Hi, please help me with this question, I am completely lost. I am taking this integral quantity as my action 'L' and trying to minimize this function. However when I set n(y,x) --> n(y), and write the Lagrangian relation d/dt(dL/dy') = dL/dy (1) d/dt(dL/dx') = dL/dx (2) nothing useful pops out of this, I don't know, someone please lead me in the direction to showing that path y(x) satisfies the relationship in a). Also note that the relation it satisfies can also be thought of as n(y) * dx/ds = constant, if that helps any bit..
With $ds=\sqrt{1+[y'(x)]^2}dx$, the Lagrangian is $L(x,y,y')=n(y)\sqrt{1+[y']^2}$. Then use Beltrami identity to get
\begin{align} \mathrm{constant}&=L-y'\frac{\partial L}{\partial y'}\\ &=n(y)\sqrt{1+[y']^2}-\frac{[y']^2n(y)}{\sqrt{1+[y']^2}}\\ &=\frac{n(y)(1+[y']^2)}{\sqrt{1+[y']^2}}-\frac{[y']^2n(y)}{\sqrt{1+[y']^2}}\\ &=\frac{n(y)}{\sqrt{1+[y']^2}} \end{align}