= Positive ideal energy of a theta pinch
{title2=$\delta W\ge0$}
For an admissible Fourier <fluid Lagrangian displacement> $\boldsymbol\xi(r)e^{im\theta+ikz}$, put $D_\perp=(r\xi_r)'/r+im\xi_\theta/r$ and $D=D_\perp+ik\xi_z$. Force balance cancels every magnetic-field-gradient cross term in the <magnetohydrodynamic energy principle>, leaving an integrand $\gamma p|D|^2+B^2[|D_\perp|^2+k^2(|\xi_r|^2+|\xi_\theta|^2)]/(4\pi)$. For $p\ge0$, $\gamma>0$, this is nonnegative. Neutral displacements can exist, so stability here means absence of negative-energy exponentially growing ideal modes, not strictly positive energy for every displacement.
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