Re: reflections on a neglected problem

[sample@lyon.edu (John D. Sample)]

At 1:46 PM on 3/27/97, <John Mallinckrodt> wrote:

> 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

You guys took the problem a level higher than it was intended (and made it
more interesting in the process).  Since the question asked for the "final
velocities" I assumed the charges were of the same sign.  I had not solved
even that problem when I posted it, but since then came up with a
"solution".  Unfortunately the algebraic expression probably can't be
solved analytically so a numerical method has to be used.  At that point I
wondered whether a brute force numerical solution to the eqns of motion
would have been just as good!

I've made no progress on the negative energy case, other than confirm for
myself Leigh's range of -q/Q which defines the case.  I would bet that the
most one could hope for is one more integral after energy
conservation,...the additional conserved quantity John M suspects exists.

So you won't get any hints from me!  But maybe someone can tell me why the
following approach won't work.

Choose new coordinates v(x,y), w(x,y) such that v=f(U(x,y)) for some nice
function f, where U is the potential energy.  For example, one choice might
be v=1/U.  Then find another function w(x,y) which makes the coordinate
system orthogonal.    Then w won't appear in the potential energy, and so
its conjugate momentum will be conserved.   I haven't been able to find a
w(x,y) yet, but I'm confident there will be at least one major stumbling
block beyond that.  It looks like it could be difficult to express the
kinetic energy in terms of the new variables.

Has anyone thought about this approach?

Chip