and* collide after a
finite (and short) period of time.

I'm about ready for a hint from Chip if he has any.  Otherwise I'll have
to put a senior on it for his or her research project ;-)

John

On Tue, 25 Mar 1997, Leigh Palmer wrote:

> Another problem, posed earlier, is much more difficult. I knew it would
> be, but I didn't get around to tackling it 'til last night. It is awful!
> That makes it worth attacking. The famous physicist Piet Hein said:
>
>   Problems worthy of attack
>   Prove their worth by fighting back.
>
> The problem (as I remember it):
>
>   Four charged particles are initially at rest at the corners
>   of a square. Particles on opposite corners are identical.
>   Two of the particles each have charge q and mass m, and the
>   other two each have charge Q and mass M. Find the subsequent
>   motion of the charges. Let the square have side length a.
....

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