How is Eikonal Equation analogous to Newton's Law?

[genxium]

I read this from a lecture note(attached) of Geometric Optics. It's said that the eikonal equation for light rays $\frac{d}{ds}(n(\vec{r})\frac{d\vec{r}}{ds})=\frac{\partial n}{\partial \vec{r}}$ is analogous to Newton's Law, however it doesn't tell which Newton's Law is referred to. (In the equation, $\vec{r}$ is position vector, $s$ is the raw path length, $n$ is the refractive index).

The equation can be rewritten to $\frac{d^2\vec{r}}{d\sigma^2}=\frac{1}{2}\frac{\partial n^2}{\partial \vec{r}}$ where $d\sigma=n^{-1}ds$. It's actually the rewritten equation that is said to be analogous to Newton's Law, but I have no idea how to interpret it.

 

[UltrafastPED]

I cannot say it better than this:

The classical roots of wave mechanics: Schrödinger's transformations of the optical-mechanical analogy

Christian Joas, Christoph Lehner
Max Planck Institute for the History of Science, Boltzmannstr. 22, 14195 Berlin, Germany
Studies In History and Philosophy of Science Part B Studies In History and Philosophy of Modern Physics (Impact Factor: 0.85). 01/2009; DOI: 10.1016/j.shpsb.2009.06.007
Source: OAI

ABSTRACT In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat's principle) and corpuscular mechanics (Maupertuis's principle). Almost a century later, this optical-mechanical analogy played a central role in the development of wave mechanics. Schrödinger was well acquainted with Hamilton's analogy through earlier studies. From Schrödinger's research notebooks, we show how he used the analogy as a heuristic tool to develop de Broglie's ideas about matter waves and how the role of the analogy in his thinking changed from a heuristic tool into a formal constraint on possible wave equations. We argue that Schrödinger only understood the full impact of the optical-mechanical analogy during the preparation of his second communication on wave mechanics: Classical mechanics is an approximation to the new undulatory mechanics, just as ray optics is an approximation to wave optics. This completion of the analogy convinced Schrödinger to stick to a realist interpretation of the wave function, in opposition to the emerging mainstream. The transformations in Schrödinger's use of the optical-mechanical analogy can be traced in his research notebooks, which offer a much more complete picture of the development of wave mechanics than has been previously thought possible.