Last time we reduced the relative motion of two particles interacting
via a central force to two variables that behave simply: r which has
the 1-d effective potential 
, and 
which can
be obtained by integrating 
.
We discussed qualitative features of the motion for a few examples:
for an attractive inverse square law force, we found two cases: 1)
when 
, the motion is bound, with r ranging between two
turning points 
and 
, at which 
. That means there is both a maximum and a minimum separation
between the two particles. Drew the motion in the plane (using the
fact that l determines 
); whether such a motion can be
periodic we save for a more quantitative discussion. 2) when 
,
the motion is not bound, since r can approach infinity. There is one
turning point , which is the point of closest approach of
the two particles. In the plane, we have a scattering orbit, where r
comes in from infinity, interacts, and goes back out to infinity.
Different potentials don't change the logic much: we considered a spring interaction, and found that we only get bound orbits.
We then
considered (quantitatively) circular and near-circular orbits. The
circular orbit corresponds to an equilibrium solution for the motion
of r. The effective potential has equilibrium points 
where the
derivative 
vanishes. A solution for the motion of r occurs
when we have just enough energy to sit at such an equilibrium point, 
, with 
forever. This constant solution for r
gives constant 
, so that we traverse a
circle in the plane at a constant rate. The motion is periodic with
period 
.
We next found near-circular orbits, where the (slightly perturbed)
radius oscillates about its equilibrium value in simple harmonic
motion. In this case increases at a nonconstant rate, which we
calculated to first order in 
by Taylor expansion. We
integrated this rate to find 
, and found that the motion was
periodic in the plane only if the expression above for 
was
a rational multiple of the period of the radial simple harmonic motion.
--
KB