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# Rays and Wave Fronts

DOI link for Rays and Wave Fronts

Rays and Wave Fronts book

# Rays and Wave Fronts

DOI link for Rays and Wave Fronts

Rays and Wave Fronts book

## 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.