A general construction of poisson brackets on exact multusymplectic manifolds

Forger, Michael; Paufler, Cornelius; Römer, Hartmann

doi:10.1016/s0034-4877(03)80012-5

Cited by 18 publications

(32 citation statements)

References 9 publications

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“…The extended Hamiltonian formalism has already been used for defining Poisson brackets in field theories [14]. It could provide new insights into some classical problems, such as: reduction of multisymplectic Hamiltonian systems with symmetry, integrability, and quantization of multisymplectic Hamiltonian field theories.…”

Section: Discussionmentioning

confidence: 99%

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Extended Hamiltonian systems in multisymplectic field theories

Echeverría-Enríquez¹,

León

Muñoz-Lecanda³

et al. 2007

Journal of Mathematical Physics

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We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of non-autonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.

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Section: Discussionmentioning

confidence: 99%

“…In this way, solutions to equations (8) are determined locally from the relations (13) and (15), and through the n independent linear equations (14). Therefore, there are n(m 2 − 1) arbitrary functions.…”

Section: Remarkmentioning

confidence: 99%

Extended Hamiltonian systems in multisymplectic field theories

Echeverría-Enríquez¹,

León

Muñoz-Lecanda³

et al. 2007

Journal of Mathematical Physics

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“…A whole literature exists about properties of such structure, which is generically referred to as multisymplectic geometry of M π (see references in [49]). In particular, efforts were made to find multisymplectic analogues of all properties of T * Q (including, for instance, the Poisson bracket [21,22,24,36,38]). Now, it is natural to wonder if it is possible to reasonably further generalize in two different directions.…”

Section: Introductionmentioning

confidence: 99%

Partial Differential Hamiltonian Systems

Vitagliano¹

2013

Can. j. math.

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Abstract. We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field theory, in our formalism, the geometric structure (kinematics) and the dynamical information on the "phase space" appear as just different components of one single geometric object.

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“…All these aspects are discussed in Sections 2 and 3 (furthermore, a quick review on multivector fields and connections is given in Appendix A.2). Moreover, multivector fields are also used in order to state generalized Poisson brackets in the Hamiltonian formalism of field theories [34,50,51,52,88].…”

Section: Introductionmentioning

confidence: 99%

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

Román‐Roy¹

2009

SIGMA

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Abstract. This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.

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A general construction of poisson brackets on exact multusymplectic manifolds

Cited by 18 publications

References 9 publications

Extended Hamiltonian systems in multisymplectic field theories

Extended Hamiltonian systems in multisymplectic field theories

Partial Differential Hamiltonian Systems

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

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