Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.3 SOME EXTENSIONS

∆x

∆y

y(x)

h(x, y)=0

y(x)+η(x)

b

Figure 22.5 Variation of the end-pointbalong the curveh(x, y)=0.

that we require

[

η

∂F

∂y′

]b

a

+

∫b

a

[

∂F

∂y

−

d

dx

(

∂F

∂y′

)]

η(x)dx=0. (22.15)

Obviously the EL equation (22.5) must still hold for the second term on the LHS

to vanish. Also, since the lower end-point is fixed, i.e.η(a) = 0, the first term on

the LHS automatically vanishes at the lower limit. However, in order that it also

vanishes at the upper limit, we require in addition that

∂F

∂y′

∣

∣

∣

∣

x=b

=0. (22.16)

Clearly if both end-points may vary then∂F/∂y′must vanish at both ends.

An interesting and more general case is where the lower end-point is again

fixed atx=a, but the upper end-point is free to lie anywhere on the curve

h(x, y) = 0. Now in this case, the variation in the value ofIdue to the arbitrary

variation (22.2) is given to first order by

δI=

[

∂F

∂y′

η

]b

a

+

∫b

a

(

∂F

∂y

−

d

dx

∂F

∂y′

)

ηdx+F(b)∆x, (22.17)

where ∆xis the displacement in thex-direction of the upper end-point, as

indicated in figure 22.5, andF(b) is the value ofFatx=b. In order for (22.17)

to be valid, we of course require the displacement ∆xto be small.

From the figure we see that ∆y=η(b)+y′(b)∆x. Since the upper end-point

must lie onh(x, y) = 0 we also require that, atx=b,

∂h

∂x

∆x+

∂h

∂y

∆y=0,

which on substituting our expression for ∆yand rearranging becomes
(
∂h
∂x

+y′

∂h

∂y

)

∆x+

∂h

∂y

η=0. (22.18)