Dirac structures in Lagrangian mechanics Part II: Variational structures

Hiroaki Yoshimura, Jerrold E. Marsden

    Research output: Contribution to journalArticle

    74 Citations (Scopus)

    Abstract

    Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange-d'Alembert principle called the Lagrange-d'Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.

    Original languageEnglish
    Pages (from-to)209-250
    Number of pages42
    JournalJournal of Geometry and Physics
    Volume57
    Issue number1
    DOIs
    Publication statusPublished - 2006 Dec 31

    Fingerprint

    Lagrangian Mechanics
    Dirac Structures
    Lagrangian Systems
    pontryagin principle
    Legendre functions
    Legendre
    Pontryagin's Principle
    Lagrange
    Rolling Disk
    Cotangent
    Euler-Lagrange equation
    Nonholonomic Systems
    Cotangent Bundle
    Constrained Systems
    Euler-Lagrange Equations
    Geometric Structure
    Variational Formulation
    variational principles
    tangents
    Variational Principle

    Keywords

    • Constrained Dirac structures
    • Degenerate Lagrangians
    • L - C circuits
    • The Hamilton-Pontryagin principle

    ASJC Scopus subject areas

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

    Cite this

    Dirac structures in Lagrangian mechanics Part II : Variational structures. / Yoshimura, Hiroaki; Marsden, Jerrold E.

    In: Journal of Geometry and Physics, Vol. 57, No. 1, 31.12.2006, p. 209-250.

    Research output: Contribution to journalArticle

    @article{c5d77272b4e744f68f62a93f5b59b22f,
    title = "Dirac structures in Lagrangian mechanics Part II: Variational structures",
    abstract = "Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange-d'Alembert principle called the Lagrange-d'Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.",
    keywords = "Constrained Dirac structures, Degenerate Lagrangians, L - C circuits, The Hamilton-Pontryagin principle",
    author = "Hiroaki Yoshimura and Marsden, {Jerrold E.}",
    year = "2006",
    month = "12",
    day = "31",
    doi = "10.1016/j.geomphys.2006.02.012",
    language = "English",
    volume = "57",
    pages = "209--250",
    journal = "Journal of Geometry and Physics",
    issn = "0393-0440",
    publisher = "Elsevier",
    number = "1",

    }

    TY - JOUR

    T1 - Dirac structures in Lagrangian mechanics Part II

    T2 - Variational structures

    AU - Yoshimura, Hiroaki

    AU - Marsden, Jerrold E.

    PY - 2006/12/31

    Y1 - 2006/12/31

    N2 - Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange-d'Alembert principle called the Lagrange-d'Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.

    AB - Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler-Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton-Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange-d'Alembert principle called the Lagrange-d'Alembert-Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L-C circuit are given to illustrate the results.

    KW - Constrained Dirac structures

    KW - Degenerate Lagrangians

    KW - L - C circuits

    KW - The Hamilton-Pontryagin principle

    UR - http://www.scopus.com/inward/record.url?scp=33749126543&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=33749126543&partnerID=8YFLogxK

    U2 - 10.1016/j.geomphys.2006.02.012

    DO - 10.1016/j.geomphys.2006.02.012

    M3 - Article

    AN - SCOPUS:33749126543

    VL - 57

    SP - 209

    EP - 250

    JO - Journal of Geometry and Physics

    JF - Journal of Geometry and Physics

    SN - 0393-0440

    IS - 1

    ER -