Light travels in a medium in which the speed of light c(x,y) is a function of position. Fermat's principle states that the time required for light to travel between two points is an extremum relative to all possible paths connecting the two points.(adsbygoogle = window.adsbygoogle || []).push({});

1) Show that the time for the light to travel from point (x1,y1) to (x2,y2) along the path y(x) is

T = INT[x1,x2] Sqrt(1+y'^2)/c(x,y) dx - Completed

2) Write down the Euler-Langrange equation for this functional

b) and its special form for when c is independent of x.

2a:

(1+y'^2)^(1/2) d(1/c)/dy - d/dx[y'/(c (1+y'^2)^(1/2))] = 0

Is as far as I can get, how should I continue this? Perhaps by noting d(1/c)/dy = (-dc/dy)/c^2 ?

2b:

(1 + y'^2)^(1/2)/c - y' (y'/c(1+y'^2)^(1/2)) = constant (A)

(1 + y'^2 - y'^2)/c(1+y'^2)^(1/2)) = A

1/c(1+y'^2)^(1/2)) = A

c(1+y'^2)^(1/2) = B = 1/A

c^2 (1+y'^2) = D = B^2 = 1/A^2; is this correct?

2c: If light is emitted from the origin making an angle t (0 <= t <= pi/2) with the positive x-axis in a medium in which c(x,y) = 1+y, show that the light travels in a circle centre (x,y) = (tan(t), -1)

Presumably for this I need to make the substitution c = 1+y into the equation derived in 2b (since there will be no dependence on x) and work from there to obtain the equation of a circle?

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# Homework Help: Calculus of Variations - Fermat's Principle

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