Editing Fermat's principle (section)

=== Analogies ===

According to the "strong" form of Fermat's principle, the problem of finding the path of a light ray from point ''A'' in a medium of faster propagation, to point ''B'' in a medium of slower propagation ([[#Fig1|Fig.{{nnbsp}}1]]), is analogous to the problem faced by a [[lifeguard]] in deciding where to enter the water in order to reach a drowning swimmer as soon as possible, given that the lifeguard can run faster than (s)he can swim.{{r|feynman-1988-51}} But that analogy falls short of ''explaining'' the behavior of the light, because the lifeguard can think about the problem (even if only for an instant) whereas the light presumably cannot. The discovery that ants are capable of similar calculations{{r|zyga-2013}} does not bridge the gap between the animate and the inanimate.

In contrast, the above assumptions (1) to (3) hold for any wavelike disturbance and explain Fermat's principle in purely [[mechanical philosophy|mechanistic]] terms, without any imputation of knowledge or purpose.

The principle applies to waves in general, including (e.g.) sound waves in fluids and elastic waves in solids.<ref>[[#deWitte|De Witte, 1959]], p.{{nnbsp}}294.</ref> In a modified form, it even works for [[matter&nbsp;wave]]s: in [[quantum mechanics]], the [[classical limit|classical path]] of a particle is obtainable by applying Fermat's principle to the associated wave &ndash; except that, because the frequency may vary with the path, the stationarity is in the [[phase&nbsp;shift]] (or number of cycles) and not necessarily in the time.{{r|ogborn-taylor-2005|vanHouten-beenakker-1995-272}}

Fermat's principle is most familiar, however, in the case of visible [[light]]: it is the link between [[geometrical optics]], which describes certain optical phenomena in terms of ''rays'', and the [[wave theory of light]], which explains the same phenomena on the hypothesis that light consists of ''waves''.