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?
