12.3: Conservation of an angular momentum
A special situation is the case where the external angular impulse \(\overrightarrow{\boldsymbol{H}}_{\text {ang }, P}\) is zero. We substitute this condition in Equation 12.4 and obtain that angular momentum does not change in this situation:
If the total angular impulse generated by external moments on a system with respect to a fixed reference point \(P\) is zero \(\left(\overrightarrow{\boldsymbol{H}}_{\mathrm{ang}, P}=\overrightarrow{\mathbf{0}}\right)\), then the total angular momentum of the system is conserved, and does not change in time.
\[\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{1}\right)=\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{2}\right) \tag{12.7} \label{12.7}\]
Whether or not conservation of angular momentum occurs can depend on the choice of the reference point \(P\). For example when a single external force \(\overrightarrow{\boldsymbol{F}}_{i}\) acts at a fixed point of action \(i\) on a system, the reference point \(P\) can be chosen at point \(i\), or at any other point on the line of action of force \(\overrightarrow{\boldsymbol{F}}_{i}\), to ensure that \(\overrightarrow{\boldsymbol{H}}_{\text {ang }, P}=\overrightarrow{\mathbf{0}}\) and the law of conservation of angular momentum holds.
An important case for conservation of angular momentum are systems on which there are no external moments and forces acting because in those systems the law of conservation of angular momentum holds for any choice of reference point \(P\).
It is important to note that when analysing systems, the boundaries of the system can be chosen arbitrarily. By expanding the system boundaries to include all objects that generate forces, all forces will become internal forces,
and conservation of (angular) momentum for this expanded system will hold. In the extreme case, the momentum and angular momentum of our whole universe are expected to be constant, since there are no external forces and moments acting on it, such that the law of conservation of momentum and angular momentum hold.