The movement of rods in an Euclidean space can be described as a field theory on a principal bundle. The dynamics of a rod is governed by partial differential equations that may have a variational origin. If the corresponding smooth Lagrangian density is invariant by some group of transformations, there exist the corresponding conserved Noether currents. Generally, numerical schemes dealing with PDEs fail to reflect these conservation properties. We describe the main ingredients needed to create, from the smooth Lagrangian density, a variational principle for discrete motions of a discrete rod, with the 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 us 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.
Skip Nav Destination
Research Article| September 05 2019
Discrete formulation for the dynamics of rods deforming in space
Ana Casimiro ;
Ana Casimiro, César Rodrigo; Discrete formulation for the dynamics of rods deforming in space. J. Math. Phys. 1 September 2019; 60 (9): 092901. https://doi.org/10.1063/1.5045125
Download citation file: