In this paper, we calculate the frequency of small oscillations of a ring confined to a plane that, along with a fixed ring of the same size, supports a soap film catenoid. The restoring force is provided by the film's surface tension. We assume that the soap film is massless and continuously assumes the shape of minimal area. Mathematically, the problem is equivalent to calculating the second derivative of the total area with respect to the displacement of the ring. The calculus of moving surfaces is extensively used in the presented calculation.
Pavel Grinfeld. "In-Plane Oscillations of a Ring Driven by a Soap Film Catenoid." J. Geom. Symmetry Phys. 20 35 - 48, 2010. https://doi.org/10.7546/jgsp-20-2010-35-48