Tensor products of dirac structures and interconnection in lagrangian mechanics

Henry O. Jacobs, Hiroaki Yoshimura

    Research output: Contribution to journalArticle

    6 Citations (Scopus)

    Abstract

    Many mechanical systems are large and complex, but are composed of simple subsystems. In order to understand such large systems it is natural to tear the system into these subsystems. Conversely we must understand how to invert this tearing procedure. In other words, we must understand interconnection of subsystems. Such an understanding has been already shown in the context of Hamiltonian systems on vector spaces via the port- Hamiltonian systems program, in which an interconnection may be achieved through the identfication of shared variables, whereupon the notion of composition of Dirac structures allows one to interconnect two systems. In this paper, we seek to extend the program of the port-Hamiltonian systems on vector spaces to the case of Lagrangian systems on manifolds and also extend the notion of composition of Dirac structures appropriately. In particular, we will interconnect Lagrange-Dirac systems by modifying the respective Dirac structures of the involved subsystems. We define the interconnection of Dirac structures via an interaction Dirac structure and a tensor product of Dirac structures. We will show how the dynamics of the interconnected system is formulated as a function of the subsystems, and we will elucidate the associated variational principles. We will then illustrate how this theory extends the theory of port-Hamiltonian systems and the notion of composition of Dirac structures to manifolds with couplings which do not require the identfication of shared variables. Lastly, we will show some examples: a mass-spring mechanical systems, an electric circuit, and a nonholonomic mechanical system.

    Original languageEnglish
    Pages (from-to)67-98
    Number of pages32
    JournalJournal of Geometric Mechanics
    Volume6
    Issue number1
    DOIs
    Publication statusPublished - 2014

    Fingerprint

    Lagrangian Mechanics
    Dirac Structures
    Hamiltonians
    Interconnection
    Tensor Product
    Tensors
    Mechanics
    Subsystem
    Vector spaces
    Hamiltonian Systems
    Mechanical Systems
    Chemical analysis
    Interconnect
    Vector space
    Large scale systems
    Lagrangian Systems
    Nonholonomic Systems
    Interconnected Systems
    Invert
    Networks (circuits)

    Keywords

    • Degenerate
    • Dirac structures
    • Interconnection
    • Lagrangians
    • Nonholonomic
    • Port systems

    ASJC Scopus subject areas

    • Applied Mathematics
    • Control and Optimization
    • Geometry and Topology
    • Mechanics of Materials

    Cite this

    Tensor products of dirac structures and interconnection in lagrangian mechanics. / Jacobs, Henry O.; Yoshimura, Hiroaki.

    In: Journal of Geometric Mechanics, Vol. 6, No. 1, 2014, p. 67-98.

    Research output: Contribution to journalArticle

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