For a particle at position $\mathbf r$ with momentum $\mathbf p$, angular momentum about an origin is $$\mathbf L=\mathbf r\times\mathbf p.$$ Its time derivative is $$\frac{d\mathbf L}{dt}=\mathbf r\times\mathbf F=\boldsymbol\tau,$$ so torque is the rate of change of angular momentum.

For a system of particles, $$\mathbf L=\sum_i \mathbf r_i\times\mathbf p_i,$$ and internal torques cancel under ordinary central pair forces. Therefore $$\frac{d\mathbf L}{dt}=\boldsymbol\tau_{\text{ext}}.$$ If the net external torque about the chosen origin vanishes, total angular momentum is conserved.

Consider a particle moving in a circle of radius $r$ with momentum tangent to the circle. Because $\mathbf r\perp\mathbf p$, $$L=rp=mvr.$$ If a central force pulls the particle inward, its torque about the force center is zero because $\mathbf r\times\mathbf F=0$. Angular momentum remains constant even though the particle's velocity can change strongly.

Angular momentum depends on the chosen origin, but conservation statements become especially useful about symmetry centers, fixed pivots, or the center of mass. It also differs from the scalar rotational relation $L=I\omega$, which is valid only in particular rigid-body situations. The vector conservation law is more general and is the foundation for central-force motion, rigid-body dynamics, and quantum angular momentum.