### Abstract

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.

Original language | English |
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Article number | 092901 |

Journal | Journal Of Mathematical Physics |

Volume | 60 |

Issue number | 9 |

DOIs | |

Publication status | Published - 1 Sep 2019 |

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### Cite this

*Journal Of Mathematical Physics*,

*60*(9), [092901]. https://doi.org/10.1063/1.5045125

}

*Journal Of Mathematical Physics*, vol. 60, no. 9, 092901. https://doi.org/10.1063/1.5045125

**Discrete formulation for the dynamics of rods deforming in space.** / Casimiro, Ana; Rodrigo, César.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Discrete formulation for the dynamics of rods deforming in space

AU - Casimiro, Ana

AU - Rodrigo, César

N1 - Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) Project No. UID/MAT/00297/2019 (Centro de Matemática e Aplicações) Project No. UID/MAT/04561/2019 (Centro de Matemática, Aplicações Fundamentais e Investigação Operacional of Universidade de Lisboa CMAF-CIO).

PY - 2019/9/1

Y1 - 2019/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85072191860&partnerID=8YFLogxK

U2 - 10.1063/1.5045125

DO - 10.1063/1.5045125

M3 - Article

VL - 60

JO - Journal Of Mathematical Physics

JF - Journal Of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 092901

ER -