Shape and symmetry of a fluid-supported elastic sheet

TL;DR

This paper explores how symmetries in higher-dimensional systems influence the shape and degeneracy of fluid-supported elastic sheets, revealing a continuous transition between symmetric and antisymmetric folds.

Contribution

It demonstrates the emergence of nontrivial symmetries and conserved quantities in embedded elastic systems, linking membrane profiles to sine-Gordon dynamics.

Findings

Identified a continuous degeneracy between symmetric and antisymmetric folds.

Derived the Hamiltonian generator and conserved momentum for the symmetry.

Connected membrane shape symmetries to sine-Gordon chain translational symmetries.

Abstract

A connection between the dynamics of a sine-Gordon chain and a certain static membrane folding problem was recently found. The one-dimensional membrane profile is a cross-section of the position-time sine-Gordon amplitude profile. Here we show that when one system is embedded in a higher-dimensional system in this way, obvious symmetries in the larger system can lead to nontrivial symmetries in the embedded system. In particular, a thin buckled membrane on a fluid substrate has a continuous degeneracy that interpolates between a symmetric and an antisymmetric fold. We find the Hamiltonian generator of this symmetry and the corresponding conserved momentum by interpreting the simple translational symmetries of the sine-Gordon chain in terms of the embedded coordinates. We discuss possible extensions to other embedded dynamical systems.