Hamilton's equations for a fluid membrane: axial symmetry

TL;DR

This paper develops a Hamiltonian framework for analyzing axially symmetric fluid membranes modeled by Helfrich-Canham energy, incorporating second derivatives and intrinsic parametrization freedom, facilitating future generalizations.

Contribution

It introduces a novel Hamiltonian formulation for fluid membranes with axial symmetry, handling second derivatives and parametrization invariance in phase space.

Findings

Phase space includes conjugate momenta for position and velocity.

Constraints are identified within the phase space.

Framework sets the stage for a field theoretical extension.

Abstract

Consider a homogenous fluid membrane, or vesicle, described by the Helfrich-Canham energy, quadratic in the mean curvature. When the membrane is axially symmetric, this energy can be viewed as an `action' describing the motion of a particle; the contours of equilibrium geometries are identified with particle trajectories. A novel Hamiltonian formulation of the problem is presented which exhibits the following two features: {\it (i)} the second derivatives appearing in the action through the mean curvature are accommodated in a natural phase space; {\it (ii)} the intrinsic freedom associated with the choice of evolution parameter along the contour is preserved. As a result, the phase space involves momenta conjugate not only to the particle position but also to its velocity, and there are constraints on the phase space variables. This formulation provides the groundwork for a field theoretical generalization to arbitrary configurations, with the particle replaced by a loop in space.