Conservation of angular momentum

Remarks

The proof: $\dot L = \sum_i \dot x_i \times m_i \dot x_i + \sum_i x_i \times m_i \ddot x_i = \sum_{i \ne j} x_i \times F_{ij}$. Grouping the terms for $(i, j)$ and $(j, i)$ gives $(x_i - x_j) \times F_{ij} = 0$, since the vectors are parallel. The same argument shows that a particle moving under a central force has constant angular momentum about the centre. So it moves in a plane and sweeps equal areas in equal times. Example: a planet around a fixed Sun has $\abs{x \times \dot x}$ constant, which is Kepler's second law. The parallel hypothesis is needed: if $F_{12} = -F_{21}$ but these forces are not along $x_1 - x_2$, they form a couple and $\dot L = (x_1 - x_2) \times F_{12} \ne 0$.

Going deeper

For a student

Angular momentum is a vector first integral; its constancy confines central-force motion to a plane and reduces it to one radial equation. More generally $\dot L$ equals the total external torque, so $L$ is conserved whenever that torque vanishes. In Lagrangian mechanics the result follows from invariance under rotations, an instance of Noether's theorem.