Monday, December 3, 2012

1211.7154 (Jemal Guven et al.)

Metric variations become a surface    [PDF]

Jemal Guven, Pablo Vázquez-Montejo
Any surface is completely characterized by a metric and a symmetric tensor field satisfying a certain set of constraints, the Gauss-Codazzi-Mainardi (GCM) equations. These equations identify the latter tensor as the surface curvature. It should thus be possible, in principle, to phrase physical questions relating to a surface described by a Hamiltonian involving only surface degrees of freedom--be it a interface or a fluid membrane --in terms of these two tensors, without any explicit reference to the environment into which it is embedded: the surface itself is an emergent entity. Introducing Lagrange multipliers to impose the GCM equations as constraints on these variables, we demonstrate how the Euler-Lagrange equation describing the stationary states of any surface Hamiltonian is derived. The behavior of these multipliers is explored in detail for surfaces minimizing area. Using a soap film between two rings as an example, it is shown how singularities in these functions correlate directly with instabilities in equilibrium surfaces. This approach to interfaces and membranes has clear relevance to a number of problems in soft matter: in particular, this framework is ideally adapted to study the recently proposed programmed swelling of thin polymer sheets.
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