Deriving fermat's principle of least time (lagrangian stuff)

[iScience]

original question: Consider light passing from one medium with index of refraction n1 into another medium with index of refraction n2. User Fermat's principle to minimize time, and derive snell's law (n1sintheta1=n2sintheta2).

here is part of the solution key

http://i.imgur.com/S1Pg9jC.png

i understand the first line but i start getting confused when it says "Although we have v = v(y), we only have dv/dy≠ 0 when y =0"

what?? I'm confused. can someone please translate that sentence for me please.

the 'v' is just c, it's a constant right? and if it's a constant, and doesn't change with respect to anything WITHIN one medium, dv/dq should equal zero all the time, so what is the text referring to when it says that?

 

[BruceW]

yeah, v is constant within one medium. But here there are two media. What will happen to v(y) at the interface between the two media?

 

[iScience]

but in the diagram the 'y' dimension I'm guessing is along the vertical. if we have y=0, then the light is just moving horizontal in which case the light would just stay in one medium wouldn't it?
i'm probably interpreting it the wrong way. how are you looking at it?

 

[BruceW]

at y=0, the light does not have horizontal velocity. That would be true if we said the light went along a path where y was constant. But that's not what the question is saying. You've got two things mixed up: 1) having y=0 at one point in the path. 2) having y=0 over the entire path.
And in this problem, it is 1) which you should think about.

 

[iScience]

i'm sorry, i don't understand how you can have an incident photon at 0 degrees to the normal and refract with some angle other than zero degrees on the other side. any incoming photons incident at the interface at 0 degrees to the normal should remain zero degrees. it is only when we have some nonzero angle that we have light actually bending. so if y=0, how is dv/dy≠ 0 valid? in other words if light is coming in at y=0 (NO horizontal component), how is the speed of light with respect to the y dimension NOT constant? the light beam pass through the material remaining at 0 degrees. how is the speed of light changing with respect to the y dimension?

unless you're saying that the light comes in at a certain angle then refracts to theta= 0 degrees. is this the scenario you are referring to?

 

[UltrafastPED]

With all the details: http://en.wikibooks.org/wiki/Optics/Fermat's_Principle

Then you can follow the brief description which you were given.

 

[BruceW]

i'm sorry, i don't understand how you can have an incident photon at 0 degrees to the normal and refract with some angle other than zero degrees on the other side. any incoming photons incident at the interface at 0 degrees to the normal should remain zero degrees.

yes, true. But if y=0, that does not mean the 'photon' is incident at zero degrees to the normal. y is position, not velocity. (and also, by saying y=0, we are not saying anything about x yet).