= Constrained second-variation stability in elasticity
For an equilibrium with $P_{Ii}=W_{,F_{iI}}+q\phi_{,F_{iI}}$ and dead loads, the constrained <second variation> on $\phi_{,F}:\delta F=0$ is
$$
\mathcal Q[\delta F]=\int_\Omega(W_{,FF}+q\phi_{,FF})[\delta F,\delta F]\,dX.
$$
Indeed, a constraint-preserving perturbation path has $\phi_{,F}:\delta^2F=-\phi_{,FF}[\delta F,\delta F]$, and stationarity of the total potential supplies the multiplier term. Linearized kinetic energy plus $\mathcal Q/2$ is conserved. Positivity gives the usual energy-norm linear stability; control in a specified displacement norm additionally requires the corresponding coercivity and treatment of rigid-motion modes.
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