From Newtonian Gravitation to Maxwell's Fish-Eye to Cassini Ovals

It is known [Needham, Am. Math. Monthly 100 (1993) 119-137] that there are only three pairs of integer exponents for dual force laws

$$[1,-2]\hspace{25pt}[-5,5]\hspace{25pt}[-4,-7].$$

 $[\tilde{a},a]$ stands for  pair of exponents in the force laws

$$\frac{d^2z}{dt^2}=-z|z|^{\tilde{a}-1},$$

and

$$\frac{d^2w}{d\tau^2}=-w|w|^{a-1},$$

where $z$ and $w$ are related by the conformal transform,

$$w=z^{\alpha},$$

where $\alpha=(\tilde{a}+3)/2$, and $(\tilde{a}+3)(a+3)=4$. The two times are related by

$$\frac{d\tau}{dt}=|z|^{\tilde{a}+1}.$$

The first pair stands for the dual Hooke-Newton's law, the second, as we have shown, to the inverse fifth power law in points of inversion in Maxwell's fish-eye, while the last one to Cassin ovals. 

You cut a cone by a plane and get conic sections; you cut a doughnut (torus) you get Cassini ovals. For an ellipse with eccentricity less than one, the sum of the distances from the two foci to any point is a constant. Cassini ovals replace the arithmetic mean by the geometric mean: the product of the distances between any point and the two foci is a constant. Therefore, the Cassini ovals are the next simplest closed curves after ellipses, and this probably drove Cassini in 1680. This is the 'next' and last generalization since it constitutes the third pair of exponents in the above list of dual laws. 

A little history. Cassini ovals are known as spiric sections (spiric=torus). In a sense they are generalizations of conic sections that were first constructed by Menaechmus in 150 BC. Two centuries later, Perseus considered slicing a torus with a plane, and obtained the spiric sections. Now most of the planets describe ellipses with low eccentricity. For such ellipses, the angular sectors created by rays from a focus are approximately equal to the corresponding angle at the other focus. Using this property Ward attempted to find the true anomaly of a planet given the mean anomaly. The ratio between the infinitely small sector and the corresponding angle is like a rectangle of two lines passing through the focus, and for an ellipse of low eccentricity this rectangle remains constant. Cassini reasoned that such a rectangle will be constant in an oval, and this gave birth to the cassinoid.

Let us begin with the first pair. A conic section is given in polar coordinates by

$$\cos\theta=\frac{p\epsilon-r}{r}=n^2<1,$$

where $p$ is the focal parameter, and $\epsilon>0$ is the eccentricity. The second equality defines the index of refraction, $n$. Expressing Kepler's area law as

$$\sin\alpha=\frac{K}{nr}=\frac{K}{\sqrt{r(p\epsilon-r)}},$$

it is easy to show that the radius of curvature, $\rho$, is given by

$$\rho\sin^3\alpha=K,$$

the angular momentum. This leads immediately to Newton's inverse square law of force. That is, introducing $-n^2r/n^{\prime}K$ for the curvature, where the prime stands for differentiation with respect to $r$ and $\sin\alpha=K/nr$, there results

$$n^2=\frac{2K}{r}+2C,$$

upon integration, where $C$ is an arbitrary constant of integration. For $C=-K/2a$, we have an elliptic orbit with semi-major axis $a$, $0$ for a parabolic one, and $C=K/2a$ for a hyperbolic orbit. The parabolic orbit equates the index of refraction with the escape velocity where $K=GM$. Finally, for circular orbits, $n^2=K/r$, which is none other than Kepler's third law, $rn^2=K$

There is little symmetry between the two angle expressions. This changes when we go to the sphere, where the angles are given by

$$\cos^{-1}\left[\frac{K}{\sqrt{1-4K^2}}\left(r-\frac{1}{r}\right)\right]=\theta,$$

and

$$\sin^{-1}\left[K\left(r+\frac{1}{r}\right)\right]=\alpha.$$

The arguments are a Jukowski transform which shifts the center of the ellipse to the left and right foci when squared. A Hookean ellipse is thus transformed into a Newtonian ellipse simply by squaring.

The index of refraction is that of Maxwell's fish-eye

$$n=\frac{1}{1+r^2},$$

and the force directed to the center of its source,

$$F=-\frac{1}{2}\frac{d}{dr}n^2=\frac{r}{(r^2+1)^3},$$

is an inverse fifth order law for both points of inversion. 

Cassini ovals appear when we square the Jukowski transforms

$$\cos 2\theta=\frac{1}{2a^2}\left(r^2+\frac{a^4-b^4}{r^2}\right).$$

and

$$\sin2\alpha=\frac{1}{2(a\epsilon)^2}\left(r^2-\frac{a^4-b^4}{r^2}\right),$$

where $a$ is the distance to either focus, and $rr'=b^2$ is the bipolar equation, proportional to the geometric average of two distances to either fixed point (foci). The first equation gives the polar equation,

$$r^4-2a^2r^2\cos 2\theta+a^4=b^4,$$

with eccentricity $\epsilon=b/a$. The constant $K=1/(a\epsilon)^2$, and

$$n=\frac{2r}{r^4-(a^4-b^4)}.$$

This gives the (known) radius of curvature,

$$\rho=\frac{2b^2r^3}{3r^4+(a^4-b^4)},$$

with inflection points located on the lemniscate $r^2=2a^2\cos 2\theta$, for which the force law will become a central force law, as we now show.

The force directed to its source is

$$F_C=2rb^2\frac{3r^4+(a^4-b^4)}{(r^4-(a^4-b^4))^3}.$$

This force governs trajectories of the motion of a central field of force whose strength is inversely proportion to the 7th power of the distance to the source. 

For $a=b$, or $\epsilon=1$, the Cassini oval transforms into a lemniscate of Bernoulli, $r^2=2a^2\cos 2\theta$, with a central law of force

$$F_C=\frac{12a^4}{r^7}.$$

This is the last pair of integer exponents in the trio given above. There are no other integer exponents for the dual laws. It remains, among other things to analyze the dual force law $F\sim 1/r^4$.

It has been fantasized [Tapia, Int J Mod Phys D 2 (1993) 413] that spiric sections may lead to a gravitational theory with a fourth-rank metric tensor. Such is the influence of general relativity. Equally fanciful is the association of Bernoulli's lemniscate  with an oriented matroid theory, which is a combinatorial structure that has been proposed as the underlying structure of $M$-theory [Nieto, hep-th/0506106]. On thing is sure: Nature is frugal and doesn't waste its integer exponents on nothing. But that nothing is still to be discovered.