American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation
  • PROC Home
  • This Issue
  • Next Article
    Bidifferential graded algebras and integrable systems
2009, 2009(Special): 220-229. doi: 10.3934/proc.2009.2009.220

Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods

Sebastián J. Ferraro 1, , David Iglesias-Ponte 2, and  D. Martín de Diego 3,

1. 

Departamento de Mateemática and Instituto de Matemática Bahía Blanca, Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía Blanca and CONICET, Argentina
2. 

Unidad asociada ULL-CSIC “Geometría Diferencial y Mecánica Geométrica”, Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain
3. 

Unidad Asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain

Received  August 2008 Revised  April 2009 Published  September 2009

Here we discuss a geometric integrator for nonholonomic mechanical systems that preserves the nonholonomic constraints, the discrete nonholonomic momentum map, and is also energy-preserving in some important cases. This method does not require a predefined discretization of the nonholonomic constraints. In Euclidean space, it yields a generalization of the classical SHAKE and RATTLE algorithms to the nonholonomic setting. This article shows that the method is second order convergent.
Keywords: RATTLE method, holonomic and nonholonomic systems, energy-momentum integrators, variational integrators.
Mathematics Subject Classification: 37J60, 37M15, 70F25.
Citation: Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220
[1]

Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137
[2]

Jorge Cortés. Energy conserving nonholonomic integrators. Conference Publications, 2003, 2003 (Special) : 189-199. doi: 10.3934/proc.2003.2003.189
[3]

Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003
[4]

Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4193-4223. doi: 10.3934/dcds.2015.35.4193
[5]

Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343
[6]

Werner Bauer, François Gay-Balmaz. Variational integrators for anelastic and pseudo-incompressible flows. Journal of Geometric Mechanics, 2019, 11 (4) : 511-537. doi: 10.3934/jgm.2019025
[7]

Luis C. garcía-Naranjo, Fernando Jiménez. The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4249-4275. doi: 10.3934/dcds.2017182
[8]

Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347
[9]

Matteo Focardi, Paolo Maria Mariano. Discrete dynamics of complex bodies with substructural dissipation: Variational integrators and convergence. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 109-130. doi: 10.3934/dcdsb.2009.11.109
[10]

Mats Vermeeren. Modified equations for variational integrators applied to Lagrangians linear in velocities. Journal of Geometric Mechanics, 2019, 11 (1) : 1-22. doi: 10.3934/jgm.2019001
[11]

Michele Zadra, Elizabeth L. Mansfield. Using Lie group integrators to solve two and higher dimensional variational problems with symmetry. Journal of Computational Dynamics, 2019, 6 (2) : 485-511. doi: 10.3934/jcd.2019025
[12]

Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455
[13]

Benedict Leimkuhler, Charles Matthews, Tiffany Vlaar. Partitioned integrators for thermodynamic parameterization of neural networks. Foundations of Data Science, 2019, 1 (4) : 457-489. doi: 10.3934/fods.2019019
[14]

A. Alamo, J. M. Sanz-Serna. Word combinatorics for stochastic differential equations: Splitting integrators. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2163-2195. doi: 10.3934/cpaa.2019097
[15]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185
[16]

Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79
[17]

Andrew D. Lewis. Nonholonomic and constrained variational mechanics. Journal of Geometric Mechanics, 2020, 12 (2) : 165-308. doi: 10.3934/jgm.2020013
[18]

Artur O. Lopes, Elismar R. Oliveira. Entropy and variational principles for holonomic probabilities of IFS. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 937-955. doi: 10.3934/dcds.2009.23.937
[19]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033
[20]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

 Impact Factor: 

Tools

Metrics

  • PDF downloads (111)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences