  Membranes are an important subject of study in physical chemistry and
biology. They can be considered as material surfaces with a surface energy
depending on the curvature tensor. Usually, mathematical models developed in
the literature consider the dependence of surface energy only on mean curvature
with an added linear term for Gauss curvature. Therefore, for closed surfaces
the Gauss curvature term can be eliminated because of the Gauss-Bonnet theorem.
In [18], the dependence on the mean and Gaussian curvatures was considered in
statics. The authors derived the shape equation as well as two scalar boundary
conditions on the contact line. In this paper-thanks to the principle of
virtual working-the equations of motion and boundary conditions governing the
fluid membranes subject to general dynamical bending are derived. We obtain the
dynamic 'shape equa-tion' (equation for the membrane surface) and the dynamic
conditions on the contact line generalizing the classical Young-Dupr{\'e}
condition.
