### Abstract

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 language | English |
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Pages (from-to) | 131-213 |

Number of pages | 83 |

Journal | Advances in Applied Mathematics |

Volume | 63 |

DOIs | |

Publication status | Published - 2015 Feb 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Dirac reduction for nonholonomic mechanical systems and semidirect products.** / Gay-Balmaz, François; Yoshimura, Hiroaki.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics*, vol. 63, pp. 131-213. https://doi.org/10.1016/j.aam.2014.10.004

}

TY - JOUR

T1 - Dirac reduction for nonholonomic mechanical systems and semidirect products

AU - Gay-Balmaz, François

AU - Yoshimura, Hiroaki

PY - 2015/2/1

Y1 - 2015/2/1

N2 - 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.

AB - 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.

KW - Dirac structures

KW - Nonholonomic systems

KW - Reduction by symmetry

KW - RivlinEricksen fluids

KW - Semidirect products

KW - Variational structures

UR - http://www.scopus.com/inward/record.url?scp=84919421403&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84919421403&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2014.10.004

DO - 10.1016/j.aam.2014.10.004

M3 - Article

AN - SCOPUS:84919421403

VL - 63

SP - 131

EP - 213

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

ER -