Oops. I just noticed that in my explanation that I previously posted
There was an error in the dimensionless function D(w) for the case
when the energy E is negative. The expression I gave for D(w) is
still correct for positive energy orbits. Last time I wrote:
D(w) == 1 - w^2 - V(L/(w*sqrt(2*m*E)))/E .
It ends up that for negative E values some of the signs change here.
The correct equation for D(w) for both positive and negative E values
is:
D(w) == sgn(E) - w^2 - V(L/(w*sqrt(2*m*|E|)))/|E|
where the algebraic sign function is given as sgn(x) == x/|x| (when
x <> 0 and sgn(0) = 0).
Even though the function D(w) above is ostensibly undefined for
the E = 0 case that case can be handled by observing that the motion
is completely insensitive to the zero level of the potential energy
function. If we need to deal with a E = 0 case we can just add a
fixed constant to the E-value and add the same constant to the
definition of the potential V(r) function and then use the shifted
definitions.
This problem with the sign of and the zero value of the energy
would not have come up at all if the integral expression I used in
defining the definite integral 'I' was not a dimensionless function
of a dimensionless integration variable where things were scaled by
a factor that included factors of |E|. We can remove this problem
altogether if we integrate over a dimensioned integration variable
u = 1/r that is just the reciprocal of the radial distance. In this
case we could write:
I == Int{u_<, u_> | du/sqrt(C(u)} .
The lower limit of the integral is u_<, the upper limit is u_>
and the function C(u) in the sqrt() in the denominator of the
integrand is defined as:
C(u) == (2*m/L^2)*(E - V(1/u)) - u^2 .
In this case the dimensions of the function C(u) is a function
whose value is an inverse square length and whose argument is an
inverse length. Also in this case the orbit's periapsis distance
r_p is given in terms of the upper limit of the integral according
to:
r_p == 1/u_>
and the orbit's apapsis distance r_a is given in terms of the upper
limit of the integral according to:
r_p == 1/u_<
In this case the annular radial bands of allowed bounded orbital
motion occur in regions of u between u_< and u_> where u_< and u_>
are adjacent positive roots of C(u) and where C(u) is positive
between them. I.e. C(u_<) = 0, C(u_>) = 0, and C(u) > 0 when
(u_<) < u < (u_>).
The scaled mapping between w and u is given by
w = sqrt(2*m*|E|)*u/L & u = L*w/sqrt(2*m*|E|)
and the scaled mapping between C(u) and D(w) is given by
C(u) = (2*m*|E|/L^2)*D(sqrt(2*m*|E|)*u/L) &
D(w) = (L^2/(2*m*|E|))*C(L*w/sqrt(2*m*|E|)) .
Everything that I said in my previous post concerning the function
D(w) also applies just as well to the C(u) function here and they
are both equivalent (except for some singularities in the
transformation at E = 0 and L = 0). The nice thing about using the
dimensioned u and C(u) rather than the dimensionless w and D(w) is
that u & C(u) have no problem at all handling the exceptional E = 0
case. But instead they both have problems dealing with the
exceptional L = 0 case. But that case gives an orbit that is
degenerate (kind of like the circular orbit situation) since when
L = 0 the orbit is just a straight line along the radial direction
where the angle [phi] doesn't change at all.)
I guess that sometimes it doesn't pay to use dimensionless variables
(i.e. in situations where the scaling factor becomes singular).