The geometry of soap films and soap bubbles

If we dip two wire rings into a solution of soapy water, then what is the surface formed by the soap film ? This question is known as the *minimal surface problem*, or the Plateau problem. This post proposes to discuss a simple solution based on variational calculus and elementary computations.

**SOAP FILM**

Consider a soap film between two coaxial circular rings of the same radius . We denote by the mono-dimensional coordinate of the central axis. The circular rings are positioned at and respectively. By symmetry arguments, neglecting the gravity, the film corresponds to a surface of revolution along the axis with radius . Clearly, we have . The length element along a meridian is , which corresponds to a displacement along both and coordinates.

**MINIMAL SURFACE**

The free energy of the soap film is where is the film tension and is the area of the film. At the thermodynamic equilibrium, the free energy is minimized. Thus, the minimization of the energy is equivalent to the minimization of the surface. The area element generates the surface as the integral

**VARIATIONAL CALCULUS**

In elementary calculus, the derivative of a function at an extremum is zero. Similarly, in variational calculus, the variation of a functional at an extremum is zero, namely

The quantity represents a small variation of the function , which must satisfy the boundary conditions

Note that the condition is necessary but not sufficient to minimize the surface, since an extremum is not always a minimum. We compute

An integration by parts on the second integral allows to use instead of

The term in square bracket cancels due to the boundary conditions, hence

The term in brace bracket cancels since is arbitrary, hence

The derivative is cumbersome but straightforward to compute, we arrive at the expression

**SOLVING**

The differential equation can be solved in two steps of integration.

First, we proceed with basic calculus, using

Second, we proceed with hyperbolic trigonometry

In particular, at we have

Since is an even function we have either or . The first case would imply , which is inconsistent. The second case implies . Thus, the final solution is

**ANALYSIS**

We analyze the solution when . This implies , which is plotted as follows

There is a minimum (positive) radius, which can be approximated with Mathematica

FindMinimum[K*Cosh[1/K], {K, 1}] {1.50888, {K -> 0.833557}}

This means that the radius has a critical value . There are two solutions above this value, and no solution below. For example, if we choose then we solve the equation for

NSolve[K*Cosh[1/K] == 2, K, Reals] {{K -> 0.47019}, {K -> 1.69668}}

We obtain a stable catenoid solution for , which is a minimal surface

and an unstable catenoid solution for , which is not a minimal surface (the soap film will pop)

**REFERENCES**

- Dierkes et al., Minimal Surfaces, Springer, 2010
- Osserman, A Survey of Minimal Surfaces, Dover Publications, 2014