Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

A (a, y(a))

dx

dy

ds

B (b, y(b))

Figure 22.2 An arbitrary path between two fixed points.

Show that the shortest curve joining two points is a straight line.

Let the two points be labelledAandBand have coordinates (a, y(a)) and (b, y(b))
respectively (see figure 22.2). Whatever the shape of the curve joiningAtoB, the length
of an element of pathdsis given by

ds=

[

(dx)^2 +(dy)^2

] 1 / 2

=(1+y′^2 )^1 /^2 dx,

and hence the total path length along the curve is given by

L=

∫b

a

(1 +y′^2 )^1 /^2 dx. (22.7)

We must now apply the results of the previous section to determine that path which makes
Lstationary (clearly a minimum in this case). Since the integral does not containy(or
indeedx) explicitly, we may use (22.6) to obtain

k=

∂F

∂y′

=

y′

(1 +y′^2 )^1 /^2

.

wherekis a constant. This is easily rearranged and integrated to give

y=

k

(1−k^2 )^1 /^2

x+c,

which, as expected, is the equation of a straight line in the formy=mx+c,with
m=k/(1−k^2 )^1 /^2. The value ofm(ork) can be found by demanding that the straight line
passes through the pointsAandBand is given bym=[y(b)−y(a)]/(b−a). Substituting
the equation of the straight line into (22.7) we find that, again as expected, the total path
length is given by

L^2 =[y(b)−y(a)]^2 +(b−a)^2 .