  Geometric continuum models for fluid lipid membranes are considered using
classical field theory, within a covariant variational approach. The approach
is cast as a higher-derivative Lagrangian formulation of continuum classical
field theory, and it can be seen as a covariant version of the field
theoretical variational approach that uses the height representation. This
novel Lagrangian formulation is presented first for a generic reparametrization
invariant geometric model, deriving its equilibrium condition equation, or
shape equation, and its linear and angular stress tensors, using the classical
Canham-Helfrich elastic bending energy for illustration. The robustness of the
formulation is established by extending it to the presence of external forces,
and to the case of heterogenous lipid membranes, breaking reparametrization
invariance. In addition, a useful and compact general expression for the second
variation of the free energy is obtained within the Lagrangian formulation, as
a first step towards the study of the stability of membrane configurations. The
simple structure of the expressions derived for the basic entities that appear
in the mechanics of a lipid membrane is a direct consequence of the well
established power of a Lagrangian variational approach. The paper is
self-contained, and it is meant to provide, besides a new framework, also a
convenient introduction to the mechanics of lipid membranes.
