= Energy conservation does not uniquely determine equations of motion
For constant symmetric mass and stiffness matrices, $E=\tfrac12\dot q^{\mathsf T}M\dot q+\tfrac12q^{\mathsf T}Kq$ 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.
Back to article page
