Angular momentum ($L$)

The angular momentum of a particle about the origin is $L=r\times p$.

The total angular momentum of an isolated system is constant. More generally, its time derivative equals the external torque.

The rate at which angular momentum crosses a surface, including advection and stresses. With positive inward mass accretion rate, the net inward flux in a steady accretion disk is $\dot{M}h-\mathcal{G}$.

For unit mass in a central force with potential $\Phi (r)$ and conserved angular momentum $L$, a radial turning point $r_{\ast }$ has zero radial speed and therefore satisfies

For unit mass, incoming speed $v>0$, impact parameter $b>0$, and potential energy $\mp \kappa /r$, put $q=\kappa /v^2>0$. At closest approach, conservation of energy and conservation of angular momentum give the displayed radii and tangential speeds $u_{\mp }=bv/p_{\mp }$. Consequently $p_{-}{p}_{+}=b^2$ and $u_{-}{u}_{+}=v^2$. These relations compare equal force magnitudes. The attractive radial case $b=0$ is a singular collision, so the speed-product statement does not extend to that endpoint.

Angular momentum transport moves angular momentum between locations through matter, stresses, waves, or fields. An accretion disk must transport angular momentum outward for most of its mass to move inward.

Torque is the moment of force, $\tau =r\times F$, and equals the rate of change of angular momentum.

Moment balance equates the rate of change of angular momentum to applied torques. A static slender rod obeys ${\partial }_s\mathbf{M}+\mathbf{t}\times \mathbf{F}+\mathbf{m}=0$.

The areal velocity is the rate at which the radius vector sweeps area. For specific angular momentum $h$, it is constant and equals