Positive ideal energy of a theta pinch ($\delta W\ge 0$)

For an admissible Fourier fluid Lagrangian displacement $\mathbf{\xi }(r)e^{im\theta +ikz}$, put $D_{\perp }=(r{\xi }_r{)}^{\prime }/r+im{\xi }_{\theta }/r$ 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 $p\ge 0$, $\gamma >0$, 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.