= Magnetohydrodynamic energy principle
For a static smooth <ideal magnetohydrodynamics> equilibrium with conservative external forces and surface terms eliminated by the <boundary conditions>, the linear displacement equation is $\rho\ddot{\boldsymbol\xi}=\mathcal F\boldsymbol\xi$, where the force operator is a <self-adjoint operator> in the appropriate integral pairing. Its restoring <quadratic form> $\delta W=-\tfrac12\int\boldsymbol\xi^*\cdot\mathcal F\boldsymbol\xi\,dV$ is real. Multiplying the <normal mode> equation by the <complex conjugate> of the <displacement> and integrating gives $\omega^2\int\rho|\boldsymbol\xi|^2dV=2\delta W$. Hence squared frequencies are real, and a negative restoring form gives an unstable mode through the variational <Rayleigh quotient>, subject to the operator domain and spectral hypotheses. Background <fluid flow> adds terms to this argument; the static hypothesis is essential.
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