Suppose we have two circular hoops of unit radius, centered on a common x-axis and a distance apart. Suppose too, that a soap films extends between the two hoops, taking the form of a surface of revolution about the x-axis (see Fig. 2). Then if gravity is negligible the film takes up a state of stable, equilibrium in which its surface area is a minimum.
Our problem is to find the function , satisfying the boundary conditions
which makes the surface area
The Euler-Lagrange equation
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
While (3) is clearly false since it claims for all , (4) gives
And so, the solution to boundary-value problem
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.
and express (7) as
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.
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,
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).