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So, I'm reading Holm's Geometric Mechanics Part I, and in it he wants to derive the Eikonal equations from Fermat's principle.
There's one part of the derivation that I don't understand. He gives the "Optical path" as:
[tex]A=\int_a^b n(\vec{r}(s))ds[/tex]
Where ds is the infinitessimal arc length.
And then proceeds to take the variation of:
[tex]\delta \int_a^b n(\vec{r}(s))\sqrt{\frac{d\vec{r}}{ds}\cdot\frac{d\vec{r}}{ds}}ds[/tex]
What's the deal with adding the square root term in there? It seems that it's legitimate to add this term since it's actually 1, but it appears that without adding this term, you don't get the Eikonal equations back by taking that variation to be 0, so what rational is there that "mandates" the adding of this term?
There's one part of the derivation that I don't understand. He gives the "Optical path" as:
[tex]A=\int_a^b n(\vec{r}(s))ds[/tex]
Where ds is the infinitessimal arc length.
And then proceeds to take the variation of:
[tex]\delta \int_a^b n(\vec{r}(s))\sqrt{\frac{d\vec{r}}{ds}\cdot\frac{d\vec{r}}{ds}}ds[/tex]
What's the deal with adding the square root term in there? It seems that it's legitimate to add this term since it's actually 1, but it appears that without adding this term, you don't get the Eikonal equations back by taking that variation to be 0, so what rational is there that "mandates" the adding of this term?