= Elastic normal-mode energy identity
{title2=$\omega^2\int\rho_0|u|^2=\int\overline{u_{i,\alpha}}c_{\alpha i\beta j}u_{j,\beta}$}
For a static base state with zero incremental <body force>, homogeneous incremental <tractions> or homogeneous <Dirichlet boundary conditions> eliminate the boundary term in <integration by parts>. The displayed <Rayleigh quotient> then gives real squared frequencies because of major symmetry. Positive moduli give strictly positive squared frequencies for modes with nonzero displacement gradient. Pure <dead loading> leaves constant translations at zero; fixed displacement conditions remove those modes.
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