The Equivalence of Fermat's Least Time with a Hypothetical Velocity

In this note we try to show directly that Fermat’s minimum time principle is equivalent to a hypothetical velocity approach where this velocity acts as a hypotenuse to the actual angled directions of light. We consider in particular problems for which there are two rays of light with possibly different speeds (at angles) and a medium with a surface of length L along the x axis. This approach applies to a fixed mirror, a mirror moving at constant v in the y direction and Snell’s law. A  goal is to find the point x on the medium surface at which the first ray hits. The distance traveled by the first ray is cta and that by the second ray ctb where t is time. For one to have a minimum of T=ta+tb:  dT/dx = 0 = dta/dx + dtb/dx.  We argue that dta/dx is of the form of a reciprocal velocity 1/H and that dtb/dx=(-1)1/H. The -1 for the tb case follows from the appearance of  d/dx (L-x) = -1 whereas d/dx x = 1.

H is a hypothetical velocity and so one may solve such problems in terms of this H instead of minimizing time which requires taking a derivative. In this note, we try to show that such a velocity approach is equivalent to a minimized time one.