Dirac reduction for nonholonomic mechanical systems and semidirect products

François Gay-Balmaz, Hiroaki Yoshimura

    Research output: Contribution to journalArticle

    8 Citations (Scopus)

    Abstract

    This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincaré-Suslov equations with advected parameters and the implicit Lie-Poisson-Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin-Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.

    Original languageEnglish
    Pages (from-to)131-213
    Number of pages83
    JournalAdvances in Applied Mathematics
    Volume63
    DOIs
    Publication statusPublished - 2015 Feb 1

    Fingerprint

    Nonholonomic Systems
    Mechanical Systems
    Paul Adrien Maurice Dirac
    Dynamical systems
    Dynamical system
    Dirac Structures
    Symmetry
    Hamiltonians
    Lie groups
    Infinite-dimensional Systems
    Poisson equation
    Poisson's equation
    Lagrange
    Equations of motion
    Euler
    Equations of Motion
    Fluid
    Fluids

    Keywords

    • Dirac structures
    • Nonholonomic systems
    • Reduction by symmetry
    • RivlinEricksen fluids
    • Semidirect products
    • Variational structures

    ASJC Scopus subject areas

    • Applied Mathematics

    Cite this

    Dirac reduction for nonholonomic mechanical systems and semidirect products. / Gay-Balmaz, François; Yoshimura, Hiroaki.

    In: Advances in Applied Mathematics, Vol. 63, 01.02.2015, p. 131-213.

    Research output: Contribution to journalArticle

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