= Stellar displacement energy identity
{title2=$\omega^2\int\rho|\boldsymbol\xi|^2=-\frac1{4\pi G}\int|\nabla\phi_1|^2+\int\left(\frac{|p_1|^2}{\gamma p}+\rho N^2|\xi_r|^2\right)$}
For a nonrotating self-gravitating star, let $p_1,\rho_1,\phi_1$ be Eulerian perturbations and $\Delta p=-\gamma p\nabla\cdot\boldsymbol\xi$ the adiabatic Lagrangian perturbation. Integration of the linear momentum equation against $\boldsymbol\xi^*$, using hydrostatic balance and the perturbed <Poisson equation>, yields the displayed <Rayleigh quotient>. The gravitational integral is over all space and the other integrals over the star. The compression denominator is $\gamma p$, not $\gamma\rho$. Surface terms vanish for compactly supported trials or the physical free-surface and exterior potential conditions.
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