# Discrete Formulation for the dynamics of rods deforming in space

## TL;DR

This paper develops a variational framework for modeling the discrete dynamics of rods in space, using geometric objects and difference operators, enabling efficient simulation of rod evolution from initial configurations.

## Contribution

It introduces a novel discrete Lagrangian formulation for rods based on Atiyah bundle elements, generalizable to other variational theories in principal bundles.

## Key findings

- Provides a geometric discrete Lagrangian density for rods
- Enables iterative computation of rod evolution over time
- Applicable to discretizations of various variational theories

## Abstract

We describe the main ingredients needed to create, from the smooth lagrangian density, a variational principle for discrete motions of a discrete rod, with corresponding conserved Noether currents. We describe all geometrical objects in terms of elements on the linear Atiyah bundle, using a reduced forward difference operator. We show how this introduces a discrete lagrangian density that models the discrete dynamics of a discrete rod. The presented tools are general enough to represent a discretization of any variational theory in principal bundles, and its simplicity allows to perform an iterative integration algorithm to compute the discrete rod evolution in time, starting from any predefined configurations of all discrete rod elements at initial times.