### Abstract

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T^{*} Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T^{*} T Q, T T^{*} Q, and T^{*} T^{*} Q. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L-C circuit, a degenerate Lagrangian system with holonomic constraints.

Original language | English |
---|---|

Pages (from-to) | 133-156 |

Number of pages | 24 |

Journal | Journal of Geometry and Physics |

Volume | 57 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Dec 31 |

### Fingerprint

### Keywords

- Dirac structures
- Implicit Lagrangian systems
- Nonholonomic systems

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology

### Cite this

*Journal of Geometry and Physics*,

*57*(1), 133-156. https://doi.org/10.1016/j.geomphys.2006.02.009

**Dirac structures in Lagrangian mechanics Part I : Implicit Lagrangian systems.** / Yoshimura, Hiroaki; Marsden, Jerrold E.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 57, no. 1, pp. 133-156. https://doi.org/10.1016/j.geomphys.2006.02.009

}

TY - JOUR

T1 - Dirac structures in Lagrangian mechanics Part I

T2 - Implicit Lagrangian systems

AU - Yoshimura, Hiroaki

AU - Marsden, Jerrold E.

PY - 2006/12/31

Y1 - 2006/12/31

N2 - This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T* Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T* T Q, T T* Q, and T* T* Q. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L-C circuit, a degenerate Lagrangian system with holonomic constraints.

AB - This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T* Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T* T Q, T T* Q, and T* T* Q. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L-C circuit, a degenerate Lagrangian system with holonomic constraints.

KW - Dirac structures

KW - Implicit Lagrangian systems

KW - Nonholonomic systems

UR - http://www.scopus.com/inward/record.url?scp=33749141694&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749141694&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2006.02.009

DO - 10.1016/j.geomphys.2006.02.009

M3 - Article

AN - SCOPUS:33749141694

VL - 57

SP - 133

EP - 156

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 1

ER -