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The dynamics of a dissipationless incompressible Hall magnetohydrodynamic(HMHD) medium is formulated using Lagrangian mechanics on a semidirect productof two volume preserving diffeomorphism groups. In the case of $\mathbb{T}^3$or $E^3$, the generalized Elsasser variables introduced by Galtier (S. Galtier2006 J. Plasma Phys. 72 721-769) yield remarkably simple expressions of basicformulas and equations such as the structure constants of Lie algebra, theequation of motion, and the conservation laws. Four constants of motion, wherethree of the four are independent, are naturally derived from the generalizedElsasser variables representation of the equation of motion for the HMHDsystem: total plasma energy, magnetic helicity, hybrid helicity, and themodified cross helicity.

journal of physics. a, mathematical and theoretical

Differential-geometrical approach to the dynamics of dissipationless  incompressible Hall magnetohydrodynamics I: Lagrangian mechanics on  semidirect product of two volume preserving diffeomorphisms and conservation  laws