A Note on Stability Analysis by the Second Variations of Lagrangian and Hamiltonian for Ideal Incompressible Plasmas

Abstract

From the viewpoint of the differential geometrical approach to the Lagrangian mechanical variational problem, a proof is presented for a general conservation law that the Lagrangian displacement type perturbation field satisfies around a stationary solution, previously derived by Hirota et al. for the Hall magnetohydrodynamic system. Additionally, this mathematical approach is applied to the Hamiltonian mechanical stability analyses of the magnetohydrodynamic and Hall magnetohydrodynamic systems.