Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.3 SOME EXTENSIONS

(a) (b) (c)

b

−b

z

ρ

a

Figure 22.4 Possible soap films between two parallel circular rings.

surface area betweenzandz+dzis

dS=2πρ

[

(dz)^2 +(dρ)^2

] 1 / 2

,

so the total surface area is given by

S=2π

∫b

−b

ρ(1 +ρ′^2 )^1 /^2 dz. (22.11)

Since the integrand does not containzexplicitly, we can use (22.8) to obtain an equation
forρthat minimisesS,i.e.

ρ(1 +ρ′^2 )^1 /^2 −ρρ′^2 (1 +ρ′^2 )−^1 /^2 =k,

wherekis a constant. Multiplying through by (1 +ρ′^2 )^1 /^2 , rearranging to find an explicit
expression forρ′and integrating we find

cosh−^1

ρ

k

=

z

k

+c.

wherecis the constant of integration. Using the boundary conditionsρ(±b)=a,we
requirec=0andksuch thata/k=coshb/k(ifb/aistoolarge,nosuchkcan be found).
Thus the curve that minimises the surface area is

ρ/k=cosh(z/k),

and in profile the soap film is a catenary (see section 22.4) with the minimum distance
from the axis equal tok.

22.3 Some extensions

It is quite possible to relax many of the restrictions we have imposed so far. For

example, we can allow end-points that are constrained to lie on given curves rather

than being fixed, or we can consider problems with several dependent and/or

independent variables or higher-order derivatives of the dependent variable. Each

of these extensions is now discussed.