# Conservation of angular momentum isn’t [very] fundamental!

What are the conservation principles (in physics)?

In the first course on engineering mechanics (i.e. the mechanics of rigid bodies) we are taught that there are these three conservation principles: Conservation of: (i) energy, (ii) momentum, and (iii) angular momentum. [I am talking about engineering programs. That means, we live entirely in a Euclidean, non-relativistic, world.]

Then we learn mechanics of fluids, and the conservation of (iv) mass too gets added. That makes it four.

Then we come to computational fluid dynamics (CFD), and we begin to deal with only three equations: conservation of (i) mass, (ii) momentum, and (iii) energy. What happens to the conservation of the angular momentum? Why does the course on CFD drop it? For simplicity of analysis?

Ask that question to postgraduate engineers, even those who have done a specialization in CFD, and chances are, a significant number of them won’t be able to answer that question in a very clear manner.

Some of them may attempt this line of reasoning: That’s because in deriving the fluids equations (whether for a Newtonian fluid or a non-Newtonian one), the stress tensor is already assumed to be symmetrical: the shear stresses acting on the adjacent faces are taken to be equal and opposite (e.g. $\sigma_{xy} = \sigma_{yx}$). The assumed equality can come about only after assuming conservation of the angular momentum, and thus, the principle is already embedded in the momentum equations, as they are stated in CFD.

If so, ask them: How about a finite rotating body—say a gyroscope? (Assume rigidity for convenience, if you wish.) Chances are, a great majority of them will immediately agree that in this case, however, we have to apply the angular momentum principle separately.

Why is there this difference between the fluids and the finite rotating bodies? After all, both are continua, as in contrast to point-particles.

Most of them would fall silent at this point. [If not, know that you are talking with someone who knows his mechanics well!]

Actually, it so turns out that in continua, the angular momentum is an emergent/derivative property—not the most fundamental one. In continua, it’s OK to assume conservation of just the linear momentum alone. If it is satisfied, the conservation of angular momentum will get satisfied automatically. Yes, even in case of a spinning wheel.

Don’t believe me?

Let me direct you to Chad Orzel; check out here [^]. Orzel writes:

[The spinning wheel] “is a classical system, so all of its dynamics need to be contained within Newton’s Laws. Which means it ought to be possible to look at how angular momentum comes out of the ordinary linear momentum and forces of the components making up the wheel. Of course, it’s kind of hard to see how this works, but that’s what we have computers for.” [Emphasis in italics is mine.]

He proceeds to put together a simple demo in Python. Then, he also expands on it further, here [^].

Cool. If you think you have understood Orzel’s argument well, answer this [admittedly deceptive] question: How about point particles? Do we need a separate conservation principle for the angular momentum, in addition to that for the linear momentum at least in their case? How about the earth and the moon system, granted that both can be idealized as point particles (the way Newton did)?

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Lyrics: M. G. Hashmat

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