The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808–1810

Marle, Charles‐Michel

doi:10.1007/s11005-009-0347-y

Cited by 15 publications

(6 citation statements)

References 9 publications

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“…The symplectic structure has been introduced in mathematics much earlier than the word symplectic, in works of the French physicist Joseph Louis Lagrange (see paper on the slow changes of the orbital elements of planets in the solar system), who showed that this geometry is a fundamental tool in the mathematical model of any problem in mechanics. Jean-Marie Souriau has shown that Lagrange’s parentheses (nowdays called Lagranges bracket) are the components of the canonical symplectic 2-form on the manifold of motions of the mechanical system, in the chart of that manifold [ 60 , 61 ].…”

Section: Seminal Idea Of Symplectic Geometry In Mechanics and In Smentioning

confidence: 99%

Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning

Barbaresco

2018

Entropy

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We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by 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 surveys, …), the 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 Ingarden. We use a 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 moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous 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. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology).

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Section: Seminal Idea Of Symplectic Geometry In Mechanics and In Smentioning

confidence: 99%

Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning

Barbaresco

2018

Entropy

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“…Jean-Marie Souriau has shown that Lagrange's parentheses are the components of the canonical symplectic 2-form on the manifold of motions of the mechanical system, in the chart of that manifold [45][46].…”

mentioning

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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“…In many texts, such as Arnold's book [7], the quantities A(γ) are called Poincaré's relative integral invariants. The invariance of A(γ) under Hamiltonian flows was known to Lagrange, who also knew of Hamilton's equations, the symplectic form, and Darboux's theorem; see [6, p. 273] and [75,99]. This is in accordance with Arnold's Principle that mathematical results are almost never called by the names of their discoverers.…”

Section: Meanings Of "Symplectic"mentioning

confidence: 83%

Untitled

2018

Bull. Amer. Math. Soc.

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We describe old and new motivations to study symplectic embedding problems, and we discuss a few of the many old and the many new results on symplectic embeddings. Contents 1. Introduction 2. Meanings of "symplectic" 3. From Newtonian mechanics to symplectic geometry 4. Why study symplectic embedding problems 5. Euclidean symplectic volume preserving 6. Symplectic ellipsoids 7. The role of J-holomorphic curves 8. The fine structure of symplectic rigidity 9. Packing flexibility for linear tori 10. Intermediate symplectic capacities or shadows do not exist 1 2 3

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The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808–1810

Abstract: We analyse articles by Lagrange and Poisson written two hundred years ago which are the foundation of present-day symplectic and Poisson geometry.

Cited by 15 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: The Bedrock for Lie Group Machine Learning

Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning

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

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