Hamilton's equations for a fluid membrane

TL;DR

This paper develops a Hamiltonian formulation for the equilibrium shape equation of a fluid membrane described by Helfrich-Canham energy, transforming a complex fourth-order differential equation into a set of coupled first-order equations.

Contribution

It introduces a Hamiltonian framework for the membrane shape equation, handling second derivatives and reparametrization invariance to facilitate analysis and solution.

Findings

Hamiltonian formulation for membrane shape equations derived.

Primary and secondary constraints identified and incorporated.

Shape equations reconstructed from Hamiltonian dynamics.

Abstract

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.