Elastic energy uniqueness with restoring boundary springs ($E=\frac{1}{2}{\int }_D(\rho |\dot{w}{|}^2+Ce:e)dV-\frac{1}{2}{\int }_{S}w\cdot kwdS$)

For a time-independent symmetric negative semidefinite boundary matrix $k$, the difference of two linear elasticity solutions with the same data has traction $kw$. Integration by parts conserves the displayed nonnegative energy. Zero initial data force zero velocity and then zero displacement. The matrix $-k$ is the stiffness of boundary springs to a prescribed support; null directions are traction-free.