Constrained second-variation stability in elasticity

For an equilibrium with $P_{Ii}=W_{,F_{iI}}+q{\varphi }_{,F_{iI}}$ and dead loads, the constrained second variation on ${\varphi }_{,F}:\delta F=0$ isIndeed, a constraint-preserving perturbation path has ${\varphi }_{,F}:{\delta }^{2}F=-{\varphi }_{,FF}[\delta F,\delta F]$, 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.