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The dynamics of an incompressible, dissipationless Hall magnetohydrodynamicmedium are investigated from Lagrangian mechanical viewpoint. The hybrid andmagnetic helicities are shown to emerge, respectively, from the application ofthe particle relabeling symmetry for ion and electron flows to Noether's firsttheorem, while the constant of motion associated with the theorem is generallygiven by their arbitrary linear combination. Furthermore, integral pathvariation associated with the invariant action is expressed by the operation ofan integro-differential operator on the reference path. The eigenfunctions ofthis operator are double Beltrami flows, i.e. force-free stationary solutionsto the equation of motion and provide a family of orthogonal function basesthat yields the spectral representation of the equation of motion with aremarkably simple form. Among the double Beltrami flows, considering theinfluence of a uniform background magnetic field and the Hall term effectvanishing limit, the generalized Elsasser variables are found to be the mostsuitable for avoiding problems with singularities in the standardmagnetohydrodynamic limit.

Helicity-based, particle-relabeling operator and normal mode expansion of the dissipationless incompressible Hall magnetohydrodynamics