Center-of-mass equation ($M{\ddot{\mathbf{r}}}_{\mathrm{CM}}=\sum {\mathbf{F}}_{\mathrm{ext}}$)

For a closed system with fixed total mass, summing the constituent Newton's second law equations makes pairwise internal forces cancel when Newton's third law applies. The remaining resultant external force is the derivative of total momentum, equal to $M{\ddot{\mathbf{r}}}_{\mathrm{CM}}$. This governs translation of the center of mass; external torque additionally governs changes of angular momentum.