Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

θ 1

θ 2

n 1

n 2

A

B

x

y

Figure 22.8 Path of a light ray at the plane interface between media with

refractive indicesn 1 andn 2 ,wheren 2 <n 1.

From Fermat’s principle deduce Snell’s law of refraction at an interface.

Let the interface be aty= constant (see figure 22.8) and let it separate two regions with
refractive indicesn 1 andn 2 respectively. On a ray the element of physical path length is
ds=(1+y′^2 )^1 /^2 dx, and so for a ray that passes through the pointsAandB,thetotal
optical path length is

P=

∫B

A

n(y)(1 +y′^2 )^1 /^2 dx.

Since the integrand does not contain the independent variablexexplicitly, we use (22.8)
to obtain a first integral, which,after some rearrangement, reads

n(y)

(

1+y′^2

)− 1 / 2

=k,

wherekis a constant. Recalling thaty′is the tangent of the angleφbetween the
instantaneous direction of the ray and thex-axis, thisgeneralresult, which is not dependent
on the configuration presently under consideration, can be put in the form

ncosφ=constant

along a ray, even thoughnandφvary individually.
For our particular configurationnis constant in each medium and therefore so is
y′. Thus the rays travel in straight lines in each medium (as anticipated in figure 22.8,
but not assumed in our analysis), and sincekis constant along thewholepath we have
n 1 cosφ 1 =n 2 cosφ 2 , or in terms of the conventional angles in the figure

n 1 sinθ 1 =n 2 sinθ 2 .

22.5.2 Hamilton’s principle in mechanics

Consider a mechanical system whose configuration can be uniquely defined by a

number of coordinatesqi(usually distances and angles) together with timetand

which experiences only forces derivable from a potential. Hamilton’s principle