On the Lagrange and Euler variational formulations of linear ideal magnetohydrodynamics

Both the Lagrangian and Eulerian variational descriptions of the flow field are employed in classical fluid mechanics. In a magnetically confined plasma, the variational formulations become more complicated due to the presence of a magnetic field and the so-called frozen-in phenomenon. The plasma compression or expansion comes along with the compression or expansion of the magnetic field. In this work, the Lagrange and Euler variational formulations of linear ideal magnetohydrodynamics (MHD) are further studied. It emphasizes examining the boundary contributions between two different variation descriptions. Although the bulk energies in the two variational descriptions remain the same, the boundary contributions differ due to the different interactions with the surrounding medium. The results not only extend the fundamental theoretical framework of magnetohydrodynamics but also are useful for analytical studies and numerical simulations of the free boundary modes, such as external kink modes and resistive wall modes.