Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

reflections on a neglected problem



It seemed to me that John Mallinckrodt polished off Ludwik's "How many
volts?" problem completely in his first post. You guys really do like to
beat a problem to death!

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.

First, some comments about the problem. I have done considerable work on
modelling triple star systems to find a stability criterion. It appears
that triple star systems are ultimately unstable; no stability criterion
has been found, either by me or by others. There does not exist a proof
that these systems are unstable, but a near proof (due to Ovenden) does
exist which seems to settle the question physically. Nonetheless, triple
star systems exist and show no signs of incipient breakup, and none has
ever been observed to break up to my knowledge, except in simulations.

All gravitating systems I have worked with have negative total energies.
I do not say that they are "bound" because that is demonstrably not the
case; they do come apart. Escape from a triple system, even by its most
massive member, is easily accomplished by drawing the necessary escape
energy from the account of the remaining pair, binding them tightly
together. The total energy of the system, then, can still be negative.

Formulation of the problem involves the solution of three coupled
nonlinear second order vector differential equations or, alternatively,
eighteen coupled nonlinear first order ordinary differential equations.
Simplifications can be affected to reduce this number considerably
before numerical integration is undertaken. No closed form of the
solution exists, but there are some well known integrals of the motion
that can be used. The solution is chaotic.

The electrodynamic problem at hand is of the same order of complexity as
the three body problem. There are elements of difference - the different
"charge to mass" ratios of the particles and the possibility of having
repulsive as well as attractive interactions, but the solution is a pair
of messy coupled second order nonlinear scalar differential equations. I
can also write a nonlinear first order differential equation based on
the one integral of the motion I am sure of, its energy. I will assume
that symmetry persists in the solution, though of course that can only
happen in an ideal, classical sense, the sense in which this problem was
proposed, I assume.

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 think I have shown that in the range of charge ratios for which

q
[sqrt(8) - sqrt(7)] < - --- < [sqrt(8) + sqrt(7)]
Q

the system has negative total energy. In this range the motion is
interesting and, I suspect, chaotic. The details can be seen in
simulations, but the parameter space is large, since various masses
must be assigned to these particles. One conclusion I will make it that
under the symmetry constraint, unlike the triple star case, negative
total energy here implies that the system is bound. The proof is easy.
By symmetry one particle escape is forbidden. Two particle escape
leaves two particles having the same charge, and these can't have the
negative energy that the remaining pair from an exploding triple have.
Thus the system must vibrate forever, if in an apparently frantic way.

One configuration does puzzle me. Consider the case q = -Q and m = M.
This is, as we have seen, a bound system. If the particles start at rest
they must initially accelerate inward. In this case symmetry demands the
accelerations be equal in magnitude; the configuration remains a square
and the charges merge at the center of the square. There exists a family
of initial configurations which meet this same fate for different mass
ratios throughout the regime of bound states. I suspect there may be
other initial conditions which lead to such ultimate collapse, but I
have been unable to find any in my brief investigation of this problem.

Systems having charge ratios outside this regime will invariably blow up
regardless of their masses. There is a small variety in the nature of
the evolution of these systems, but none shows the sort of interesting
motion we see in the interesting regime.

Did anyone else play with this problem? Is it isomorphic with anything
found in Nature?

Leigh