Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance

Details Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance

2004-05-27

arxiv.org/abs/cond-mat/0405655v1

J. of Phys. A: Math. and Gen. 37 (2004) 5983

Physics

Condensed Matter

Soft Condensed Matter

20 pages

Scientific paper

10.1088/0305-4470/37/23/003

A covariant approach towards a theory of deformations is developed to examine both the first and second variation of the Helfrich-Canham Hamiltonian -- quadratic in the extrinsic curvature -- which describes fluid vesicles at mesoscopic scales. Deformations are decomposed into tangential and normal components; At first order, tangential deformations may always be identified with a reparametrization; at second order, they differ. The relationship between tangential deformations and reparametrizations, as well as the coupling between tangential and normal deformations, is examined at this order for both the metric and the extrinsic curvature tensors. Expressions for the expansion to second order in deformations of geometrical invariants constructed with these tensors are obtained; in particular, the expansion of the Hamiltonian to this order about an equilibrium is considered. Our approach applies as well to any geometrical model for membranes.

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