C[omp]ute

From: "andrew cooke" <andrew@...>

Date: Wed, 11 Jan 2006 10:44:21 -0300 (CLST)

A while back, on AskMe, someone asked aboutthe behaviour of the following
physical system:

 Two points, of equal mass, are stationary in an otherwise empty universe.
 They are attracted to each other via gravity.

Now to solve this directly you know that acceleration is proportional to
gravity and so inversely proportional to the separation.  And you end up
with something like
  x'' = -k / x^2
Which is rather nasty to solve (something like a tan substitution?).

Now vacapinta observed that the system is the "usual" gravitational two
body problem, so the time before the two objects collide (which was
requested by the original poster) was half the period given by Kepler's
equations.

A bit of a recap on the two body problem: it reduces to the classical
single body problem (eg planet in orbit around much more massive star)
with a simple transformation (centre of mass, major axis etc).

This made my brain explode.  Because solving the single body problem is
easy.  Yet I just said that solving the problem above was hard.

Why is the single body problem easy?  Indirectly, because angular momentum
is not zero.  So the particle always misses the centre, avoiding the nasty
infinities you get with the two points described above (lim 1/x^2 as x ->
infinity).

Another way of looking at this is to see that you get simple harmonic
motion (or something very like it; I haven't looked in detail) when you
resolve along the x/y axes.  This is because when x=0, y is non-zero and
the force is perpendicular to the x axis.

That also helps show why angular momentum doesn't affect the period of the
orbit - because it's "orthogonal" to the forces involved.

So by introducing angular momentum into the problem I first described you
end up with a system that is simpler, mathematically.  You can then look
at the limit as the angular momentum tends to zero to see the solution for
the original system.

This seems very odd to me.

And a final comment - if you think about this in terms of Lagrangians
(which I was never taught, despite having a degree in physics from
Cambridge - something that seems *very* odd in retrospect) then you get a
path integral (I think).  Now is there some relationship between this and
all that fuss about poles in the complex plane and path integrals that go
around them?  I wish I could remember all that stuff....

Andrew