We continued the discussion of 2-body central force motion, whose
Lagrangian we had previously shown to decouple when written in terms
of the center of mass coordinate 
and relative coordinate
. Moreover, 
is just the Lagrangian for a free
particle: the E-L equations for just tell us that
must remain constant -- the center of mass continues
along at constant velocity, insensitive to the interactions between
the two particles.
The interesting dynamics is therefore all in the relative motion
, governed by the Lagrangian 
. This is now an equivalent 1-body
problem for the relative motion, with 3 degrees of freedom .
Here we start exploiting our knowledge of conservation laws. L is rotationally-invariant, since it depends only on the magnitudes of vectors, which do not change under rotation. Thus the corresponding Noether charge, angular momentum, is conserved. Since angular momentum is a vector, that actually tells us a lot:
1) The direction of 
is conserved. is perpendicular to
the plane determined by and 
; thus and
must remain in that same plane forever to preserve the direction of
. This means that our 3-d motion is really only motion in a
plane, the plane perpendicular to .
2) The magnitude l of is conserved. This becomes clearer when
we pick coordinates for our motion in the plane perpendicular to
. For polar coordinates, 
. 
is cyclic -- the Lagrangian does not
explicitly depend on -- so that the momentum conjugate to
is conserved. This conjugate momentum is 
-- precisely equal to l, the magnitude of the angular momentum.
Thus the E-L equation is solved by forcing 
to remain constant.
This conservation of l by the motion in 
has a geometric
interpretation known as Kepler's second law. As the system
evolves, its radial vector sweeps out a wedge in the plane. The rate
at which the area of that wedge increases is constant: dA for a little
piece of wedge is just 
(that is, 1/2 base times
height), so that 
, proportional to l. l
conservation means that this rate is constant. Kepler derived this law
for the elliptical orbits caused by gravity; here we see that it is a very
general consequence of gravity being a central force.
So angular momentum conservation reduces us to only one variable,
r. Substituting into the E-L equations for r
gives us an E-L for r that depends only on r and 
. This E-L has
an effective force driving 
of 
,
equivalent to the force produced by the effective potential 
. Thus we have reduced the problem to a
1-dimensional problem for r, with 
. This L has no
explicit time-dependence, so the energy 
is
conserved. We thus have a problem of 1-d motion in a potential, where
the original potential picks up an extra piece 
, which
acts as a "centrifugal barrier" to bringing the particles together by
reducing r. (This makes sense because 
must spin faster to
carry the same angular momentum as r is decreased -- which costs more
and more rotational energy, until the system's entire energy is
rotational and r can decrease no further.)
--
KB