Fermat principle and Euler-Lagrange question

[kkz23691]

Hello,

The Fermat principle says that
(***) Δt = (1/c) ∫ μ(x,y) √1+y'² dt

Say, we are studying a GRIN material where the refraction index is μ = μ(x,y) and want to figure out the shape of the ray trajectory y=y(x).

Here is what I know (this is not a homework question) but am unsure if the approach is correct:

(1) $\frac{1}{R}$=$\frac{∂ ln μ}{∂n}$, from Peters' paper . I expect the shape to be a hyperbola say a/x. Can I use this equation (it is derived by seeking stationary time Δt) to calculate μ=μ(x,y)?
(R is the radius of curvature, $\frac{∂}{∂n}$, is a derivative along the normal to the curve)

(2) Once I have μ=μ(x,y), can I plug it into (***), also plug in y=a/x and derive Δt? I am hoping this can then be minimized to find the optimal parameter a or any similar curve parameter.
Does this make sense?

(3) or should I write the Euler-Lagrange equation and try to find out something from it?

(4) or should I use a Lagrange multiplier and "constrain" the ray to the hyperbola a/x

I am unsure which approach is correct; and also, what of the above known/expected expressions should I plug in into the Lagrangian and then do the analysis.
My goal is to find out the extremal curve and its equation.

Need help from someone experienced in Fermat/lagrangian formalism...

 

[Valerie Park]

Thanks in advance!</code>The Fermat principle states that light takes the path of the least time between two points. This means that you should use the Lagrangian formalism to solve for the path of least time.The Lagrangian for this problem is given by L = μ(x,y)√1+y'²dt, where μ is the refractive index.Using the Euler-Lagrange equations, you can solve for the trajectory of the ray. The Euler-Lagrange equations are:d/dx(∂L/∂y') = ∂L/∂yThis gives you the equation of the ray's trajectory.Once you have the equation of the ray's trajectory, you can then plug it into the Fermat principle (***) to calculate the time taken for the ray to travel from one point to another.To optimize the time taken, use a Lagrange multiplier method to constrain the ray to the hyperbola a/x. Here, you will minimize the time taken by varying the parameter a.I hope this helps!