RE: reflections on a neglected problem

[Leigh Palmer <palmer@sfu.ca>]

> High Leigh-

High? Me? Everyone thinks, just because I was in Berkeley in the sixties...

> 	I've been looking at the 4-charges problem also.  You write:
> ******************************************************************
> The solutions here will be chaotic in some regimes. If q and Q are of
> the same sign then the subsequent motion will be simple (and dull) and
> not chaotic. If q and Q are of opposite sign things get interesting. It
> is only this case that I will discuss below.
> ************************************
>
> 	I haven't found the same sign problem "dull".  The issue is,
> how do the two pairs of particles split up the total energy?  I'm convinced
> that John is correct; there is another constant of the motion other than
> the total energy, but I haven't found it yet either.  I'm not quite ready
> to give up, though.

I guess I should expand on why I think same sign problems will be dull.
I have parametrized the solutions with two quantities, charge ratio and
mass ratio. Changing the absolute values of these quantities does not
change the character of the motion; only the time scale is affected.

The question of how the particles split up the total energy is a good
measure of how interesting the problem is, I agree. First I will note
that the idea only makes sense if one asks it after the particles
separate to infinity. Before that time there is a shared potential
energy which cannot be apprortioned. Thus the division of energy (the
ratio of the final kinetic energies) is a single scalar result for any
given values of the parameters defined above. As such it can be
represented by a surface in three-space. My conjecture that the same
charge solutions are dull is equivalent to saying that this surface
will be monotonic in the other two coordinates in the positive charge
ratio half-space. In the negative charge ratio half-space the energy
ratio may be somewhat more interesting, but I expect that, where it is
defined, it will be only slightly more interesting. Of course the
energy ratio is undefined for bound states.

The triple star problem is of a higher order of complexity. I can't
construct a "phase diagram" for it nearly so easily as I can for the
four charge problem. My conjecture there is that triples always split
into a pair and a single. Two mass ratio parameters are needed to
characterize the systems to start with, and it would be highly
unrealistic to examine only the class of solutions in which the
initial condition was one of rest. You can see that the space which
can be visualized grows rapidly in dimension as one develops the
problem. I give up above three dimensions as far as visualization
goes. Some of my work was done with three equal mass stars. Even
exploration of that simplified manifold defies characterization of a
system of solutions. I did find some lovely periodic solutions (as
did Joseph Lagrange) and also quasiperiodic solutions.

Leigh