Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

we can use (22.8) to obtain a first integral of the EL equation fory,namely

y(1−y′^2 )^1 /^2 +yy′^2 (1−y′^2 )−^1 /^2 =k,

wherekis a constant. On rearranging this gives

ky′=±(k^2 −y^2 )^1 /^2 ,

which, usingy(0) = 0, integrates to

y/k= sin(s/k). (22.10)

The other end-point,y(l/2) = 0, fixes the value ofkasl/(2π) to yield

y=

l

2 π

sin

2 πs

l

.

From this we obtaindy=cos(2πs/l)dsand since (ds)^2 =(dx)^2 +(dy)^2 we find also that
dx=±sin(2πs/l)ds. This in turn can be integrated and, usingx(0) = 0, givesxin terms
ofsas

x−

l

2 π

=−

l

2 π

cos

2 πs

l

.

We thus obtain the expected result thatxandylie on the circle of radiusl/(2π)givenby
(
x−

l

2 π

) 2

+y^2 =

l^2

4 π^2

.

Substituting the solution (22.10) into the expression for the total area (22.9), it is easily
verified thatA=l^2 /(4π). A much quicker derivation of this result is possible using plane
polar coordinates.

The previous two examples have been carried out in some detail, even though

the answers are more easily obtained in other ways, expressly so that the method

is transparent and the way in which it works can be filled in mentally at almost

every step. The next example, however, does not have such an intuitively obvious

solution.

Two rings, each of radiusa, are placed parallel with their centres 2 bapart and on a

common normal. An open-ended axially symmetric soap film is formed between them (see

figure 22.4). Find the shape assumed by the film.

Creating the soap film requires an energyγper unit area (numerically equal to the surface
tension of the soap solution). So the stable shape of the soap film, i.e. the one that
minimises the energy, will also be the one that minimises the surface area (neglecting
gravitational effects).
It is obvious that any convex surface, shaped such as that shown as the broken line in
figure 22.4(a), cannot be a minimum but it is not clear whether some shape intermediate
between the cylinder shown by solid lines in (a), with area 4πab(or twice this for the
double surface of the film), and the form shown in (b), with area approximately 2πa^2 , will
produce a lower total area than both of these extremes. If there is such a shape (e.g. that in
figure 22.4(c)), then it will be that which is the best compromise between two requirements,
the need to minimise the ring-to-ring distance measured on the film surface (a) and the
need to minimise the average waist measurement of the surface (b).
We take cylindrical polar coordinates asin figure 22.4(c) and let the radius of the soap
film at heightzbeρ(z)withρ(±b)=a. Counting only one side of the film, the element of