Angular Momentum

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Section 21.5 Angular Momentum

Consider the angular momentum of the reduced mass system \(\LLv=\rr\times\pp=\rr\times\mu\vv\text{.}\) How does \(\LLv\) change with time? We have:

\begin{align} \frac{d\LLv}{dt} \amp= \frac{d}{dt}\left(\rr\times \mu\vv\right)\tag{21.5.1}\\ \amp= \rr\times \mu\dot{\vv}+\cancelto{0}{\vv\times \mu\vv}\tag{21.5.2}\\ \amp= \rr\times \mu\aa\tag{21.5.3}\\ \amp= \rr\times\FF\tag{21.5.4}\\ \amp= \cancelto{0}{r\rhat\times f(r)\rhat}\tag{21.5.5}\\ \amp= 0\tag{21.5.6} \end{align}

To get from (21.5.1) to (21.5.2), use the product rule, which is valid for cross products as long as you don’t change the order of the factors. The second term in (21.5.2) is zero since \(\vv\times\vv=0\text{.}\) To get from (21.5.4) to (21.5.5), impose the assumption that the force is a central force. The expression (21.5.5) is zero because \(r\rhat\times \rhat=0\text{.}\)

Question 21.6.

Which of the steps in (21.5.1)-(21.5.6) are valid only for central forces and which are true more generally?

Definition 21.7. Torque.

Recall that \(\rr\times\FF\) which occurs in (21.5.4) is called the torque \(\tauv\text{.}\)

To Remember.

We have shown that in the case of central forces the time derivative of the angular momentum, and hence the torque, is zero. Therefore:

\begin{equation} \tauv = \frac{d\LLv}{dt}=0 \qquad\Longrightarrow\qquad \LLv = \hbox{constant}\tag{21.5.7} \end{equation}

Angular momentum is conserved in central force motion.

The central force \(\FF(r)\) depends only on the distance of the reduced mass from the center of mass and not on the orientation of the system in space. Therefore, this system is spherically symmetric; it is invariant (unchanged) under rotations. Noether’s theorem states that whenever the laws of physics are invariant under a particular motion or other operation, there will be a corresponding conserved quantity. In this case, we see that the conservation of angular momentum is related to the invariance of the physical situation under rotations. Noether’s theorem, in general, is most easily discussed using Lagrangian techniques.