Proof of Snell's law using Fermat's Principle and the Euler-Lagrange equation

[@theboombody]

I'm usually not a big fan of step functions but I really enjoyed this demonstration. At this point I think I'll enjoy anything involving the Euler Lagrange Equation that doesn't just parrot the derivation you can find on Wikipedia.

[@SimulatingPhysics]

For the ones interested, I leave here an alternative derivation of Snell Law (very simple), using the conservation of momentum of photons: If in medium 1 the index of refraction is n1 and in medium 2 the index of refraction is n2 and for the definition of index of refractions and wavelength we have: n1 = c / v1 , v1 = λ1 * f n2 = c / v2 , v2 = λ2 * f Dividing the above equations we obtain: n2 / n1 = λ1 /λ2 So if n2 is higher than n1, the wavelength in medium 2 is smaller than in medium 1. Now because the photon momentum is: p = h / λ Applying the conservation of momentum of the incident and transmitted photon along the parallel line of medium separation: p1 = p2 → h/λ1 *sin(θ1) = h/λ2 *sin(θ2) → n1 *sin(θ1) = n2 *sin(θ2) This is the Snell Law!! So the refraction is just a consequence of the conservation of momentum of photons! There are a lot of ways of derive this law!

[@AndrewNicoll]

Yes there are a lot of ways. But I would argue the one I've demonstrated in the most concrete and relies on little assumption.

[@SimulatingPhysics]

@@AndrewNicoll Yes, this is true! Fermat Principle is beautiful.

[@gerardomorales5181]

Amazing proof! Greetings from Mexico!

[@henrymochi]

Thank you for going through that, each step was very clearly explained!

[@chahn987]

It is explained very well. Thank you.

[@AndrewNicoll]

Any suggestions for a video topic? If so let me know.

[@alonso7327]

thanks for the video Andrew, it was really useful, it was quite clear method use step function for the speed as well. But I have been thinking that is not necessary for the speed to be constant, actually you can get the same results, I know speed only changes at the boundary, but I think is important to say it. Regards :)

[@DroneGirl1]

Ohh my Goodness this is genius :) Thumbs up from Ireland :) !

[@AndrewNicoll]

Drone Girl Thanks very much! love your videos as well, I'm a keen drone flyer my self.

[@8Soul-Light8]

Andrew Nicoll 😂😂😂

[@rabbitpi]

At 7:34: Surely V(x) being constant relies on v_1 and v_2 being both constant *and* equal? Otherwise V(x) is not constant, as it changes value at x=a? Unless I'm mistaking what you mean by "V(x) is constant", it can't be constant if it's taking 2 different values at different values of x.

[@DEEPAKKUMAR-xi2hu]

Love from India🇮🇳

[@godzillazumagod9146]

We are in either that is the gas particle by which light is slow and limited in space. N1 that is.

[@karthikyadavalli4798]

Is there any reason the x-axis is the vertical one? I followed your entire derivation, setting the x-axis as the horizontal axis and my final answer ends up with cosines instead of sines.

[@МартинДимитров-ь9р]

I believe that when you change the axes, you also change the dependent and the independent variables. So for example, when you calculate dl, you have to take out not dx, but dy from the square root and you will be integration w.r.t y now (i.e. you have x as a function of y, not the other way around). However, since you've swapped the variables, you also have to do that in all the steps leading to the Euler-Langrange equation when deriving it. So in the new case, at the end you should have the partial derivative of G w.r.t. dx/dy equal to a constant (not dy/dx as it is in the video). After performing the calculations you should arrive at the correct version of the Snell's law. I find it also confusing so I might be wrong but I believe that Marion and Thornton (classical dynamics of particles) made a comment on that. Usually when it is not predictable which choice of coordinate system is the easiest one, I arrive at something wrong. A change of the axes fixes that but I don't really have a proper strategy towards harder problems, i.e. I proceed by trial and error.

[@karthikyadavalli4798]

I think I understand now. My initial issue was the fact that I did not understand why the vertical axis needed to be the independent axis, but I understand now. Thanks!

[@silverman6296]

That’s not the explanation, Christ! The reason is that when you switch axis, you get a changed normal vector of the surface, so you get the complementary angle, which is the cosine.

[@abdulrahmanmohamed8298]

@@karthikyadavalli4798 could you explain that part to me?

[@marychrisgo2073]

What is v is not constant?

[@godzillazumagod9146]

Like it does between the galaxys.

[@Andy-tf2ph]

At 3:20 why is it dl_1=dx√ *1* +(dy/dx)^(2 ) and not dl_1=dx√ *dx* +(dy/dx)^(2 )

[@rabbitpi]

Think about the fact that dx = √(dx^2), maybe then you'll see it? Taking dx^2 out of the square root leaves 1 inside it, and dx on the outside.

[@mohammadjavadsoltani5316]

Tank's 😍🤍