Hamiltonian multivector fields and Poisson forms in multisymplectic field theory

Forger, Michael; Paufler, Cornelius; Römer, Hartmann

doi:10.1063/1.2116320

Cited by 35 publications

(67 citation statements)

References 12 publications

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“…41); alternatively, note that D M is of the form described in Theorem IV.4 and hence automatically determines a (multi-)Dirac structure. Now, let X : M → T Z be a partial vector field as in (18). In local coordinates, we have…”

Section: B Electromagnetismmentioning

confidence: 99%

“…To see why the previous theorem implies energy conservation, decompose the multivector field X as in (18). Using Lemma A.2, the interior product can then be written as…”

Section: Theorem 46: For Any Lagrange-dirac System (X E D N+1 ) mentioning

confidence: 99%

“…We will refer to multivector fields X which are locally given by (18) as partial multivector fields.…”

Section: F Implicit Field Equations Using Multivector Fieldsmentioning

confidence: 99%

“…(23) is somewhat involved and depends on a number of coordinate identities, listed here. First, recall that X is an (n + 1)-multivector field with local expression (18), and that the multi-symplectic form is locally given by Eq. (11).…”

Section: Proof Of (23)mentioning

confidence: 99%

“…Theorem 3.4: Let E be the generalized energy (14) and consider a multivector field X on M of the form (18) such that…”

Section: F Implicit Field Equations Using Multivector Fieldsmentioning

confidence: 99%

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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 Electromagnetismmentioning

confidence: 99%

“…To see why the previous theorem implies energy conservation, decompose the multivector field X as in (18). Using Lemma A.2, the interior product can then be written as…”

Section: Theorem 46: For Any Lagrange-dirac System (X E D N+1 ) mentioning

confidence: 99%

“…We will refer to multivector fields X which are locally given by (18) as partial multivector fields.…”

Section: F Implicit Field Equations Using Multivector Fieldsmentioning

confidence: 99%

Section: Proof Of (23)mentioning

confidence: 99%

“…Theorem 3.4: Let E be the generalized energy (14) and consider a multivector field X on M of the form (18) such that…”

Section: F Implicit Field Equations Using Multivector Fieldsmentioning

confidence: 99%

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

Vankerschaver

Yoshimura

Leok

2012

Journal of Mathematical Physics

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Covariant Poisson Brackets in Geometric Field Theory

Forger

Romero

2005

Commun. Math. Phys.

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133

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We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnkovic-Witten and Zuckerman, from the multisymplectic form. The main result is that the Poisson bracket associated with this symplectic structure, according to the standard rules, is precisely the covariant bracket due to Peierls and DeWitt.Comment: 42 page

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On the geometry of multi-Dirac structures and Gerstenhaber algebras

Vankerschaver

Yoshimura

Leok

2011

Journal of Geometry and Physics

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We dedicate this paper to the memory of Jerrold E. Marsden, our friend and mentor. AbstractIn a companion paper, we introduced a notion of multi-Dirac structures, a graded version of Dirac structures, and we discussed their relevance for classical field theories. In the current paper we focus on the geometry of multi-Dirac structures. After recalling the basic definitions, we introduce a graded multiplication and a multi-Courant bracket on the space of sections of a multi-Dirac structure, so that the space of sections has the structure of a Gerstenhaber algebra. We then show that the graph of a k-form on a manifold gives rise to a multi-Dirac structure and also that this multi-Dirac structure is integrable if and only if the corresponding form is closed. Finally, we show that the multi-Courant bracket endows a subset of the ring of differential forms with a graded Poisson bracket, and we relate this bracket to some of the multisymplectic brackets found in the literature.

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Hamiltonian multivector fields and Poisson forms in multisymplectic field theory

Cited by 35 publications

References 12 publications

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

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

Covariant Poisson Brackets in Geometric Field Theory

On the geometry of multi-Dirac structures and Gerstenhaber algebras

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