= Areal velocity
{title2=$\dot A$}
{wiki}
Areal velocity measures how fast a <position> <vector> from a chosen origin sweeps <signed swept area>. It connects a geometric picture of motion to <angular momentum>: a long <position> <vector> with slow transverse motion can sweep the same <area> per unit <time> as a short one with fast transverse motion. Constancy requires a dynamical condition, given by the <torque criterion for constant areal velocity>.
Choose a fixed origin $O$ in an <inertial frame> and a fixed <plane> through $O$, with constant <unit vector> $\mathbf n$ normal to it. For a differentiable trajectory in that <plane>, let $\mathbf r(t)$ be <position> relative to $O$ and $\mathbf v=\dot{\mathbf r}$ its <velocity>. Positive swept <area> follows the <right-hand rule> about $\mathbf n$. In positively oriented Cartesian coordinates in the <plane>, and in <plane polar coordinates> with <angle> $\theta$ in <radians>,
$$
\dot A=\frac12\mathbf n\cdot(\mathbf r\times\mathbf v)
=\frac12(x\dot y-y\dot x)
=\frac12r^2\dot\theta.
$$
Here the <dot product> selects the normal component of the <cross product>. Motion away from or towards $O$ alone contributes no <area>. The signed <specific angular momentum> component $h_n=\mathbf n\cdot(\mathbf r\times\mathbf v)$ obeys
$$
\dot A=\frac{h_n}{2}=\frac{L_n}{2m}
$$
for a constant <mass> $m$. If $\mathbf n$ is chosen along a nonzero conserved <specific angular momentum> <vector>, then $h_n=h=|\mathbf r\times\mathbf v|$ and the familiar positive formula is $\dot A=h/2$. Reversing the orientation changes the sign; the instantaneous unsigned swept-<area> rate is $|\dot A|$.
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