Properties of compressible elastica from relativistic analogy

TL;DR

This paper extends Kirchhoff's kinetic analogy to compressible elastica, revealing a relativistic analogy that uncovers symmetries in elastic deformations and connects to relativistic nonlinear pendulum dynamics.

Contribution

It introduces a novel extension of Kirchhoff's analogy to compressible filaments, linking elastic deformation to relativistic effects and deriving explicit solutions for related nonlinear systems.

Findings

Reveals symmetry in compressible elastica deformations

Derives explicit relativistic nonlinear pendulum solutions

Suggests applications to other soft matter systems

Abstract

Kirchhoff's kinetic analogy relates the deformation of an incompressible elastic rod to the classical dynamics of rigid body rotation. We extend the analogy to compressible filaments and find that the extension is similar to the introduction of relativistic effects into the dynamical system. The extended analogy reveals a surprising symmetry in the deformations of compressible elastica. In addition, we use known results for the buckling of compressible elastica to derive the explicit solution for the motion of a relativistic nonlinear pendulum. We discuss cases where the extended Kirchhoff analogy may be useful for the study of other soft matter systems.