The Günther’s formalism in classical field theory: momentum map and reduction

Munteanu, Florian; Rey, Ángel Martín del; Salgado, Modesto

doi:10.1063/1.1688433

Cited by 47 publications

(99 citation statements)

References 21 publications

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“…Initiated by C. Gunther [12] and [13] based on n-symplectic model [14,15], it has been shown that the symplectic structure on the phase space remains true, if we replace the symplectic form by a vector valued form, that is called polysymplectic. This extension defines an action of G over…”

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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“…Concerning reduction theory in general, only partial results about reduction by foliations are currently being studied [49]. The corresponding reduction theorem has been stated and proved for the k-symplectic formulation [84], but the theory of reduction of multisymplectic Lagrangian and Hamiltonian systems under the action of groups of symmetries is still under research, and only partial results have been achieved [17,18,19,81].…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…For instance, we have the so-called k-symplectic formalism which uses the k-symplectic forms introduced by Awane [4,5,6], and which coincides with the polysymplectic formalism described by Günther [46] (see also [84]). A natural extension of this is the k-cosymplectic formalism, which is the generalization to field theories of the cosymplectic description of non-autonomous mechanical systems [75,76].…”

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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“…First, several Hamiltonian models can be stated, and the equivalence among them is not always clear (see, for instance, [2], [12], [19], [21], [22], [23], [40], [43]). Furthermore, there are equivalent Lagrangian models with non-equivalent Hamiltonian descriptions [26], [27], [28].…”

Section: Introductionmentioning

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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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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The Günther’s formalism in classical field theory: momentum map and reduction

Abstract: The polysymplectic formalism in local field theory, developed by Günther [J. Diff. Geom. 25, 23 (1987)], is revised. A new approach and new results on momentum maps and reduction are given.

Cited by 47 publications

References 21 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

Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories

Extended Hamiltonian systems in multisymplectic field theories

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