Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How is Eikonal Equation analogous to Newton's Law?

  1. Jul 16, 2014 #1
    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.
     

    Attached Files:

  2. jcsd
  3. Jul 16, 2014 #2

    UltrafastPED

    User Avatar
    Science Advisor
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: How is Eikonal Equation analogous to Newton's Law?
  1. Newton's Laws (Replies: 2)

  2. Newtons Laws? (Replies: 3)

  3. Newton's laws (Replies: 9)

Loading...