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Geometric aspects of the isentropic liquid dynamics and vorticity invariants

Balinsky, Alexander A., Blackmore, Denis, Kycia, Rados?aw and Prykarpatski, Anatolij K. 2020. Geometric aspects of the isentropic liquid dynamics and vorticity invariants. Entropy 22 (11) , 1241. 10.3390/e22111241

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We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented. Keywords: liquid flow; hydrodynamic Euler equations; diffeomorphism group; Lie-Poisson structure; isentropic hydrodynamic invariants; vortex invariants; charged liquid fluid dynamics; symmetry reduction

Item Type: Article
Date Type: Published Online
Status: Published
Schools: Mathematics
Additional Information: Attribution 4.0 International (CC BY 4.0)
Publisher: MDPI
ISSN: 1099-4300
Date of First Compliant Deposit: 2 November 2020
Date of Acceptance: 26 October 2020
Last Modified: 03 Nov 2020 09:00

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