  The dynamical phase-space of axisymmetric Canham-Helfrich (CH) cells is
constructed from a Hamiltonian field recapitulating membrane
curvature-elasticity and systemic restrictions. Guiding principles are
reparametrization to convert a static geometric system into a dynamical system,
and Galilean transformation, to build a transformed Lagrangian invariant with
respect to the action described by the CH free-energy. Building on the fluidity
postulate, this Lagrangian describes the cellular membrane as an inverted
harmonic oscillator driven by bending elasticity and effective friction
governed by Gaussian curvature. To close the spring-mass interaction, we
explicit the mass of the membrane and establish a dimensionally-minimal
Lagrangian. Then, the canonical Hamiltonian is constructed in generalized
coordinates $H(p, q, t)$, and the equations of motion derived in accordance
with the principle of minimal action. The derived phase-space is used as a
global predictor of the cellular shapes for different mechanical settings with
a biological significance.
