Dirac reduction for nonholonomic mechanical systems and semidirect products

François Gay-Balmaz*, Hiroaki Yoshimura

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincaré-Suslov equations with advected parameters and the implicit Lie-Poisson-Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin-Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.

Original languageEnglish
Pages (from-to)131-213
Number of pages83
JournalAdvances in Applied Mathematics
Publication statusPublished - 2015 Feb 1


  • Dirac structures
  • Nonholonomic systems
  • Reduction by symmetry
  • RivlinEricksen fluids
  • Semidirect products
  • Variational structures

ASJC Scopus subject areas

  • Applied Mathematics


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