In the last blog we observed that Newton's expression for the tangent to the curve
r'/r=z=\tan\theta (1)
is valid for only a straight line. Hence, that contradicted his result of the resulting conic through integration of the denominator of the radius of curvature in polar coordinates
\rho=r(1+z^2)^(3/2)/[1+z^2-z', (2)
where the prime denotes the derivative with respect to the angle \theta. The derivation of the formula for \rho rests on the expression for the interval as
ds = n\sqrt{E dr^2+G d\theta^2}, (3)
where n is the index of refraction, assumed unity, and the metric coefficients E and G are unity and r^2, respectively. Hence, Newton was dealing with a flat metric.
The integration of (1) leads to
r=\sec\theta,
apart from an arbitrary constant of integration, that is a straight line. Had we chosen the negative sign in (1), we would have come out with a circle whose circumference passes through the origin, What can we say with more general expressions that are derived from (3)?
The expression for the slope (1) generalizes to
\sqrt{E/G} r' = \tan\theta. (4)
If E is a power law as we saw in the last blog, E=1/r^n, n>0, the integration of (4) leads to
r=1/[ln \cos\theta + A]. (5)
This expression for the radial coordinate captures the behaviour of both circles and hyperbolas without explicity mentioning eccentricity and directrix. It provides a unified description of these curves in terms of a single constant, A. Several values of A are shown in the figure below.
For A=1, we get a red circle whose circumference intersects the origin. In contrast for A=-1, we get a pair of green hyperbolae with the center of one at the origin. As A drecreases from 1, yet remains positive we obtain blue ellipses with incresing eccentricity. Likewise, as A increases from -1, yet remains negative we get purple ellipses.
Rather, if we treat the conformal factor of the index of refraction, and treat, otherwise, a flat metric we obtain expressions for both the eccentricity and directrix. If we choose an index of refraction given by the Eaton lens (see, Seeing Gravity)
n^2 =2\mu/r- 1/a
where \mu is the gravitational parameter, and a is a distance, we come out with
r=1/[\mu/h^2+\Delta\sin\theta]
where h^2/mu is the semi-latus rectum, and \Delta is the discriminant of the quadratic form
r^2/a h^2-2\mu r/h^2 -1,
where h is the conserved angular momentum. It arises from treating an ignorable variable, \theta, in the expression for the lagrangian
n\sqrt{1+r^2\theta'^2}
It's variation with respect to \theta' gives rise to a first integral, h.
Summary: So here we have two complementary waves of deriving expressions for general conics: the generalization of the expression for the tangent to the curve, (4), and the method of the index of refraction which provides more specific information regarding the eccentricity and directrix that depends on the particular expression that the index of refraction may assume.
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