After drawing several periodic near-circular orbits, we went on to
consider the more general solution of the central force motion
problem: to find all orbits, with no approximations. We found we could use
our old one-dimensional tricks, and get an integral equation relating
the duration of the motion to an integral over r, giving us the
function t(r). This we could invert to get r(t). We could then
integrate 
to get 
, by substituting our solution
r(t) into 
. Alternatively, if we were only
interested in the orbit's shape 
, we found an integral
equation relating the change in polar angle during the motion to an integral
over r, giving 
, which we could invert to give the shape
.
The above is an entirely deterministic algorithm to find the orbits for a general central force. It could be fed into a computer program, since integrating is trivial on a computer (just add all the area slices under the curve). However, this procedure is often hard for us humans to do analytically.
Instead, we return to the E-L equation for r(t). If we're only
interested in the shape of the orbit, we can convert our
t-derivatives to 
-derivatives using the chain rule: 
. We know , so we have
. Substituting this expression in
everywhere we see a d/dt gives a new differential equation, which
simplifies with the substitution 
. This substitution gives
us the orbit equation for 
.
We pointed out the following qualitative feature of the orbit
equation. At a turning point P, where 
(and 
by choice of starting point), we have a reflection symmetry of the
orbit equation. That is, if obeys the orbit equation,
must also obey it, as neither the equation nor the initial
condition changes form under the reflection. Thus if we've solved for
the orbit PQ between turning points P and Q, we can extend the orbit
indefinitely through time by iteratively reflecting the previous piece
of orbit about the current turning point.
We stated Bertrand's theorem about whether such a procedure gives a closed (periodic) orbit, i.e. about whether such a trajectory ever returns to its beginning point. Only for two possible force laws -- inverse square and simple harmonic -- are all bound orbits periodic.
We then specialized to the case of planetary motion, using the orbit
equation for for an attractive inverse square law
force. The orbit equation then looks like the differential equation for a
simple harmonic oscillator with a constant driving force. We solved
this, getting 
.
We related the parameter e (called the "eccentricity") to the
orbit's energy. Discussed expectations for the solutions' behavior
for different energies, based on the effective potential for r. We
expect bound orbits for 
, barely free orbits (where the velocity
decays to zero as r goes to infinity) for E= 0, and scattering orbits
for 
.
--
KB