The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory

Abstract

We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.

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References in corpus (8)

• Canonical Structure of Classical Field Theory in the Polymomentum Phase Space
• On Field Theoretic Generalizations of a Poisson Algebra
• Covariant Poisson Brackets in Geometric Field Theory
• Geometry of Hamiltonean n-vectors in Multisymplectic Field Theory
• Finite dimesional Hamiltonian formalism for gauge and field theories
• Geometric Structures in Field Theory
• A Poisson Bracket on Multisymplectic Phase Space
• A vertical exterior derivative in multisymplectic geometry and a graded Poisson bracket for nontrivial geometries

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