  The formal stability analysis of Eulerian extended magnetohydrodynamics
(XMHD) equilibria is considered within the noncanonical Hamiltonian framework
by means of the energy-Casimir variational principle and the dynamically
accessible stability method. Specifically, we find explicit sufficient
stability conditions for axisymmetric XMHD and Hall MHD (HMHD) equilibria with
toroidal flow and for equilibria with arbitrary flow under constrained
perturbations. The dynamically accessible, second-order variation of the
Hamiltonian, which can potentially provide explicit stability criteria for
generic equilibria, is also obtained. Moreover, we examine the Lagrangian
stability of the general quasineutral two-fluid model written in terms of
MHD-like variables, by finding the action and the Hamiltonian functionals of
the linearized dynamics, working within a mixed Lagrangian-Eulerian framework.
Upon neglecting electron mass, we derive a HMHD energy principle, and in
addition, the perturbed induction equation arises from Hamilton's equations of
motion in view of a consistency condition for the relation between the
perturbed magnetic potential and the canonical variables.

Plasma Physics;Fluid Dynamics