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etotheipi
Take a medium with ##n = n(y)## and define the functional as usual$$T[y] = \int_{\mathcal{C}} \frac{ds}{cn^{-1}} = \frac{1}{c} \int_{x_1}^{x_2} dx \, n(y)\sqrt{1+ y'^2}$$along ##\mathcal{C}## between ##\mathcal{P}_1 \overset{.}{=} (x_1, y_1)## and ##\mathcal{P}_2 \overset{.}{=} (x_2, y_2)##. Because the integrand does not depend explicitly on ##x##, Beltrami identity gives$$ \frac{y'^2 n(y)}{\sqrt{1+y'^2}} - n(y) \sqrt{1+y'^2} = \frac{n(y)}{\sqrt{1+y'^2}} = K \in \mathbb{R}$$Consider a ray entering the medium horizontally. When ##n(y)## is not a constant function, the trajectory ##y = C## where ##C## is a constant will not make the functional extremal. Instead you must solve ##y' = \sqrt{\left(\frac{n(y)}{K}\right)^2 - 1}## with the IC ##(x_1, y_1) = (0,C)## or something, which gives the nice bendy trajectory.
Physically [without appealing to variational calculus] if a ray enters such a medium horizontally, what causes the ray to bend away from a possible horizontal path ##y=C## with constant refractive index ##n(C)## [which would, at first glance, seem like the more reasonable path]? How does EM theory describe the same behaviour, for instance?
Physically [without appealing to variational calculus] if a ray enters such a medium horizontally, what causes the ray to bend away from a possible horizontal path ##y=C## with constant refractive index ##n(C)## [which would, at first glance, seem like the more reasonable path]? How does EM theory describe the same behaviour, for instance?
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