Editing Calculus of variations (section)

==== Snell's law ====
There is a discontinuity of the refractive index when light enters or leaves a lens. Let
<math display="block">n(x,y) = \begin{cases}
n_{(-)} & \text{if} \quad x<0, \\
n_{(+)} & \text{if} \quad x>0,
\end{cases}</math>
where <math>n_{(-)}</math> and <math>n_{(+)}</math> are constants. Then the Euler–Lagrange equation holds as before in the region where <math>x < 0</math> or <math>x > 0,</math> and in fact the path is a straight line there, since the refractive index is constant. At the <math>x = 0,</math> <math>f</math> must be continuous, but <math>f'</math> may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form
<math display="block">\delta A[f_0,f_1] = f_1(0)\left[ n_{(-)}\frac{f_0'(0^-)}{\sqrt{1 + f_0'(0^-)^2}} - n_{(+)}\frac{f_0'(0^+)}{\sqrt{1 + f_0'(0^+)^2}} \right].</math>

The factor multiplying <math>n_{(-)}</math> is the sine of angle of the incident ray with the <math>x</math> axis, and the factor multiplying <math>n_{(+)}</math> is the sine of angle of the refracted ray with the <math>x</math> axis.  [[Snell's law]] for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.