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Open AccessFeature PaperArticle

Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Department of Mathematics, Gebze Technical University, 41400 Gebze-Kocaeli, Turkey
École Polytechnique de Montréal, C.P.6079 suc. Centre-ville, Montréal, QC H3C 3A7, Canada
Department of Mathematics, Yeditepe University Atasehir, 34755 Istanbul, Turkey
Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague, Czech Republic
Authors to whom correspondence should be addressed.
Entropy 2019, 21(9), 907;
Received: 5 August 2019 / Revised: 10 September 2019 / Accepted: 13 September 2019 / Published: 18 September 2019
(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)
Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. This pathway is free from Poisson brackets and Hamiltonian functionals. Momentum realizations (sections on T * T * Q ) of (both compressible and incompressible) Euler’s fluid and Vlasov’s plasma are derived. Poisson mappings relating the momentum realizations with the usual field equations are constructed as duals of injective Lie algebra homomorphisms. The geometric pathway is then used to construct the evolution equations for 10-moments kinetic theory. This way the entire Grad hierarchy (including entropic fields) can be constructed in a purely geometric way. This geometric way is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets. View Full-Text
Keywords: diffeomorphisms group; cotangent lift; vertical representative; fluids; kinetic theory; entropy diffeomorphisms group; cotangent lift; vertical representative; fluids; kinetic theory; entropy
MDPI and ACS Style

Esen, O.; Grmela, M.; Gümral, H.; Pavelka, M. Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy. Entropy 2019, 21, 907.

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