  The dynamics of an incompressible, dissipationless Hall magnetohydrodynamic
medium are investigated from Lagrangian mechanical viewpoint. The hybrid and
magnetic helicities are shown to emerge, respectively, from the application of
the particle relabeling symmetry for ion and electron flows to Noether's first
theorem, while the constant of motion associated with the theorem is generally
given by their arbitrary linear combination. Furthermore, integral path
variation associated with the invariant action is expressed by the operation of
an integro-differential operator on the reference path. The eigenfunctions of
this operator are double Beltrami flows, i.e. force-free stationary solutions
to the equation of motion and provide a family of orthogonal function bases
that yields the spectral representation of the equation of motion with a
remarkably simple form. Among the double Beltrami flows, considering the
influence of a uniform background magnetic field and the Hall term effect
vanishing limit, the generalized Elsasser variables are found to be the most
suitable for avoiding problems with singularities in the standard
magnetohydrodynamic limit.
