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What are the Euler-Lagrange equations in cylindrical coordinates $(r,\theta,z)$ for light moving at speed $v(r,\theta)$, where $r$ and $\theta$ depend on $z$?

I.e. for the problem of minimising the time taken for light to get from one point to another, where the speed is $v(r,\theta)$, what are the Euler-Lagrange equations for $r(z)$ and $\theta(z)$? I need both the general Euler-Lagrange equations and a version for when $v(r,\theta)=A(\lambda^2+r^2)$.

By Fermat's principle, the time taken is stationary w.r.t. variations of the path leaving the endpoints fixed. By converting to Cartesian coordinates, we get that $v^2 = \big(\frac{dz}{dt}\big)^2 (r'^2+r^2\theta'^2+1)$. (The question asks for a proof that $v^2(r'^2+r^2\theta'^2+1)$ is constant! Is this a mistake, or am I missing something?)

The formula for the time taken between points $z=z_1$ and $z=z_2$ should be $\int_{z_1}^{z_2} \frac{1}{v(r,\theta)}\sqrt{r'^2+r^2\theta'^2+1} \; dx$, which makes the Euler-Lagrange equations pretty ghastly.

What am I missing? Are my formulae for speed and time correct? Is there a way of simplifying the Euler-Lagrange equations?

Many thanks for any help with this!

share|improve this question
    
Please clarify what the question you're trying to solve is. –  nbubis Apr 28 '12 at 14:07
    
@nbubis I have edited the question to clarify. –  Harry Macpherson Apr 28 '12 at 16:03

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