Preprint Review Version 1 Preserved in Portico This version is not peer-reviewed

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Version 1 : Received: 20 September 2020 / Approved: 23 September 2020 / Online: 23 September 2020 (03:45:21 CEST)

How to cite: Balinsky, A.A.; Blackmore, D.; Kycia, R.; Prykarpatski, A.K. Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants. Preprints 2020, 2020090530 (doi: 10.20944/preprints202009.0530.v1). Balinsky, A.A.; Blackmore, D.; Kycia, R.; Prykarpatski, A.K. Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants. Preprints 2020, 2020090530 (doi: 10.20944/preprints202009.0530.v1).

Abstract

We review a modern differential geometric description of the fluid isotropic motion and featuring it the diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. There is analyzed the adiabatic liquid dynamics, within which, following the general approach, there is explained in detail, 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. We also present a modification of the Hamiltonian analysis in case of the isotermal liquid dynamics. We study the differential-geometric structure of the adiabatic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and invariant theory. In particular, we construct an infinite hierarchies of different kinds of integral magneto-hydrodynamic invariants, generalizing those before constructed in the literature, and analyze their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, some generalization of the canonical Lie-Poisson type bracket is obtained.

Subject Areas

liquid flow; hydrodynamic Euler equations; diffeomorphism group; Lie-Poisson structure; isentropic hydrodynmaic invarinats; vortex invariants; charged liquid fluid dynamics; symmetry reduction

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