Variational Analysis of Shape Equations for Axisymmetric Vesicles

In derivation of the shape equation for axisymmetric vesicles from the Helfrich free energy the variation must be taken with respect to the contour. It is pointed out that the widely used Helfrich shape equation and the one proposed by Peterson were derived by improper variations, hence both equations are generally incorrect. The validity of the two equations is analyzed.