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A soap film is formed between two parallel rings of radius ... separated by a distance ... . To minimize the surface-tension energy of the soap film, its total area ... seeks a minimum value. The derivation of the shape of the film involves a problem in the calculus of variations. Let ... represent the functional form of the film in cylindrical coordinates. The area is then given by ... . The integrand ... is determined by the Euler–Lagrange equation ... , which can be reduced to its first integral ... , a constant. The solution works out to ... , a catenary of revolution, with the boundary condition ... . When ... , the film collapses to disks within the two rings.
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