### Abstract

Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange-d'Alembert principle called the Lagrange-d'Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.

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

Pages (from-to) | 209-250 |

Number of pages | 42 |

Journal | Journal of Geometry and Physics |

Volume | 57 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Dec 31 |

### Fingerprint

### Keywords

- Constrained Dirac structures
- Degenerate Lagrangians
- L - C circuits
- The Hamilton-Pontryagin principle

### ASJC Scopus subject areas

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

### Cite this

*Journal of Geometry and Physics*,

*57*(1), 209-250. https://doi.org/10.1016/j.geomphys.2006.02.012

**Dirac structures in Lagrangian mechanics Part II : Variational structures.** / Yoshimura, Hiroaki; Marsden, Jerrold E.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 57, no. 1, pp. 209-250. https://doi.org/10.1016/j.geomphys.2006.02.012

}

TY - JOUR

T1 - Dirac structures in Lagrangian mechanics Part II

T2 - Variational structures

AU - Yoshimura, Hiroaki

AU - Marsden, Jerrold E.

PY - 2006/12/31

Y1 - 2006/12/31

N2 - Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange-d'Alembert principle called the Lagrange-d'Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.

AB - Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange-d'Alembert principle called the Lagrange-d'Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.

KW - Constrained Dirac structures

KW - Degenerate Lagrangians

KW - L - C circuits

KW - The Hamilton-Pontryagin principle

UR - http://www.scopus.com/inward/record.url?scp=33749126543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749126543&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2006.02.012

DO - 10.1016/j.geomphys.2006.02.012

M3 - Article

VL - 57

SP - 209

EP - 250

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 1

ER -