Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems

Yoshimura, Hidetoshi; Marsden, Jerrold E.

doi:10.1016/j.geomphys.2006.02.009

Cited by 117 publications

(173 citation statements)

References 31 publications

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“…On the Lagrangian side, it was shown in Ref. 41 that Dirac structures induced from distributions on configuration manifolds naturally yield a notion of implicit Lagrangian systems, allowing for the description of mechanical systems with degenerate Lagrangians and nontrivial constraint distributions.…”

Section: B Dirac Structures and Lagrange-dirac Systems In Mechanicsmentioning

confidence: 99%

The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories

Vankerschaver

Yoshimura

Leok

2012

Journal of Mathematical Physics

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We introduce a variational principle for field theories, referred to as the HamiltonPontryagin principle, and we show that the resulting field equations are the EulerLagrange equations in implicit form. Second, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit EulerLagrange equations for fields obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Finally, we show a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields, Maxwell's equations, and elastostatics. C 2012 American Institute of Physics.

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Section: B Dirac Structures and Lagrange-dirac Systems In Mechanicsmentioning

confidence: 99%

The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories

Vankerschaver

Yoshimura

Leok

2012

Journal of Mathematical Physics

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“…Before going into the details of Dirac reduction, we shall briefly review how to define an induced Dirac structure and, given a Lagrangian, an associated implicit Lagrangian system, following Courant [23] and Yoshimura and Marsden [50].…”

Section: Review Of Implicit Lagrangian Systemsmentioning

confidence: 99%

“…Let us recall the definition of implicit Lagrangian systems; once again, for further details, see [50].…”

Section: Implicit Lagrangian Systemsmentioning

confidence: 99%

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Reduction of Dirac structures and the Hamilton-Pontryagin principle

Yoshimura

Marsden

2007

Reports on Mathematical Physics

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This paper develops a reduction theory for Dirac structures that includes in a unified way, reduction of both Lagrangian and Hamiltonian systems. It includes the reduction of variational principles and in particular, the Hamilton-Pontryagin variational principle. It also includes reduction theory for implicit Lagrangian systems that could be degenerate and have constraints.In this paper we focus on the special case in which the configuration manifold is a Lie group G. In our earlier papers we established the link between the Hamilton-Pontryagin principle and Dirac structures. We begin the paper with the reduction of this principle. The traditional view of Poisson reduction in this case is to reduce T * G with its natural Poisson structure to g * with its Lie-Poisson structure. However, the basic step of reducing Hamilton's phase space principle already shows that it is important to use g ⊕ g * for the reduced space, rather than just g * . In this way, our construction includes both Euler-Poincaré as well as Lie-Poisson reduction. The geometry behind this procedure, which we call Lie-Dirac reduction starts with the standard (i.e., canonical) Dirac structure on T * G (which can be viewed either symplectically or from the Poisson viewpoint) and for each µ ∈ g * , produces a Dirac structure on g ⊕ g * . This geometry then simultaneously supports both Euler-Poincaré and Lie-Poisson reduction.In the last part of the paper, we include nonholonomic constraints, and illustrate this construction with Suslov systems in nonholonomic mechanics, both from the Euler-Poincaré and Lie-Poisson viewpoints.

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“…For an arbitrary manifold, M , there is a bracket on the sections of T M ⊕ T * M , the Courant bracket, which is skew-symmetric but does not satisfy the Jacobi identity. The vector bundle T M ⊕ T * M is called the generalized tangent bundle of M , and is also called the standard Courant algebroid, or the Pontryagin bundle of M , because of its rôle in control theory (see [YM06]). …”

Section: Infinitesimals Of Multiplicative Forms On Lie Groupoidsmentioning

confidence: 99%