Physics Help Forum snell's law and variational calculus

 Aug 26th 2017, 07:03 PM #1 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 518 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. Attached Thumbnails
 Aug 27th 2017, 04:56 AM #2 Senior Member   Join Date: Apr 2015 Location: Somerset, England Posts: 684 Does this help? Attached Thumbnails
 Aug 27th 2017, 06:52 AM #3 Senior Member   Join Date: Aug 2010 Posts: 232 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.
 Aug 27th 2017, 12:35 PM #4 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 518 Thank you. Somehow I missed $\displaystyle n_1 = \frac{c}{v_1}$ and $\displaystyle n_2 = \frac{c}{v_2}$

 Tags calculus, law, snell, variational

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