Dirac structures in vakonomic mechanics

JOURNAL OF GEOMETRY AND PHYSICS

卷:94

Article

Jimenez, Fernando¹  Yoshimura, Hiroaki^2,3 

[1] Tech Univ Munich, Zentrum Math, D-85747 Garching, Germany

[2] Waseda Univ, Dept Appl Mech & Aerosp Engn, Shinjuku Ku, Tokyo 1698555, Japan

[3] Waseda Univ, Inst Nonlinear Partial Differential Equat, Shinjuku Ku, Tokyo 1698555, Japan

关键词: Dirac structures;    Vakonomic mechanics;    Nonholonomic mechanics;    Variational principles;    Implicit Lagrangian systems;   

DOI  :  10.1016/j.geomphys.2014.11.002

来源: Elsevier

【 摘 要 】

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 x 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 circle plus T*Q) x V*. Associated with this variational principle, we establish a Dirac structure on (TQ circle plus T*Q) x 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 x 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. (C) 2015 Published by Elsevier B.V.

【 授权许可】

Free