(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let y(x) represent the path of light through a variable transparent medium. The speed of light at some point (x,y) in the medium is a function of x alone and is written c(x). Write down an expression for the time T taken for the light to travel along some arbitrary path y(x) from the point (a,yI) to the point (b,yF) in the form:

T=[tex]\int L(y,y')dx[/tex]

(between a and b, can't seem to add in limits on the integral sign)

where L is a function you should determine.

2. Relevant equations

S = [tex]\int \sqrt{1+y'^{2}}dx[/tex]

I derived this though, don't think its meant to be a known equation.

3. The attempt at a solution

Well using calculus of variation I know you can formulate a path length as:

S = [tex]\int \sqrt{1+y'^{2}}dx[/tex]

So the time for the light to travel along this path should just be:

T= [tex]\int \frac{ \sqrt{1+y'^{2}}}{c(x)}dx [/tex]

However, this has a dependence on x where the question states it must be only in terms of y and y'. This is because in the first part of the question I had to derive the Euler-Legrange equation and as light takes the path which minimises time, you must plug this function L into the equation.

So, how do I get rid of the x dependence? If there's a trick or a technique involved could someone just inform me of the name so I can look it up for myself? I've read the classical mechanics textbook which is meant to accompany this course and can find nothing which helps with this.

Thank.

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# Homework Help: Path of Light- Calculus of Variation

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