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

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, $C^{m}_{jk}$, 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 also discussed.