Particle-relabeling symmetry, generalized vorticity, and normal-mode expansion of ideal incompressible fluids and plasmas in three-dimensional space

Abstract

The Lagrangian mechanical consideration of the dynamics of ideal incompressible hydrodynamic, magnetohydrodynamic, and Hall magnetohydrodynamic media, which are formulated as dynamical systems in appropriate Lie groups equipped with Riemannian metrics, leads to the notion of generalized vorticities, as well as generalized coordinates, velocities, and momenta. The action of each system is conserved against the integral path variation in the direction of the generalized vorticity, and this invariance is associated with the particle relabeling symmetry. The generalized vorticities are formulated by the operation of integro-differential operators on the generalized velocities. The eigenfunctions of the operators provide sets of orthogonal functions, and we obtain common mathematical expressions concerning these dynamical systems using the orthogonoal functions. In particular, we find that the product of the Riemannian metric, $g_{im}$, and the structure constants of the Lie group, $C_{jk}^m$, is given by the product of the eigenvalue of the operator, $\Lambda (i)$, and a certain totally antisymmetric tensor, $T_{ijk}$: $g_{i\alpha }{C}_{jk}^{\alpha }=\Lambda (i)T_{ijk}.$ Its physical implications, including the weak interaction conjecture of MHD turbulence, are also discussed.

14 pages, 2 figures. The manuscript is fully revised. Consideration about the influence of external uniform fields and the discussion of physical implications of our findings are added

Cited by in corpus (3)

• Derivation of the Hall and Extended Magnetohydrodynamics Brackets
• On Magnetohydrodynamic Gauge Field Theory
• Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: II. Geodesic formulation and Riemannian curvature analysis of hydrodynamic and magnetohydrodynamic stabilities