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Old Aug 26th 2017, 07:03 PM   #1
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snell's law and variational calculus

Working through some exercises on Variational Calculus from Taylor's Classical Mechanics.

I'm not entirely sure how to set this up.

I had a look at past post "Does Fermat principle explain refraction inside water?" but the problem seemed to be subtlely different.

I can see that the distance $\displaystyle \left | \overrightarrow{P_1 Q} \right | = \sqrt{x^2+h_1^2+z^2}$ and $\displaystyle \left | \overrightarrow{Q P_2} \right | = \sqrt{(x_2-x)^2+h_2^2+z^2}$

I imagine we set up the Lagrangian as $\displaystyle \mathcal{L} = \left | \overrightarrow{P_1 Q } \right | + \left | \overrightarrow{Q P_2} \right | $ and then find stationary conditions with respect to x and z. (Bit fuzzy about that part).

However, my immediate problem is that I am not sure how to work $\displaystyle n_1$ and $\displaystyle n_2$ into the problem.

Any thoughts.
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Old Aug 27th 2017, 04:56 AM   #2
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Does this help?
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Old Aug 27th 2017, 06:52 AM   #3
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The basic idea behind "Snell's law" is that light will follow the fastest path between two points. The idea here is to show that minimizing the time required gives the ratio of sines equal to the ratio of the speed of light in each medium.
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Old Aug 27th 2017, 12:35 PM   #4
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Thank you. Somehow I missed $\displaystyle n_1 = \frac{c}{v_1}$ and $\displaystyle n_2 = \frac{c}{v_2}$
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