Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1523–1562 | Cite as

Energy-Preserving Integrators Applied to Nonholonomic Systems

  • Elena Celledoni
  • Marta Farré Puiggalí
  • Eirik Hoel Høiseth
  • David Martín de DiegoEmail author


We introduce energy-preserving integrators for nonholonomic mechanical systems. We will see that the nonholonomic dynamics is completely determined by a triple \(({{\mathcal {D}}}^*, \varPi , \mathcal {H})\), where \({{\mathcal {D}}}^*\) is the dual of the vector bundle determined by the nonholonomic constraints, \(\varPi \) is an almost-Poisson bracket (the nonholonomic bracket) and \( \mathcal {H}: {{\mathcal {D}}}^*\rightarrow \mathbb {R}\) is a Hamiltonian function. For this triple, we can apply energy-preserving integrators, in particular, we show that discrete gradients can be used in the numerical integration of nonholonomic dynamics. By construction, we achieve preservation of the constraints and of the energy of the nonholonomic system. Moreover, to facilitate their applicability to complex systems which cannot be easily transformed into the aforementioned almost-Poisson form, we rewrite our integrators using just the initial information of the nonholonomic system. The derived procedures are tested on several examples: a chaotic quartic nonholonomic mechanical system, the Chaplygin sleigh system, the Suslov problem and a continuous gearbox driven by an asymmetric pendulum. Their performance is compared with other standard methods in nonholonomic dynamics, and their merits verified in practice.


Nonholonomic mechanical systems Almost-Poisson bracket Geometric numerical integration Energy preservation Discrete gradients 

Mathematics Subject Classification

65P99 37J60 70F25 



This work has been partially supported by MINECO (Spain) MTM2013-42870-P, MTM2015-69124-REDT, the ICMAT Severo Ochoa project SEV-2015-0554 and the Nils-Abel project 010-ABEL-CM-2014ANILS. It has also received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 691070. M. Farré has been financially supported by a FPU scholarship from MECD.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNTNUTrondheimNorway
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain

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