Phase-space dynamics of minimal Canham-Helfrich cells

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.