Energy conservation does not uniquely determine equations of motion

For constant symmetric mass and stiffness matrices, $E=\frac{1}{2}{\dot{q}}^{\mathsf{T}}M\dot{q}+\frac{1}{2}{q}^{\mathsf{T}}Kq$ satisfies $\dot{E}={\dot{q}}^{\mathsf{T}}(M\ddot{q}+Kq)$. Every equation $M\ddot{q}+Kq=G\dot{q}$ with antisymmetric $G$ conserves this same mechanical energy, because ${\dot{q}}^{\mathsf{T}}G\dot{q}=0$. Thus a scalar conservation law alone cannot select the dynamics of several degrees of freedom. The specified conservative-force or Lagrangian structure supplies the missing information.