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How a Joukowsky Ellipse Emerges in the Non-Euclidean Plane of Constant Curvature

In previous blogs we saw that Newton's slope of the tangent line to the curve of the trajectory,

r'/r =\pm \tan\theta, (1)

where the prime denotes differentiation with respect to \theta gives either a straight line, r=\sec\theta (+ sign) or a circle or a circle r=\cos\theta (- sign) depending on orientation.


Instead of considering the flat metric,

                                      ds^2=dr^2+r^2d\theta^2 (2)

which leads to the expression for the tangent to the curve (1), we consider a non-Euclidean analog of constant curvature replacing r^2 by G^2 where G is either R\sinh(r/R) or R\sin(r/R), where R is a constant scale factor, in the hyperbolic or elliptic plane, respectively. Considering the former, (1) is replaced by

r'/R\sinh(r/R) =\pm\tan\theta (3)

Integration gives

\tanh(r/2R)=\frac{\cosh\r/R)-1}{\sinh(r/R)= \cos\theta (4)

or

\tanh(r/2R)=\frac{\sinh(r/R)}{\cosh(r/R)-1=\frac{1}{\sec\theta}. (5)

Since the inverse radial coordinate is u=\coth(r/R)/R, (4) and (5) give

cos\theta=R(u-\sqrt{u^2-1} (6)

and

\sec\theta= R(u+\sqrt{u^2-1} (7)

Adding (6) and (7) gives the desired expression

u=(1/2)(\cos\theta+\sec\theta). (8)



Since u=coth(r/R), we have

r=R\coth^{-1}(.5\cos\theta+.5\sec\theta)) (9)

which should be compared to the Joukowski transformation

w=1/z+z (10)

that stretches a circle in the complex z-plane into an ellipse in the complex

w-plane where z is a complex variable. The Joukowski (small black rhs) ellipse (9) is shown in the figure



It is compared to the Euclidean (blue ellipse)

r=\frac{2}{\sec\theta)+\cos\theta}) (11)

which has no cusp. Moreover, the square of a Joukowski ellipse (9) are two Joukowski ellipses on either side of the vertical axis. It thus maintains its reproduce property.



In contrast, the Joukowski ellipse (10) has two foci at (-2,2) located on the boundary of the ellipse. The square of a Joukowski ellipse is, again, a Joukowski ellipse except that the foci have been transferred 2 units, i.e. w^2=1/z^2+z^2+2. This has the effect of transferring the foci to the center, and equivalently transferring the Hookean force into Newton's inverse square law. This is known as a Bohlin transform and is one of the three dual laws.


The same property of shifting foci may not apply in the hyperbolic plane. Things become more complicated due to curvature.


But the circle was originally described by an inverse fifth law, and any distortions of the circle into a more complicated shape should requires modifications of the inverse-fifth law rather than the emergence of Newton's or Hooke's laws,




The Joukowski transformation can be considered as the arithemtic average of the straight line and the circle, (9). The corresponding Binet equation is

u" + u = \sec^3\theta= (u+\sqrt{u^2-1|)^3. (12)

If it is a circle that we are describing then the Binet equation reduces to

u" + u = u^3. (13)

Since the right-hand side of the Binet equation for a conic is f/h^2u^2, where f is the force, and h is the angular momentum (conserved. Comparing the last two equations gives f~u^5, which is the central force for the trajectory of a particle gong through the origin,.


For a straight line, on the other hand, \sec\theta=r, and the corresponding force is

f ~ u^2/u^3 ~ r,

which is a Hookean force, i.e., the magnitude of the force being proportional to the diplacement of the particle from its equilibrium position. Qualitatively speaking, the inverse square law would emerge when there is a balance betwee the circle and straight line contributions to the force. Let us explore this further.


The dual laws are the inverse-square law and Hooke's law, and the inverse-fourth and seventh laws with the inverse-fifth being its own dual. Apart from the first pair, the phyiscal significance of the other duals remain obscure. And even the first pair is now not so clear since the line and the circle have Hookean and inverse-fifth laws, respectively, instead of inverse-square as the dual laws claim.


For large r, the ellipse approaches a circle with foci at \pm 2 in the w-plane.

As r decreases, distortions occur, capturing the complex behavior of lift and drag forces around airfoils, which is a realization of the Joukowski ellipse.


In the previous blog we showed that the radial distance, r\rightarrow R\sinh(r/R) in the hyperbolic plane. Since

\frac{1}{R\sinh(r/R)=\frac{u^2-1} (14)

the second term in last expression of (12) decays to zero with distance, while the first term assumptotically approaches R in the same limit. Distortions, in this limit, are imperceptible and the symmetry of the circle is restored.


What is not clear is how Newton's law emerges in this picture. In contrast, if we began with the conventional expression for a conic

\frac{1}{r} = A + B\cos\theta, (15)

where the constant A is related to the semi-latus rectum, containing the ratio of the gravitational parameter \mu= GM and the square of the angular momentum, and B contains the eccentricity, it is clear that (15) is incompatible with Newton's expression for the slope since A must vanish, leaving the equation of a straight line, r=\sec\theta/B. There is no need to introduce Newton's expression for the radius of curvature into the expression for the centripetal acceleration and to claim the invariance of \rho\sin^3\alpha in order to get the inverse-square law, where \rho is the radius of curvature and \alpha is the angle that the slope makes with the trajectory of the particle.


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