It is an experimental fact that light travels at different speeds, depending on the material it is travelling through - it's $3 \times 10^8$m/s only in a vacuum.

Fermat postulated that the path between two points taken by a light ray is the curve that minimises the total time it takes to travel between the two points. In optics, this is called the **principle of least time**.

We are supposing that the speed of light depends on position, $$c = c(x, y, z)$$

In principle, we can parametrize any curve in space, between two points $P, Q$ say, by the distance travelled along it, $s$, hence we can write $$\begin{align*}x &= x(s)\\ y &= y(s)\\ z &= z(s)\end{align*}$$

Also, note that we can also write the speed as $$\frac{\mathrm{d}s}{\mathrm{d}t} = c(x, y, z)$$ and so $$\mathrm{d}t = \frac{\mathrm{d}s}{c(x, y, z)}$$ which we integrate to get the total time it takes for a light ray to travel from $P$ to $Q$, $$T = \int_{P}^{Q} \frac{1}{c(x, y, z)} \mathrm{d}s$$

We would need to know what the function $c(x, y, z)$ actually is to proceed further, but we can still make the identifications for the Euler-Lagrange equations, $$\begin{align*}t & \to s\\ q_i & \to (x, y, z)\\ \dot{q_i} & \to (x', y', z')\\ A & \to T\\ L(q, \dot{q_i}) & \to I(x, y, z, x', y', z') = \frac{1}{c(x, y, z)}\end{align*}$$