TY - JOUR

T1 - Dirac structures in vakonomic mechanics

AU - Jiménez, Fernando

AU - Yoshimura, Hiroaki

N1 - Funding Information:
The research of F. J. was supported in its first part by Institute of Nonlinear Partial Differential Equations at Waseda University and was partially developed during his staying there in 2012 as a visiting Postdoctoral associate. In its second stage, the research of F. J. was supported by the DFG Collaborative Research Center , TRR 109 , ‘Discretization in Geometry and Dynamics’. The research of H. Y. is partially supported by JSPS Grant-in-Aid 26400408 , JST-CREST , Waseda University Grant for SR 2012A-602 and SR 2014B-162 and IRSES project Geomech-246981.
Publisher Copyright:
© 2015 Published by Elsevier B.V.

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

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U2 - 10.1016/j.geomphys.2014.11.002

DO - 10.1016/j.geomphys.2014.11.002

M3 - Article

AN - SCOPUS:84946174882

SN - 0393-0440

VL - 94

SP - 158

EP - 178

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

ER -