ABSTRACT
For the Lagrangian L = L(x1, x2, x3, x′ 1, x 2′ ) used in Equation 10.51, we can write
dL___ dx3
∂L___∂xk
x′ k+ ∂L___∂ x′ k d x′ k
___ dx3
+ ∂L___∂x3 (11.1)
Considering the Euler Equations 10.52, we get1
dL___ dx3
d___
dx3 (∂L___ ∂x′ k) xk′ +
∂L___∂ x′ k dxk′___ dx3
+ ∂L___∂x3 = d___
dx3 (∑ k=1
∂L____∂ x′ k
x′ k ) + ∂L___∂x3 (11.2)
and therefore,
d___ dx3
(L − ∑ k=1
∂L____∂ x′ k
x′ k ) = ∂L___∂x3 (11.3)
Considering Equation 10.49 and dx3/dx3 = x′ 3 = 1, we can now write
∂L___∂ x′ k
x′ k = n √
n x′ k2_____________
√ _____________
√ _____________
2 (11.4)
This relation can be written as
∂L___∂ x′ k x′ k
= n x′ 3______________
√ _____________
= ∂L____∂ x′ 3 (11.5)
Replacing it into Equation 11.3, we can fi nally write
d___ dx3(
∂L____∂ x′ 3) = ∂L___∂x3
(11.6)
Combining this equation with the Euler equations (Equation 10.52), we get
d___ dx3(
∂L____∂ x′ k) = ∂L___∂xk
(k = 1, 2, 3) (11.7)
Since pk = ∂L/∂ x′ k, we can also write
dpk___ dx3
= ∂L___∂xk (k = 1, 2, 3) (11.8)
equations in the form just mentioned, we can write
∂S___∂xk = ∂___∂xk
∫ Ldx3 = ∫ ∂L___∂xk dx3 = ∫ dpk___ dx3
dx3 = pk (11.9)
or pk = ∂S/∂xk, which can be written as
p = ∇S (11.10)
From this expression, it can be concluded that vector p is perpendicular to the surfaces S = constant, S being the optical path length. Since p is tangent to the rays of light, it can be concluded that the surfaces S = constant are perpendicular to the rays of light. Such surfaces are called wave fronts.
