### Abstract

In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoints of Dirac structures as well as Lagrangian submanifolds. Namely, we clarify that Lagrangian submanifold theory cannot represent nonholonomic mechanics properly, but vakonomic mechanics instead. Second, in order to represent vakonomic mechanics, we employ the space TQ×V*, where a vakonomic Lagrangian is defined from a given Lagrangian (possibly degenerate) subject to nonholonomic constraints. Then, we show how implicit vakonomic Euler-Lagrange equations can be formulated by the Hamilton-Pontryagin variational principle for the vakonomic Lagrangian on the extended Pontryagin bundle (TQ⊕T*Q)×V*. Associated with this variational principle, we establish a Dirac structure on (TQ⊕T*Q)×V* in order to define an intrinsic vakonomic Lagrange-Dirac system. Furthermore, we also establish another construction for the vakonomic Lagrange-Dirac system using a Dirac structure on T*Q×V*, where we introduce a vakonomic Dirac differential. Finally, we illustrate our theory of vakonomic Lagrange-Dirac systems by some examples such as the vakonomic skate and the vertical rolling coin.

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

Pages (from-to) | 158-178 |

Number of pages | 21 |

Journal | Journal of Geometry and Physics |

Volume | 94 |

DOIs | |

Publication status | Published - 2015 Aug 1 |

### Fingerprint

### Keywords

- Dirac structures
- Implicit lagrangian systems
- Nonholonomic mechanics
- Vakonomic mechanics
- Variational principles

### ASJC Scopus subject areas

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

### Cite this

*Journal of Geometry and Physics*,

*94*, 158-178. https://doi.org/10.1016/j.geomphys.2014.11.002

**Dirac structures in vakonomic mechanics.** / Jiménez, Fernando; Yoshimura, Hiroaki.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 94, pp. 158-178. https://doi.org/10.1016/j.geomphys.2014.11.002

}

TY - JOUR

T1 - Dirac structures in vakonomic mechanics

AU - Jiménez, Fernando

AU - Yoshimura, Hiroaki

PY - 2015/8/1

Y1 - 2015/8/1

N2 - In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoints of Dirac structures as well as Lagrangian submanifolds. Namely, we clarify that Lagrangian submanifold theory cannot represent nonholonomic mechanics properly, but vakonomic mechanics instead. Second, in order to represent vakonomic mechanics, we employ the space TQ×V*, where a vakonomic Lagrangian is defined from a given Lagrangian (possibly degenerate) subject to nonholonomic constraints. Then, we show how implicit vakonomic Euler-Lagrange equations can be formulated by the Hamilton-Pontryagin variational principle for the vakonomic Lagrangian on the extended Pontryagin bundle (TQ⊕T*Q)×V*. Associated with this variational principle, we establish a Dirac structure on (TQ⊕T*Q)×V* in order to define an intrinsic vakonomic Lagrange-Dirac system. Furthermore, we also establish another construction for the vakonomic Lagrange-Dirac system using a Dirac structure on T*Q×V*, where we introduce a vakonomic Dirac differential. Finally, we illustrate our theory of vakonomic Lagrange-Dirac systems by some examples such as the vakonomic skate and the vertical rolling coin.

AB - In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoints of Dirac structures as well as Lagrangian submanifolds. Namely, we clarify that Lagrangian submanifold theory cannot represent nonholonomic mechanics properly, but vakonomic mechanics instead. Second, in order to represent vakonomic mechanics, we employ the space TQ×V*, where a vakonomic Lagrangian is defined from a given Lagrangian (possibly degenerate) subject to nonholonomic constraints. Then, we show how implicit vakonomic Euler-Lagrange equations can be formulated by the Hamilton-Pontryagin variational principle for the vakonomic Lagrangian on the extended Pontryagin bundle (TQ⊕T*Q)×V*. Associated with this variational principle, we establish a Dirac structure on (TQ⊕T*Q)×V* in order to define an intrinsic vakonomic Lagrange-Dirac system. Furthermore, we also establish another construction for the vakonomic Lagrange-Dirac system using a Dirac structure on T*Q×V*, where we introduce a vakonomic Dirac differential. Finally, we illustrate our theory of vakonomic Lagrange-Dirac systems by some examples such as the vakonomic skate and the vertical rolling coin.

KW - Dirac structures

KW - Implicit lagrangian systems

KW - Nonholonomic mechanics

KW - Vakonomic mechanics

KW - Variational principles

UR - http://www.scopus.com/inward/record.url?scp=84946174882&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946174882&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2014.11.002

DO - 10.1016/j.geomphys.2014.11.002

M3 - Article

AN - SCOPUS:84946174882

VL - 94

SP - 158

EP - 178

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -