A canonical structure for classical field theories

Kijowski, Jerzy; Szczyrba, Wiktor

doi:10.1007/bf01608496

Cited by 118 publications

(115 citation statements)

References 9 publications

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“…We can also formalize the Multisymplectic Geometry with an extension of the Poincaré-Cartan invariant integrals. Frédéric Hélein has observed the fact that different theories could cohabitate was considered jointly by T. Lepage [97], P. Dedecker [98][99] and J. Kijowski [32][33][34]. The Lepage-Dedecker theory was developed by F. Hélein [101], and the modern formulation using the multisymplectic (n + 1)-form as the fundamental structure of the theory starts with J. Kijowski papers.…”

Section: R M a H Rmentioning

confidence: 99%

Higher Order Geometric Theory of Information and Heat based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures

Barbaresco¹

2018

Preprint

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We introduce poly-symplectic extension of Souriau Lie groups Thermodynamics based on higher-order model of statistical physics introduced by R.S. Ingarden. This extended model could be used for small data analytics and Machine Learning on Lie groups. Souriau Geometric Theory of Heat is well adapted to describe density of probability (Maximum Entropy Gibbs density) of data living on groups or on homogeneous manifolds. For Small Data Analytics (Rarified Gases , sparse statistical survey,…), density of maximum entropy should consider Higher Order Moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by R.S. Ingarden. We use Poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of momentum mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of Ghomogeneous systems. The formalism is covariant, i.e. no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures.Keywords: Higher Order Thermodynamics; Lie Groups Thermodynamics; Homogeneous Manifold; Poly-Symplectic Manifold; Dynamical Systems; Non-equivariant Cohomology « … inviter les savants géomètres à traiter nos problèmes avec le soucis de la commodité et de l'agrément : qu'ils écartent tout ce qui n'a rien à voir avec la pénétration de l'esprit, seule qualité dont nous faisons grand cas et que nous nous sommes proposé d'éprouver et de couronner » -Blaise Pascal -Deuxième Lettre sur la roulette, Paris, 19 Juillet 1658 « Nous avons fait de la Dynamique un cas particulier de la Thermodynamique, une Science qui embrasse dans des principes communs tous les changements d'état des corps, aussi bien les changements de lieu que les changements de qualités physiques » -Pierre Duhem, Sur les équations générales de la Thermodynamique, 1891 [25] « Nous prenons le mot mouvement pour désigner non seulement un changement de position dans l'espace, mais encore un changement d'état quelconque, lors même qu'il ne serait accompagné d'aucun déplacement… De la sorte, le mot mouvement s'oppose non pas au mot repos, mais au mot équilibre. » -Pierre Duhem, Commentaire aux principes de la Thermodynamique, 1894 [26, 27, 115]

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Section: R M a H Rmentioning

confidence: 99%

Higher Order Geometric Theory of Information and Heat based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures

Barbaresco¹

2018

Preprint

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“…The analysis of this problem can be dealt by looking for all vector fields ξ 0 = X µ (x, φ, e, p) ∂ ∂x µ + Φ a (x, φ, e, p) ∂ ∂φ a + E(x, φ, e, p) ∂ ∂e + P satisfying (24) and (25). For simplicity we will assume that X µ = 0 (this will exclude stress-energy tensor observable forms X µ ∂ ∂x µ θ 0 , for X µ constant).…”

Section: Example 8 (Complex Scalar Fields)mentioning

confidence: 99%

The Notion of Observable in the Covariant Hamiltonian Formalism for the Calculus of Variations with Several Variables

Hélein¹,

Kouneiher²

2004

Advances in Theoretical and Mathematical Physics

113

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This papers is concerned with multisymplectic formalisms which are the frameworks for Hamiltonian theories for fields theory. Our main purpose is to study the observable (n − 1)-forms which allows one to construct observable functionals on the set of solutions of the Hamilton equations by integration. We develop here two different points of view: generalizing the law {p, q} = 1 or the law dF/dt = {H, F }. This leads to two possible definitions; we explore the relationships and the differences between these two concepts. We show that -in contrast with the de Donder-Weyl theory -the two definitions coincides in the Lepage-Dedecker theory.e-print archive:

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“…In the Lagrangian framework, a variety of approaches have been proposed: pseudotensors [9], Komar integrals [10], Noether's method applied to linearized field equations [11,12,13], a quasi-local approach [14,15] and covariant phase space methods [16,17,18,19,20,21,22,23]. Recent related work can be found for instance in [24,25,26,27,28,29,30,31,32,33].…”

Section: Introductionmentioning

confidence: 99%

Surface charge algebra in gauge theories and thermodynamic integrability

Barnich¹,

Compère²

2008

Journal of Mathematical Physics

278

468

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ABSTRACT. Surface charges and their algebra in interacting Lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus. In the case of exact solutions and symmetries, the surface charges are interpreted as a Pfaff system. Integrability is governed by Frobenius' theorem and the charges associated with the derived symmetry algebra are shown to vanish. Surfaces charges reproduce well-known Hamiltonian and covariant phase space expressions. In the asymptotic context, we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended. Comparision with Hamiltonian and covariant phase space methods are done in appendix.

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A canonical structure for classical field theories

Cited by 118 publications

References 9 publications

Higher Order Geometric Theory of Information and Heat based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures

Higher Order Geometric Theory of Information and Heat based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures

The Notion of Observable in the Covariant Hamiltonian Formalism for the Calculus of Variations with Several Variables

Surface charge algebra in gauge theories and thermodynamic integrability

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