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

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