Angular momentum about the centre of mass

The derivative of angular momentum about the moving centre of mass equals the total external torque about that centre, because total relative momentum vanishes and internal central torques cancel.

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- Angular momentum decomposition about the centre of mass Angular momentum about the centre of mass

Angular momentum decomposition about the centre of mass ( $L = M X \times V + \sum_{i} m_{i} r_{i}^{'} \times v_{i}^{'}$ )

Write each position and velocity as its centre of mass value plus a relative value. Expanding the total angular momentum gives the displayed split into orbital angular momentum of the centre and internal angular momentum about it. The cross terms vanish because

\sum_{i} m_{i} r_{i}^{'} = \sum_{i} m_{i} v_{i}^{'} = 0

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