Analysis of one-dimensional structures using Lie groups

This article presents a formulation for one-dimensional structures. It uses the structure of Lie groups and the associated differential calculus to describe deformations and the dynamic equation of the structure. Three levels of equation setting are explored: level one is the most abstract where a single equation is obtained using the Lie algebra of the displacement group, level two refers to the semidirect product decomposition of the displacement group into a rotation and a translation group, and the third level is obtained by selecting a suitable basis within the Lie algebra, which leads to scalar equations. Equations at each level are derived, and a comparison with the literature is made for the static equilibrium.

The article also addresses perturbations of the linear dynamic around an equilibrium position. The symmetry of the deformation operator is examined, which has implications for the study of instability.