Geometric aspects of the isentropic liquid dynamics and vorticity invariants

Alexander A. Balinsky, Denis Blackmore, Radosław Kycia, Anatolij K. Prykarpatski

Research output: Contribution to journalReview articlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Article number1241
Pages (from-to)1-26
Number of pages26
Issue number11
StatePublished - Nov 2020

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Mathematical Physics
  • Physics and Astronomy (miscellaneous)
  • Electrical and Electronic Engineering


  • Charged liquid fluid dynamics
  • Diffeomorphism group
  • Hydrodynamic Euler equations
  • Isentropic hydrodynamic invariants
  • Lie-Poisson structure
  • Liquid flow
  • Symmetry reduction
  • Vortex invariants


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