# On the planar elastica, stress, and material stress

## TL;DR

This paper revisits the classical planar elastica problem, classifies shapes via conserved quantities, compares different representations, and highlights the importance of tension information often overlooked in shape equations.

## Contribution

It provides a comprehensive classification of elastica shapes using symmetry-based conserved quantities and compares various modeling approaches, emphasizing the physical significance of tension.

## Key findings

- Classification of elastica shapes via conserved quantities.
- Comparison of director, variational, and dynamical representations.
- Highlighting the importance of tension information in physical analysis.

## Abstract

We revisit the classical problem of the planar Euler \emph{elastica} with applied forces and moments, and present a classification of the shapes in terms of tangentially conserved quantities associated with spatial and material symmetries. We compare commonly used director, variational, and dynamical systems representations, and present several illustrative physical examples. We remark that an approach that employs only the shape equation for the tangential angle obscures physical information about the tension in the body.