View source for Hamilton's principle/Applications/Fermat's principle

'''Fermat's principle''' states that the path taken by light between two points
is stationary with respect to time. It is the oldest surviving variational
principle in physics, it precedes the calculus of variations by nearly a century,
and it is the historical bridge to mechanics: the analogy Hamilton exploited runs
from this principle to his own.

== Statement ==

[[File:OPL-HP-F-08.png|thumb|right|400px|Snell's law from stationary time.]]

In a medium of refractive index <math>n(x,y,z)</math>, light travels at speed
<math>c/n</math>. The transit time along a path is

:<math>T = \int \frac{ds}{v} = \frac{1}{c}\int n(x,y,z)\; ds
\qquad\qquad\mathrm{(1)}</math>

The quantity <math>\int n\,ds</math> is the '''optical path length'''. Fermat's
principle is that light follows a path for which (1) is '''stationary'''.

Fermat himself said ''least'', and for the cases he treated the two coincide. The
correction to ''stationary'' is required, and the failures are exhibited at the
end of this page.

== Snell's law ==

Take two media, indices <math>n_1</math> above and <math>n_2</math> below a flat
interface. Light goes from <math>A = (0, a)</math> to <math>B = (d, -b)</math>
with <math>a, b > 0</math>, crossing at <math>(x, 0)</math>. Straight segments
within each medium — the index is constant there, so each leg is a straight line
— give the time as a function of the single unknown <math>x</math>:

:<math>cT(x) = n_1\sqrt{a^2 + x^2} + n_2\sqrt{b^2 + (d-x)^2}
\qquad\qquad\mathrm{(2)}</math>

This is ordinary calculus, not the calculus of variations: the variational
problem has collapsed to one variable because the path shape within each medium
is already known. Differentiate:

:<math>c\,\frac{dT}{dx} = \frac{n_1 x}{\sqrt{a^2+x^2}}
- \frac{n_2 (d-x)}{\sqrt{b^2+(d-x)^2}}
\qquad\qquad\mathrm{(3)}</math>

The two fractions are the sines of the angles from the normal, so setting (3) to
zero gives

:<math>n_1 \sin\theta_1 = n_2 \sin\theta_2
\qquad\qquad\mathrm{(4)}</math>

'''Snell's law''', from a stationarity condition in three lines.

The second derivative of (2) is positive for all <math>x</math>, so this
stationary point is a genuine minimum — refraction at a flat interface really is
least time. That will not survive curved interfaces.

The law of reflection follows the same way with <math>n_1 = n_2</math> and both
points on the same side, giving equal angles.

== The continuous case ==

For a medium with varying index the problem is a genuine variational one. With
the path written <math>y(x)</math> in the plane,

:<math>J[y] = \int n(x,y)\,\sqrt{1 + y'^2}\; dx
\qquad\qquad\mathrm{(5)}</math>

Two special cases are worth having.

'''Stratified medium, <math>n = n(y)</math>.''' The integrand carries no explicit
<math>x</math>, so the Beltrami identity of
[[Hamilton's principle/Mathematics/The Beltrami identity]] applies. The same
collapse as in
[[Hamilton's principle/Applications/The brachistochrone]] gives

:<math>\frac{n(y)}{\sqrt{1+y'^2}} = \text{const}
\qquad\qquad\mathrm{(6)}</math>

Since <math>1/\sqrt{1+y'^2} = \sin\theta</math> with <math>\theta</math> measured
from the vertical, (6) reads <math>n\sin\theta = \text{const}</math> — Snell's
law again, now holding continuously along the ray rather than at an interface.
The discrete law is the first integral of the continuous problem.

This is the mirage. Hot air near a road surface has lower index; a ray descending
into it bends until <math>\sin\theta = 1</math> and turns back up, and the
observer sees sky below the horizon.

'''Constant index.''' Then (5) is arc length and the rays are straight lines, the
case worked on
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]].

== Rays are geodesics ==

Equation (5) is the arc-length functional of
[[Hamilton's principle/Applications/Geodesics]] with the metric conformally
rescaled by <math>n^2</math>:

:<math>d\tilde s^2 = n^2(q)\,\bigl( dx^2 + dy^2 + dz^2 \bigr)
\qquad\qquad\mathrm{(7)}</math>

So '''light rays are geodesics of the optical metric''', and every result about
geodesics transfers. In particular the geodesic equation (6) of that page
supplies the ray equation, and conjugate points supply the foci.

Compare this with Jacobi's form of Maupertuis' principle on
[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]],
where a particle of energy <math>E</math> in potential <math>V</math> follows a
geodesic of the metric rescaled by <math>2(E - V)</math>. The two statements are
the same statement under

:<math>n(q) \;\longleftrightarrow\; \sqrt{2\bigl(E - V(q)\bigr)}
\qquad\qquad\mathrm{(8)}</math>

That correspondence is the optical-mechanical analogy, and it is developed on
[[Hamilton's principle/Applications/The optical-mechanical analogy]].

== Stationary, not least ==

Three standard cases show that ''least'' fails, and none of them is contrived.

'''A concave mirror at its focus.''' Place source and observer at the two foci of
an ellipsoidal mirror. Every path via the surface has the '''same''' optical
length. The time is stationary and constant — neither minimum nor maximum, and
not isolated.

'''A mirror curved more sharply than the ellipse.''' Replace the ellipsoidal
surface by one curving away more sharply through the same reflection point. The
reflected path now has '''greater''' optical length than neighbouring paths: the
actual light path is a local '''maximum''' of time. The reflection law (4) still
holds, and the light still goes that way.

'''Beyond a focus.''' A ray passing through a focus and continuing is stationary
but no longer minimal, for exactly the reason a great-circle arc past the antipode
is not shortest. The focus is a conjugate point of
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]],
and the optical and mechanical statements are the same statement under (7).

'''This is the cleanest place in the volume to see why the correction to
"stationary" is forced.''' A maximum-time path is not an edge case of least time;
it is the opposite, and the principle as Fermat stated it is simply false there.

== Historical note ==

Fermat proposed the principle in the 1650s and defended it against Descartes'
derivation of the refraction law from an assumption about light travelling
''faster'' in denser media. Fermat's principle requires the opposite, and the
question was settled experimentally by Foucault in 1850, two centuries later, in
Fermat's favour.

The metaphysical framing — that nature acts by the shortest paths — was contested
at the time and is the same framing Maupertuis would use for mechanics ninety
years later. Both principles survived; both framings did not. The corrected
statement is stationarity, and it carries no claim about nature's economy.

<blockquote>
Provenance: the historical paragraph is editorial summary. The Fermat row of the
survey on [[Hamilton's principle/Sources]] is UNCOMPUTED — no full text of the
''Synthesis ad refractiones'' has been located for this volume — so no words are
attributed to Fermat here, and every equation on this page is derived on it.
</blockquote>

== Next ==

[[Hamilton's principle/Applications/The optical-mechanical analogy]] makes the
correspondence (8) the subject rather than a remark.

== References ==

<references />