Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.2 SPECIAL CASES

dx

y ds dy

x

Figure 22.3 A convex closed curve that is symmetrical about thex-axis.

22.2.2Fdoes not containxexplicitly

In this case, multiplying the EL equation (22.5) byy′and using

d

dx

(

y′

∂F

∂y′

)

=y′

d

dx

(

∂F

∂y′

)

+y′′

∂F

∂y′

we obtain

y′

∂F

∂y

+y′′

∂F

∂y′

=

d

dx

(

y′

∂F

∂y′

)

.

But sinceFis a function ofyandy′only, and not explicitly ofx,theLHSof

this equation is just the total derivative ofF, namelydF/dx. Hence, integrating

we obtain

F−y′

∂F

∂y′

= constant. (22.8)

Find the closed convex curve of lengthlthat encloses the greatest possible area.

Without any loss of generality we can assume that the curve passes through the origin
and can further suppose that it is symmetric with respect to thex-axis; this assumption
is not essential. Using the distancesalong the curve, measured from the origin, as the
independent variable andyas the dependent one, we have the boundary conditions
y(0) =y(l/2) = 0. The element of area shown in figure 22.3 is then given by

dA=ydx=y

[

(ds)^2 −(dy)^2

] 1 / 2

,

and the total area by

A=2

∫l/ 2

0

y(1−y′^2 )^1 /^2 ds; (22.9)

herey′stands fordy/dsrather thandy/dx. Since the integrand does not containsexplicitly,