# Calculus of Variations - Fermat's Principle

• SunGod87

#### SunGod87

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.

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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I actually have this exact same problem and I'm stuck in exactly the same places.