= Cowling energy principle for a rotating barotropic star
{c}
{title2=$\omega^2\int\rho|\xi|^2dV=\int[|\delta p|^2/(\gamma p)+\rho\mathcal N^2|\xi\cdot\nabla\Psi|^2+\rho\kappa^2|\xi_R|^2]dV$}
With vanishing surface <pressure>/<mass density> and regular admissible displacements, <integration by parts> makes the <axisymmetric adiabatic displacement operator> symmetric in the <mass density>-weighted <inner product>. Completing the <pressure> square gives the displayed energy form. Nonnegative form for every admissible displacement excludes exponentially growing axisymmetric adiabatic modes in the <Cowling approximation>. A negative trial <Rayleigh quotient> proves negative spectrum by the <Rayleigh-Ritz variational principle>. Nonnegative stratification and epicyclic coefficients give a simple sufficient stability condition.
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