Dirac structures in vakonomic mechanics

Fernando Jiménez*, Hiroaki Yoshimura

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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 languageEnglish
Pages (from-to)158-178
Number of pages21
JournalJournal of Geometry and Physics
Publication statusPublished - 2015 Aug 1


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

ASJC Scopus subject areas

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


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