  Consider a homogenous fluid membrane described by the Helfrich-Canham energy,
quadratic in the mean curvature of the membrane surface. The shape equation
that determines equilibrium configurations is fourth order in derivatives and
cubic in the mean curvature. We introduce a Hamiltonian formulation of this
equation which dismantles it into a set of coupled first order equations. This
involves interpreting the Helfrich-Canham energy as an action; equilibrium
surfaces are generated by the evolution of space curves. Two features
complicate the implementation of a Hamiltonian framework: (i) The action
involves second derivatives. This requires treating the velocity as a phase
space variable and the introduction of its conjugate momentum. The canonical
Hamiltonian is constructed on this phase space. (ii) The action possesses a
local symmetry -- reparametrization invariance. The two labels we use to
parametrize points on the surface are themselves physically irrelevant. This
symmetry implies primary constraints, one for each label, that need to be
implemented within the Hamiltonian. The two lagrange multipliers associated
with these constraints are identified as the components of the acceleration
tangential to the surface. The conservation of the primary constraints imply
two secondary constraints, fixing the tangential components of the momentum
conjugate to the position. Hamilton's equations are derived and the appropriate
initial conditions on the phase space variables are identified. Finally, it is
shown how the shape equation can be reconstructed from these equations.
