Editing Fermat's principle (section)

=== Relation to Hamilton's principle ===

If {{math|''x''}}, {{math|''y''}}, {{math|''z''}} are Cartesian coordinates and an overdot denotes differentiation with respect to {{mvar|s}}, Fermat's principle '''(2)''' may be written<ref>Cf. [[#Chaves16|Chaves, 2016]], pp.{{nnbsp}}568–9.</ref>
<math display="block">\begin{align}
 \delta S
 &=\,\delta\int_A^B\!n_{\mathrm{r}}\,\sqrt{dx^2+dy^2+dz^2}\\
 &=\,\delta\int_A^B\!n_{\mathrm{r}}\,\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}~ds
 \,=\,0\,.
 \end{align}</math>
In the case of an isotropic medium, we may replace {{math|''n''<sub>r</sub>}} with the normal refractive index{{hsp}} {{math|''n''(''x'', ''y'', ''z'')}}, which is simply a [[scalar field]]. If we then define the ''optical [[Lagrangian mechanics|Lagrangian]]''<ref>[[#Chaves16|Chaves, 2016]], p.{{nnbsp}}581.</ref> as
<math display="block">L(x,y,z,\dot{x},\dot{y},\dot{z}) \,=\, n(x,y,z)\,\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}\,,</math>
Fermat's principle becomes<ref>[[#Chaves16|Chaves, 2016]], p.{{nnbsp}}569.</ref>
<math display="block">\delta S =\, \delta\int_A^B\!L(x,y,z,\dot{x},\dot{y},\dot{z})\,ds\,=\,0\,.</math>
If the direction of propagation is always such that we can use {{mvar|z}} instead of {{mvar|s}} as the parameter of the path (and the overdot to denote differentiation w.r.t.&nbsp;{{mvar|z}} instead of {{mvar|s}}), the optical Lagrangian can instead be written<ref>Cf. [[#Chaves16|Chaves, 2016]], p.{{nnbsp}}577.</ref>
<math display="block">L\big(x(z),y(z),\dot{x}(z),\dot{y}(z),z\big) = n(x,y,z)\,\sqrt{1+\dot{x}^2+\dot{y}^2}\,,</math>
so that Fermat's principle becomes
<math display="block">\delta S =\, \delta\int_A^B\!  L\big(x(z),y(z),\dot{x}(z),\dot{y}(z),z\big)\,dz\,=\,0.</math>
This has the form of [[Hamilton's principle]] in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics.<ref>Cf.{{tsp}} [[#BW|Born & Wolf, 2002]], pp.{{nnbsp}}853–4,{{px2}}868; [[#Chaves16|Chaves, 2016]], p.{{nnbsp}}669.</ref> The optical Lagrangian is the function which, when integrated w.r.t.&nbsp;the parameter of the path, yields the OPL; it is the foundation of [[Hamiltonian optics|Lagrangian and Hamiltonian optics]].<ref>[[#Chaves16|Chaves, 2016]], ch.{{nnbsp}}14.</ref>