Fig. 1

Illustrated in Fig. 2 are two circular hoops of unit radius, centered on a common x-axis and a distance apart. There is also a soap films extends between the two hoops, taking the form of a surface of revolution about the x-axis. If gravity is negligible, the film takes up a state of stable, equilibrium in which its surface area is a minimum.

Fig. 2

Our problem is to find the function , satisfying the boundary conditions

which makes the surface area

a minimum.

Let

We have

and

The Euler-Lagrange equation

becomes

Fig. 3

Using Omega CAS Explorer (see Fig. 3), it can be simplified to:

This is the differential equation solved in “A Relentless Pursuit” whose solution is

We must then find and subject to the boundary condition (1), i.e.,

The fact that is an even function implies either

or

While (3) is clearly false since it claims for all , (4) gives

And so, the solution to boundary-value problem

is

To determine , we deduce the following equation from the boundary conditions that at

This is a transcendental equation for that can not be solved *explicitly*. Nonetheless, we can examine it qualitatively.

Let

and express (7) as

Fig. 4

A plot of (8)’s two sides in Fig. 4 shows that for sufficient small , the curves and will intersect. However, as increases, , a line whose slope is rotates clockwise towards -axis. The curves will not intersect if is too large. The critical case is when , the curves touch at a single point, so that

and is the tangent line of i.e.,

Dividing (9) by (10) yields

What the mathematical model (5) predicts then is, as we gradually move the rings apart, the soap film breaks when the distance between the two rings reaches , and for , there is no more soap film surface connects the two rings. This is confirmed by an experiment (see Fig. 1).

We compute the value of , the maximum value of that supports a minimum area soap film surface as follows.

Fig. 5

Solving (11) for *numerically *(see Fig. 5), we obtain

By (10), the corresponding value of

.

We also compute the surface area of the soap film from (2) and (6) (see Fig. 6). Namely,

Fig. 6

*Exercise-1 *Given , solve (7) numerically for

*Exercise-2* Without using a CAS, find the surface area of the soap film from (2) and (6).