Conservation of angular momentum and central forces

We review the statement and derivation of the law of conservation of angular momentum of a system of particles.

$$\textit{Theorem:}$$ Consider the system of particles $$S := \{ P_i | P_i \; \text{is a material particle,} \; 1 \leq i \leq n \}$$ with the mass of particle $$i$$, the net force, the net force due to external entities and the force due to particle $$j$$ on the particle $$i$$ denoted by $$m_i$$, $$\vec{f}_i$$, $$\vec{f}_{i\;ext}$$ and $$\vec{f}_{ij}$$ respectively. Let $$v_i^I := \frac{d^I}{dt} \vec{s}_{P_i O},$$ $$\vec{G}_O^{SI} := \vec{s}_{P_i O} \times (m_i v_i^I),$$ $$\frac{d^I}{dt} \vec{G}_O^{SI} := \vec{H}_O^{SI} = \sum_{i=1}^n \vec{s}_{P_i O} \times \vec{f}_{i}$$ (the final equality follows on applying Newton's second law of motion) denote the velocity of particle $$i$$ calculated using the inertial reference frame $$I$$, the angular momentum of the system $$S$$ referred to the point $$O$$ calculated using the inertial reference frame $$I$$ and the moment of forces acting on the system $$S$$ referred to point $$O$$ using the reference frame $$I$$ respectively, where we denote the displacement vector of point $$A$$ with respect to point $$B$$ by $$\vec{s}_{AB}$$ and the summation operator $$\sum_i := \sum_{i=1}^n$$. The law of conservation of angular momentum states that, if $$\vec{f}_{ii} = 0$$ and $$\vec{f}_{i \; ext} = 0$$ for all $$i$$, then $$\frac{d^I}{dt} \vec{G}_O^{SI} = \vec{H}_O^{SI} \\= \sum_i \vec{s}_{P_i O} \times \vec{f}_{i\;ext} = 0.$$

$$\textit{Proof:}$$ To prove the conservation law, we first observe that $$\vec{H}_O^{SI} = \sum_i \vec{s}_{P_i O} \times \vec{f}_{i} = \sum_i \vec{s}_{P_i O} \times \vec{f}_{i\;ext} + \sum_i \vec{s}_{P_i O} \times \sum_j \vec{f}_{ij}.$$ Next, recalling our assumption that $$\vec{f}_{ii} = 0$$ for all $$i$$ and applying Newton's third law of motion $$\vec{f}_{ij} = - \vec{f}_{ji}$$ for all $$i \neq j$$, the latter term $$\sum_i \sum_j \vec{s}_{P_i O} \times \vec{f}_{ij}$$ can be written as the sum, with $$i \neq j$$, $$\frac{1}{2} \sum_i \sum_j \vec{s}_{P_i O} \times \vec{f}_{ij} + \vec{s}_{P_j O} \times \vec{f}_{ji} \\= \frac{1}{2} \sum_i \sum_j \vec{s}_{P_i O} \times \vec{f}_{ij} - \vec{s}_{P_j O} \times \vec{f}_{ij} \\= \frac{1}{2} \sum_i \sum_j (\vec{s}_{P_i O} - \vec{s}_{P_j O}) \times \vec{f}_{ij} \\= \frac{1}{2} \sum_i \sum_j \vec{s}_{P_i P_j} \times \vec{f}_{ij}.$$ The proof is completed if the expression which we are summing vanishes. In this expression, we observe that

• if the particles $$P_i$$ and $$P_{j}$$ have the same location, that is to say that $$\vec{f}_{ij}$$, $$\vec{f}_{ji}$$ are forces between $$P_i$$ and $$P_j$$ at the same location $$P_i \equiv P_j$$ then $$\vec{s}_{P_i P_j} = 0$$
• this implication shows that the contribution to angular moment acting on $$S$$ due to internal (non-conservative) contact forces (similar to friction) vanishes
• if $$0 < \|\vec{s}_{P_i P_j}\|$$ then $$\vec{f}_{ij}$$ is acted on $$P_i$$ by $$P_j$$ at a distance and on further assuming that they are central forces represented as $$\vec{f}_{ij} = -k(\|\vec{s}_{P_i P_j}\|) \vec{s}_{P_i P_j}$$, then $$\vec{s}_{P_i P_j} \times \vec{f}_{ij} = 0$$
• this implication shows that the contribution of angular moment acting on $$S$$ due to internal (conservative) central forces vanishes

Since all the constituents of the $$i, j$$ pair expression vanish, this completes the proof on observing that the expression for $$\vec{G}_O^{SI}$$ is equal to $$\sum_i \vec{s}_{P_i O} \times \vec{f}_{i\;ext}$$.

• In the proof above, it was assumed that forces which act at a distance, acted on a particle by another in the system $$S$$ are central forces. Is this assumption required to prove the law of conservation of angular momentum of the system $$S$$?
• Why is this assumption not stated explicitly in the textbooks on classical mechanics?
• If this is indeed a required assumption, does it mean that material particles behave in this manner? That is to say, forces-at-a-distance between two material particles act along the line connecting them, while forces-at-location or contact forces (such as friction) do not have this constraint.
• Is the above assumption not a comment on the fundamental physical nature of force? As such, does the evidence support this (perhaps empirical) view?
• That formatting is very hard on my eyes, but I suspect the result you're looking for is Noether's theorem. Jan 21 '21 at 4:55
• The analysis you are looking for is made in the references given here: physics.stackexchange.com/a/603347/132418 especially the one by Truesdell Jan 21 '21 at 5:27
• @RichardMyers, I apologize for the formatting which has resulted from a explicit-is-better-than-implicit policy on notation. The question and related analysis I'm referring to is not related to Noether's theorem, but thanks for the comment. Jan 22 '21 at 23:27
• @kb314 The point is that with contact forces the notion of centrality does not even appear, and yet the law of conservation of rotational momentum is needed. This shows that centrality, for systems where it makes sense, such as particles, can be seen as a consequence of it (and of frame-invariance), rather than vice versa. Jan 23 '21 at 8:29
• @kb314 This is even clearer when we consider general relativity and fields. Again "centrality" is undefined there. The balance of rotational momentum there is expressed by the symmetry of the stress-energy-momentum tensor (and it includes the equivalence of energy and momentum as a special case). But maybe I'm now misunderstanding your original question? Jan 23 '21 at 8:29

In the proof above, it was assumed that forces which act at a distance, acted on a particle by another in the system S are central forces. Is this assumption required to prove the law of conservation of angular momentum of the system S?

Why is this assumption not stated explicitly in the textbooks on classical mechanics?

Answering both the questions, from the textbook on Classical Mechanics by Goldstein, Poole and Safko (Goldstein et al), states this in section 1.2 of chapter 1.

The conservation of the total angular momentum of the system in the absence of applied torques requires the validity of the strong law of action and reaction - that the internal forces in addition be central.

(Strong law of action and reaction is followed when the internal forces between two particles are equal and opposite and lie along the line joining the particles).

So yes, the forces must be central and the book by Goldstein et al mentions it.

Yes, to prove conservation of angular momentum (and also conservation of linear momentum) you need the internal interaction forces between the particles to be central. Well, the forces which do not obey Newton's 3rd law are non-central, like the magnetic force acting between two moving point different magnitude charges along mutually perpendicular paths are non-central. (You can easily see this, the direction of the magnetic forces are not along their line of separation and also do not align parallely). The fact is that when we have these magnetic forces coming into play, the total momentum is conserved, but may not the mechanical momentum since there may be an inter-conversion between mechanical and electromagnetic field momentum. Hence, Newton's 3rd law and conservation of linear and angular momentum fails. Well, I don't know why majority of classical mechanics textbooks don't mention about it, but in my physics lecture I was told of this exception. However, this effect is very negligible, since magnetic force between two moving point charges are far far weaker than the central electric force operating in between them. Hence, to an impressingly good approximation, even in electromagnetic situations, the conservation laws work. However, there may be some cases where we have to take into account these subtleties.