Balinsky Alexander A, Blackmore Denis, Kycia Radosław, Prykarpatski Anatolij K
School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK.
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA.
Entropy (Basel). 2020 Oct 31;22(11):1241. doi: 10.3390/e22111241.
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.
我们回顾了流体等熵运动的现代微分几何描述及其特征，包括微分同胚群结构、对相关动力学进行建模，以及它与准静态热力学约束的兼容性。我们分析了绝热液体动力学，在此框架内，按照一般方法，详细解释了流体运动相空间上相关泊松结构的性质，它是半直积巴拿赫群，以及其余切空间上的典范辛结构自然约化为相应半直李代数积的伴随空间上的经典李 - 泊松括号。我们还给出了等温液体动力学所支配流动情况下哈密顿分析的一种修正。我们研究了等熵磁流体动力学超流体相空间的微分几何结构及其在哈密顿分析和相关不变量理论中的相关运动。特别地，我们构造了不同类型的积分磁流体动力学不变量的无穷层次，推广了文献中先前构造的那些不变量，并分析了它们的微分几何起源。我们还研究了关于阿贝尔规范群变换不变的相空间上的带电液体动力学，并给出了典范李 - 泊松型括号的一些推广。
