Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.4 CONSTRAINED VARIATION

wherekis a constant. Lettinga=k^2 and solving fory′we find

y′=

dy

dx

=

√

a−y

y

,

which on substitutingy=asin^2 θintegrates to give

x=

a

2

(2θ−sin 2θ)+c.

Thus the parametric equations of the curve are given by

x=b(φ−sinφ)+c, y=b(1−cosφ),

whereb=a/2andφ=2θ; they define a cycloid, the curve traced out by a point on
the rim of a wheel of radiusbrolling along thex-axis. We must now use the end-point
conditions to determine the constantsbandc. Since the curve passes through the origin,
we see immediately thatc=0.Nowsincey(x 0 ) is arbitrary, i.e. the upper end-point can
lie anywhere on the curvex=x 0 , the condition (22.20) reduces to (22.16), so that we also
require

∂F

∂y′

∣

∣

∣∣

x=x 0

=

y′

√

y(1 +y′^2 )

∣∣

∣

∣

∣x=x

0

=0,

which implies thaty′=0atx=x 0. In words, the tangent to the cycloid atBmust be
parallel to thex-axis; this requiresπb=x 0 .

22.4 Constrained variation

Just as the problem of finding thestationary values of a functionf(x, y) subject to

the constraintg(x, y) = constant is solved by means of Lagrange’s undetermined

multipliers (see chapter 5), so the corresponding problem in the calculus of

variations is solved by an analogous method.

Suppose that we wish to find the stationary values of

I=

∫b

a

F(y, y′,x)dx,

subject to the constraint that the value of

J=

∫b

a

G(y, y′,x)dx

is held constant. Following the method of Lagrange undetermined multipliers let

us define a new functional

K=I+λJ=

∫b

a

(F+λG)dx,

and find itsunconstrainedstationary values. Repeating the analysis of section 22.1

we find that we require

∂F

∂y

−

d

dx

(

∂F

∂y′

)

+λ

[

∂G

∂y

−

d

dx

(

∂G

∂y′

)]

=0,