In-situ observation of a soap film catenoid - a simple educational physics experiment

The solution to the Euler-Lagrange equation is an extremal functional.To understand that the functional is stationary at local extrema (maxima or minima), we propose a physics experiment that involves using soap film to form a catenoid. A catenoid is a surface that is formed between two coaxial circular rings and is classified mathematically as a minimal surface.Using soap film, we create catenoids between two rings and characterize the catenoid in-situ while varying distance between rings. The shape of the soap film is very interesting and can be explained using dynamic mechanics. By observing catenoid, physics students can observe local extrema phenomena. We stress that in-situ observation of soap film catenoids is an appropriate physics experiment that combines theory and experimentation.

💡 Research Summary

The paper presents a low‑cost, hands‑on physics experiment that makes the abstract concepts of the Euler‑Lagrange variational principle and minimal surfaces tangible for students. By exploiting the property of soap films to minimize surface tension, the authors generate a catenoid—a classic minimal surface—between two coaxial circular rings and study its shape and stability as the ring separation (h) is varied.

Theoretical background. Assuming axial symmetry, the surface area of a surface of revolution can be written as a functional (S=\int 2\pi r\sqrt{1+r’^2},dz). Treating the integrand as a Lagrangian (L(r,r’)) and applying the Euler‑Lagrange equation yields the well‑known catenoid profile (r(z)=a\cosh(z/a)), where (a) is the neck radius. The geometric parameters satisfy (h=2a\cosh^{-1}(R/a)) (Eq. 3), linking the ring distance (h), the ring radius (R), and the neck radius (a). Two solutions exist for a given (h<R,1.33): a “thick‑neck” and a “thin‑neck” catenoid. Their surface areas are given by Eq. 4, and a comparison shows that the thick‑neck catenoid always has the smaller area, making it the absolute minimum and dynamically stable, while the thin‑neck catenoid is only a local minimum (unstable). The critical separation at which no catenoid can exist is (h_c\approx1.33,R). The area of two flat disks (the soap film attached directly to each ring) equals the catenoid area at (h_0\approx1.05,R).

Experimental setup. The authors mount two identical rings on the jaws of a slide caliper, introduce soap solution when the rings touch ((h=0)), and then pull a string attached to the movable jaw to increase (h) while recording the film with a high‑speed camera (120 fps). The apparatus allows simultaneous measurement of (h) and the neck radius (a).

Observations. As (h) grows from zero, the film initially follows the thick‑neck catenoid curve predicted by theory. Contrary to the simple energy‑minimum picture, the transition from the stable catenoid to the two‑disk configuration does not occur at (h_0). Instead, the film remains on the thick‑neck branch up to a value close to (h_c). Just before the critical distance, a fleeting thin‑neck (unstable) catenoid appears; the authors capture this moment in a high‑speed frame. The unstable catenoid then collapses, and the film snaps into two separate disks attached to each ring. The full sequence can be summarized as: (i) stable thick‑neck catenoid (absolute minimum), (ii) stable thick‑neck catenoid that has become a local minimum, (iii) unstable thin‑neck catenoid (local maximum), (iv) rupture, (v) two disks (absolute minimum).

Dynamic interpretation. To explain the observed pathway, the authors introduce a one‑dimensional “surface‑energy potential” (V(x)) with (x) representing the neck radius. (V(x)) possesses three extrema: a global minimum at (x=R) (initial film), a local minimum corresponding to the thick‑neck catenoid, and a local maximum corresponding to the thin‑neck catenoid. As (h) increases, the global minimum rises, the local minimum deepens, and a barrier separates the two minima. When the barrier is lowered enough (near (h_c)), thermal or mechanical perturbations allow the system to cross the hill (the thin‑neck state) and roll down to the global minimum at (x=0) (the two disks). This picture emphasizes that the transition is governed not merely by static area comparison but by the shape of the energy landscape and the ability to overcome the intervening barrier.

Limitations and practical notes. The experiment does not control soap concentration or pulling speed precisely; however, the authors kept the pulling speed slow enough to avoid dynamic effects that could mask the quasi‑static behavior. They acknowledge that variations in surfactant concentration would alter surface tension and thus the quantitative values of (h_c) and (h_0), but the qualitative sequence of extrema remains unchanged.

Educational significance. By integrating differential geometry, calculus of variations, and dynamical systems into a single, inexpensive demonstration, the experiment provides students with a concrete visualization of functional extrema, stability analysis, and energy‑landscape concepts. The authors argue that this approach can be readily incorporated into undergraduate physics curricula to bridge theory and observation.

In summary, the paper successfully combines (1) a clear derivation of the catenoid as a minimal surface, (2) a simple yet effective experimental apparatus, (3) high‑speed video documentation of the full transition pathway, and (4) a dynamic potential‑energy interpretation that accounts for the observed hysteresis and sudden collapse. The work demonstrates that soap‑film catenoids are an excellent pedagogical tool for exploring local extrema, stability, and the interplay between geometry and physics.