Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Entropy

Denis Blackmore¹  AnatolijK. Prykarpatski²  Radosław Kycia³  AlexanderA. Balinsky⁴ 

[1] Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA;Department of Physics, Mathematics and Computer Science, Cracov University of Technology, 31-155 Krakow, Poland;Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, 31-155 Cracow, Poland;School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK;

关键词: liquid flow;    hydrodynamic Euler equations;    diffeomorphism group;    Lie-Poisson structure;    isentropic hydrodynamic invariants;    vortex invariants;   

DOI  :  10.3390/e22111241

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown