Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

Ciaglia, Florio M.; Cosmo, Fabio Di; Ibort, Alberto; Marmo, Giuseppe; Schiavone, Luca; Zampini, Alessandro

doi:10.3390/sym14010070

Cited by 6 publications

(2 citation statements)

References 47 publications

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“…For gauge field theories, constraints again appear. For instance, for the free Maxwell Lagrangian L = − 1 4 F µν F µν we have the canonical momentum tensor π µν ≡ ∂ L/∂ (∂ µ A ν ) = −F µν satisfying the constraint equation π µν + π νµ = 0 which leads to complications related to the gauge symmetry [4,5,54,88,91].…”

Section: Energy-momentum Tensormentioning

confidence: 99%

Covariant canonical formulations of classical field theories

Gieres

2023

SciPost Phys. Lect. Notes

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We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories and general relativity) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws (notably related to geometric symmetries) within the different approaches. Moreover, for each of these approaches, the impact of space-time boundaries is also addressed. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.

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Section: Energy-momentum Tensormentioning

confidence: 99%

Covariant canonical formulations of classical field theories

Gieres

2023

SciPost Phys. Lect. Notes

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“…For a large class of field theories, this infinite dimensional space admits a Poisson (or a presymplectic) structure. This is not a new theory and we will quote an old paper by Peierls [63] (see [17,18,28,29]; see also the recent papers [13][14][15][16]). In this setting, the boundary conditions of the theory play an important role [57].…”

Section: Introductionmentioning

confidence: 99%

A new canonical affine bracket formulation of Hamiltonian classical field theories of first order

Gay-Balmaz,

Marrero,

Martínez Alba

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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It has been a long standing question how to extend, in the finite-dimensional setting, the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field Theories of first order on an arbitrary configuration bundle. It is obtained via the construction of the appropriate field-theoretic analogues of the Hamiltonian vector field and of the space of observables, via the introduction of a suitable canonical Lie algebra structure on the space of currents (the observables in field theories). This Lie algebra structure is shown to have a representation on the affine space of Hamiltonian sections, which yields an affine analogue to the Jacobi identity for our bracket. The construction is analogous to the canonical Poisson formulation of Hamiltonian systems although the nature of our formulation is linear-affine and not bilinear as the standard Poisson bracket. This is consistent with the fact that the space of currents and Hamiltonian sections are respectively, linear and affine. Our setting is illustrated with some examples including Continuum Mechanics and Yang–Mills theory.

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Reductions: precontact versus presymplectic

Grabowska

Grabowski

2023

Annali di Matematica

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We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden–Weinstein–Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds.

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Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

Cited by 6 publications

References 47 publications

Covariant canonical formulations of classical field theories

Covariant canonical formulations of classical field theories

A new canonical affine bracket formulation of Hamiltonian classical field theories of first order

Reductions: precontact versus presymplectic

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