The Fermat principle defines the rays. To any pair of points A, B in a connected region we can assign a ray as follows: given an arbitrary curve, C, starting from the initial point A and ending at the end point B, there belongs a "propagation time," T(C), defined by the formula:
T(C) = integral over the curve C (ds/v) eq 2.1
Fermat's principle states that this propagation time is a minimum for the actual path of propagation.
Ray: A curve connecting A with B for which:
Tc = minimum eq 2.1
Fronts: surfaces orthogonal to rays.
Remarks:
1) the rays defined here are the extremals of the variational problem 2.2 (see: variational calculus)
2) the admissible set of curves should satisfy some mathematical requirements (rectifiability) and must lie entirely in the admissible region.
3) Between a given pair of points it is possible there exists a unique ray, but it is also possible to have several different paths which produce the same minimum. This problem is related to singularity theory (V.I. Arnold; Singularities of Caustics and Wave Fronts; Kluwer Academic Publishers, 1990) and conflict sets (Dirk Siersma)
4) In the geometric theory, rays are primary, while fronts are derived as surfaces orthogonal to the rays, compared to the former structure where the wave function was given, the wave fronts were primary, and rays were defined as their orthogonal trajectories.
5) The eikonal defined here differs from the one used in optics [Max Born & Emil Wolf; Principles of Optics; Pergamon Press;Sixth (Corrected) Edition, 1980] g* with a constant factor c: g*=cg, where c is the light velocity in vacuum. Our eikonal has the time dimension while the eikonal in optics has the length dimension.
6) As stated in many books Fermat principle requires
that the propagation time must be stationary, that is the propagation time
for paths closely adjacent to the actual path differs from that of the
actual path by a second order term. This is the case in optics, where the
laws of reflection and refraction satisfy this requirement. There are known
examples when the propagation time is stationary or even maximun
for the actual path. E.g., for an elliptic mirror, if a point source is
in one focal point F1, the image point will be on the
other focal point F2, consequently, the propagation time
is equal for any ray starting from F1, reflectiong at the mirror
and reaching F2 /the propagation time is stationary/. If we
use a mirror having common tangent with the elliptic mirror at a point,
the actual ray reflected at this point is the same, but the adjacent rays
have greater time or lesser time according to the ratio of curvatures of
mirrors [Jenkins, F.A. & H.E. White; Fundamentals of Optics;
McGraw-Hill Publishing Co. 1951]. Notwithstanding this fact, we formulated
the Fermat principle as minimum principle. Really, for waves of excitations
as well as for chemical waves the minimum condition is necessarily satisfied.
One can see the importance of minimum for the case of praire fire
model: the fire propagates according to the minimum time requirement,
since after the fire reaches a point, all the other paths ending at that
point are irrelevant.