Lie Symmetry Analysis for Cosserat Rods

TL;DR

This paper applies Lie symmetry analysis to a subsystem of the Cosserat theory of rods, deriving a general analytic solution dependent on arbitrary functions, enhancing understanding of slender structure dynamics.

Contribution

It provides an explicit, general solution to a subsystem of the Cosserat rods equations using Lie symmetry methods, including proof of analyticity under certain conditions.

Findings

Derived explicit general solutions depending on arbitrary functions

Proved the analyticity of solutions under analyticity assumptions

Enhanced understanding of the dynamic equilibrium of slender structures

Abstract

We consider a subsystem of the Special Cosserat Theory of Rods and construct an explicit form of its solution that depends on three arbitrary functions in (s,t) and three arbitrary functions in t. Assuming analyticity of the arbitrary functions in a domain under consideration, we prove that the obtained solution is analytic and general. The Special Cosserat Theory of Rods describes the dynamic equilibrium of 1-dimensional continua, i.e. slender structures like fibers, by means of a system of partial differential equations.