The seemingly innocuous equations of the Kepler equation for the eccentric anomaly, E,
r=a(1-\cos(E)), where a is the semi-major axis of the ellipse and e is the eccentricity, and the equation for the true anomaly, \nu, r=\frac{a(1-e^2)}{1-e\cos(\nu)}, lead to nonlineary critical phenomena when r is eliminated between the two equations. The Kepler equation is a limacon, and with e<1, there can only appear a dimple but no loops.
When the two equations are combined, there results
\cos(\nu)=\frac{e-\cos(E)}{1-e\cos(E)},
which is nothing other than the double angle formula for the hyperbolic tangent
\tanh(r_0-r)=\frac{\tanh(r_0)-\tanh(r)}{1-\tanh(r_0)\tanh(r}}.
This association leads to
\cos(E)=\tanh(r) or r=\frac{1}{2}\ln(1+\cos(E)}{1-\cos(E)}.
This is a Joukowski ellipse, as shown in the figure.
Juxtaposed in the figure is the limacon which is just beginning to form a dimple with e=0.9.
The true anomaly is given by
\cos(\nu)=\tanh(r_0-r),
and the eccentricity is
e=\tanh(r_0)<1,
showing that the conic section is indeed an ellipse.
The second derivative of the eccentric anomaly with respect to r, is
E_{rr}+\sin(E)\cos(E)=0
is the sine Gordon equation for the propagation of the solitary wave along the trajectory of the Joukowski ellipse.
At the point, r_0, on the ellipse, the critical phenomenon of the transition between kinks and anti-kinks occur. This can be seen by defining the potential
V=\frac{GM}{\tanh(r)}=GM\sec(E).
The form of the potential is a well between \pm\pi/2, and repeating itself as shown in the figure. The well denotes the "stable" particle or kink, while the mountain, the anti-kink.
This happens at regular intervals along the trajectory of the ellipse when r=r_0. As seen from one of the foci, the potential is
V =\frac{GM}{\tanh(r_0-r}=GM\sec{\nu).
So e=\tanh(r_0) would now have a larger role to play than a mere eccentricity insofar as it determines the limits of hyperbolic plane in much the same way as the rim on the Poincare disc. However, whereas the rim of the disc is at infinity, and would take an infinitely long time to arrive at the rim, here we have the propagation of solitary waves transforming from kinks to anti-kinks as E varies between -n\pi/2 to +n\pi/2 where n is an integer. Having once transformed from a kink to an anti-kink, it reaches the cusp at E=\pi.
The force,
F=-\frac{GM}{\sinh^2(r_0-r)}=GM\left(\frac{\sin(E)\sqrt{1-e^2}}{\cos(E)+e}\right)^2,
becomes infinite at e+\cos(E)=0, or r=r_0, on the curve of the Joukowski ellipse. Thus, when the observer is at the center of the ellipse, he doesn't observe the transitions between kinks and anti-kinks, and the motion is continuous until it reaches the cusp E=\pi where e=1, or r_0=\infty. We have taken the negative branch of the hyperbolic sine of r_0, i.e. \sinh(r_0)=-\frac{e}{\sqrt{1-e^2}.
Whereas the source is located at the center in terms of the eccentric anomaly, it becomes transferred to the trajectory at r=r_0 when observed from the focus where \nu=\pi/2. So unlike the Poincare disc model, all points in the interior of the Joukowski ellipse are not equivalent. At the focus the picture of the motion becomes much more detailed, since it is dependent on the eccentricity, which is no longer a mere constant.
And to think all this happens merely by transferring to the hyperbolic plane of constant negative curvature, just like the pseudosphere. Thus, the Joukowski ellipse is another model of hyperbolic geometry equivalent to the pseudosphere, yet giving a more dynamic interpretation akin to the Poincare disc model.
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