A simultaneous variational principle for elementary excitations of fluid lipid membranes

TL;DR

This paper introduces a novel simultaneous variational principle that describes both equilibrium shapes and elementary excitations of fluid lipid membranes, simplifying the derivation of their governing equations and excitation energies.

Contribution

It presents a new variational approach that yields both equilibrium and excitation equations simultaneously without second variations, applicable to bending-dominated systems.

Findings

Derivation of Euler-Lagrange and Jacobi equations from a single variational principle

Elementary excitation energies obtained without second variations

Applicable to systems with dominant bending modes

Abstract

A simultaneous variational principle is introduced that offers a novel avenue to the description of the equilibrium configurations, and at the same time of the elementary excitations, or undulations, of fluid lipid membranes, described by a geometric continuum free energy. The simultaneous free energy depends on the shape functions through the membrane stress tensor, and on an additional deformation spatial vector. Extremization of this free energy produces at once the Euler-Lagrange equations and the Jacobi equations, that describe elementary excitations, for the geometric free energy. As an added benefit, the energy of the elementary excitations, given by the second variation of the geometric free energy, is obtained without second variations. Although applied to the specific case of lipid membranes, this variational principle should be useful in any physical system where bending modes are dominant.