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How is Eikonal Equation analogous to Newton's Law?

by genxium
Tags: analogous, eikonal, equation, newton
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Jul16-14, 08:54 PM
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I read this from a lecture note(attached) of Geometric Optics. It's said that the eikonal equation for light rays [itex]\frac{d}{ds}(n(\vec{r})\frac{d\vec{r}}{ds})=\frac{\partial n}{\partial \vec{r}}[/itex] is analogous to Newton's Law, however it doesn't tell which Newton's Law is referred to. (In the equation, [itex]\vec{r}[/itex] is position vector, [itex]s[/itex] is the raw path length, [itex]n[/itex] is the refractive index).

The equation can be rewritten to [itex]\frac{d^2\vec{r}}{d\sigma^2}=\frac{1}{2}\frac{\partial n^2}{\partial \vec{r}}[/itex] where [itex]d\sigma=n^{-1}ds[/itex]. 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.
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Jul16-14, 09:48 PM
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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.

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