Chapter 5

(a) Write down the integral which must be extremized, according to Fermat's principle, if the light paths are not restricted to plane curves, and with \(u=u(x, y, z)\). Let \(x\) be the independent variable. (b) Write down the pair of Euler-Lagrange equations (again with \(x\) as independent variable) which describe light paths in three dimensions if \(u=u(x, y, z)\).

Describe the plane paths of light in the (two-dimensional) media in which the light velocities are given respectively by (i) \(u=a y\); (ii) \((a / y)\); (iii) \(a y^{\frac{4}{4}}\); (iv) \(a y^{-\frac{1}{2}}\); where \(a>0, y>0\).