A Weberian Force For the Generalized Newtonian Force

There are more parallels between Newton's force and the Weberian force of electrodynamics than meets the eye. Newton derived his inverse square law from the centrifugal force

$$F_c=F_s\sin\alpha=\frac{v^2}{\varrho},$$

where $F_s$ is the force directed at the source, $v$ the velocity and $\varrho$, the radius of curvature. Newton then introduced Kepler's II,

$$rv\sin\alpha=L,$$

to obtain

$$F_s=\frac{L^2}{r^2\varrho\sin^3\alpha}.$$

Newton developed, early in his carrier an expression for the radius of curvature, which reads

$$\varrho=\frac{(1+z^2)^{3/2}}{1+z^2-z^{\prime}},$$

where the prime denotes differentiation with respect to $\theta$. Since $z$ is the slope of the curve, $\tan\theta$, the numerator can be expressed in terms of the complementary angle, $\alpha$, as

$$1+z^2=1/\sin^2\alpha$$

so that the product $\varrho\sin^3\alpha$ can be replaced by the inverse of $1+z^2-z^{\prime}$, and, consequently

$$F_s=\frac{L^2}{r^3}\left\{1+\frac{2}{r^2}\left(\frac{dr}{d\theta}\right)^2-\frac{1}{r}\frac{d^2r}{d\theta^2}\right\}.$$

From the way we have written the centrally directed force,

it is very strange that Newton didn't consider a centripetal force varying as the inverse cube of the distance. Introducing the transform $u=1/r$, the force becomes

$$F_s=L^2u^2\left[u+\frac{d^2u}{d\theta^2}\right].$$

In this form, it is appear that the equation of the trajectory,

$$\frac{d^2u}{d\theta^2}+u=\mbox{const.},$$

equivalent to $\varrho\sin^3\alpha=\mbox{const}$, gives his inverse square law for the particular conic section of an ellipse. But, this is not the only conic section that can be obtained, and will explain the dual force laws for the magic pairs

$$[1,-2]\hspace{20pt}[-4,-7]\hspace{20pt}[-5,-5]$$

that we have discussed in previous blogs. The first pair is the Hooke-Newton pair, while the other two are unknown. This is where out generalized force law comes in, that is not limited to these pairs, but, moreover, to the elusive inverse cubic law as well.

There are many parallelisms between Newton's and Weber's forces which was to follow a century and a half later. Newton believed that the manifestation of force was curvature so that gravity is a force. Einstein said that gravity is not a force but geometry so he turned the argument on its head. In other words, geometry is the manifestation of gravity so that a universe can be empty while still containing gravity. This means that gravity is no longer related to mass in Einstein's interpretation.

An empty universe, according to Einstein, means that the sum of the sectional curvatures in a given direction should vanish. Newton's force, however, is not related to sectional curvature. Newton's force is proportional to the difference between the square of the speed, $r^{\prime 2}+\omega^2r^2$ and Gaussian curvature, $K=-\Delta\ln r$. Weber replaces the angular velocity $\omega r$ by his constant $c_W=\surd 2 c$, and added $-2K$

 

 

Other differences exist between Newton and Weber:  the azimuthal angle $\theta$ is replaced by current elements and then by charges and time, 

$$F_W=\frac{ee^{\prime}}{r^2}\left(c_W^2-\dot{r}^2+2\ddot{r}r\right).$$

 

We can also appreciate  that Einstein's equation for the advance of the perihelion of Mercury,

$$\frac{d^2u}{d\theta^2}+u=3u^2.$$

This corresponds to an inverse quartic force, and corresponds to a cardoid,

$$r=1+\cos\theta.$$

Third, the inverse seventh corresponds to a Cassini oval, and, in particular the lemniscate,

$$r=\surd\sin 2\theta.$$

This turns out give the force,

$$F_s=\frac{3L^2}{r^7},$$

one of the magic trio, since

$$\frac{d^2u}{d\theta^2}+u=3u^5.$$

Finally, the unstable spirals, $r=a/\theta$,and $r=a\theta$, the logarithmic and Archimedes spirals, respectively, correspond to the inverse cubic

$$F_s=\frac{L^2}{r^3},$$

and the combination of inverse cubic and fifth,

$$F_s=\frac{L^2}{r^3}\left(1+\frac{2a}{r^2}\right),$$

respectively. The latter belongs to one of the magic trio, but seems to have no problems to appear with the solo inverse cubic.

According to de Sitter's appraisal of GR, "gravity is not a force." This, in essence, relegates gravity to the time dimension which it has no right of being there. That is why Einstein's determination of the "gap" in the numerical value of the advance of the perihelion, was just that---gap fitting. Here we can appreciate it as the result of the combination of the inverse square and inverse quartic forces. We will have more to say about this in coming blogs.