If the eccentricity is treated as a relative velocity, the relation between the true and eccentric anomalies is comparable to the addition law for the hyperbolic tangent. If \phi denotes the eccentric anomaly then the relation is
\tanh r= \cos\phi. (1)
This strange relation between hyperbolic and trigonometric functions has appeared in the literature before, namely when Legendre derived his first elliptic integral. There is no analytic solution to such an equation, and Legendre left it at that.
If we invert the relation we get
\phi=2\arctan(e^r). (2)
This is known in literature as a single soliton solution. It satisfies the sinh-Gordon equation
\phi_rr=\sech(r)\tanh(r). (3)
The right hand side has a hump at the origin but the hyperbolic tangent is zero there so the is a negative hump to the left and a positive hump to the right of the origin. This indicates the localized particle nature of the solution. But because of (1) \sech(r)=\sin\phi, and sinh-Gordon equation is transformed into the sine-Gordon equation.
2\phi_rr=\sin(2\phi) (4)
This represents the wave-like aspect of the solution,
Equation (1),
r=\arctanh(\cos\phi) (5)
represents a pair of solitary waves, one extending from \phi=0 to \phi=\pi/2, and the other from \phi=\pi/2 to \phi=\pi. The head of the resulting Joukowski ellipse has its head at the origin and the asymptotic cusp along the symmetry axis. Rotating the ellipse gives a Joukowski drop which has constant negative curvature and is the pseudosphere with added angle dependencies. Both have the metric
ds^2=dr^2+\sinh^2(r)d\phi^2. (6)
So what we have done is to take a circle through the origin, r=cos\phi, applied a hyperbolic transform of a Lobschevskian staight line, \tanh(r), and have come out with a Joukowski ellipse with constant negative curvature.
We can do the same thing with an ellipse passing through the origin,
r =(\cos\theta+\sec\theta)^{-1} (7)
and come out again with a Joukowski ellipse
r=\coth^{-1}[A(\cos\theta+\sec\theta)] (8) for the special value A=1/2, the arithmetic average of the Joukowki transform
w=z+1/z,
where z=\cos\theta. For A<1/2 the trajectories are closed and for A>1/2 they are open.
Why the passage through the origin? Newton is accredited with displacing the source at the center of an ellipse to the focus. This transformed the harmonic force into the inverse square law. The next question should have been what happens when the source is transferred to the boundary of the ellipse at either apside? What happens is that the point is no longer part of the euclidean plane but becomes a point at infinity in prespective geometry where two parallel rail lines meet. The force at this point is an inverse-fifth force, have much shorter range than either of the two other points where the source can be located.
Rotating the Joukowski (J-) ellipse about its axis of symmetry and results in a J-drop. This is a new model of hyperbolic geometry with only one ideal point. The pseudosphere has two ideal points where the geodesics, which are tractrix curves meet. The circles of constant latitude are horocycles. Horocycles came into prominence with the Poincare disc model where the whole hyperbolic plane is contained in a unit disc. The boundary of the disc is infinitely far away and would take an infinite amount of time to be reached. The boundary represents a locus of ideal points where the geodesics, sections of circular arcs, cut the disc orthogonally.
The seemingly contorted description of the hyperbolic plsne is due to the fact that the Poincare disc model is a stereographic projection of the pseudosphere, or bugle shaped surface onto the euclidean plane. The geodesics, tractrix curves get bent in the process and the two ideal points of the ends of the bugle become a loculs of equidistant points on the rim. The horocycle is an euclidean circle that touches the boundary in one point. It is an ideal point because all geodesics that cut the horocycle normally approach the ideal point in a parallel fashion. This corresponds to the circle or ellipse passing through the origin that we discussed previously.
The J-drop is a much simpler model of the hyperbolic plane because it has only one ideal point at the cusp. The meridians are solitary waves that traverse the J-drop, and the horocycles are the latitude of the drop all points being equidistant from the ideal point. The boundary of the J-drop is no longer taboo for particles to travel one since the ideal point has been isolated at the cusp. They are free to propagate back and forth, even on the surface of the drop.
Turning to the soliton solution, we see that this becomes a model of elliptic geometry. Until now only the sphere has been a model of constant positive curvature with one of the two poles removed. The metric is
ds^2=\sin^2\phi dr^2+d\phi^2 (9)
In comparison with (6) we can appreciate that the radial coordinate, r, and angle, \phi, have swapped roles. \sin\phi modulates the distance in the r-direction, creating a wave-like, or helical structure as r varies,much like a barbershop pole. The geodesics, (2), are spirals that wind around the surface reflecting the continuous winding pattern inherent in the sine function.
Propagation of solitons by kinks and anti-kinks, follows this helical path maintaining their shape as they travel from end to end of the pinched cylinder which consists of two ideal points at \phi=0 and \phi=\pi.
So here we have united elliptic geometry with soliton dynamics creating a visually intuitive and mathematically consistent framework of the elliptic surfasce and the behavior of the geodesics. The potential
V=1-\cos(2\phi)=2\sin^2(\phi) (10)
has ideal points, or "resting" points of geodesics at \phi=0 and \phi=\pi. It has a maximum at \phi=\pi/2, where a kink starting at \phi=0, transforms into a kink and propagates till it reaches \phi=\pi. On the reverse trip, the wave starts out as kink but transforms into an anti-kink at \phi=3\pi/2.
In contrast to measuring distance on great circles as d=\arccos(r_1r_2) between points r_1 and r_2, we now have the relative value
\phi_1-\phi_2=2[\arctan(e^{r_1})-\arctan(e^{r_2})]. (11)
Geodesic paths spiral along the surface governing this relative difference and creating a stable, continuous propagation. There is no geodesic intersection and looping around, since there are two ideal points which are unheard of in elliptic geometry. Geodesics in the barbershop pole model spiral and propagate dynamically from one end to the other.
It seems quite amazing that when we consider r=r(\phi) we get a hyperbolic model while inverting it, \phi=\phi(r), we obtain a model for the elliptic plane!
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