= Unit-direction kinetic-energy identity
{title2=$v^2=(\mathbf u\cdot\mathbf v)^2-r^2\mathbf u\cdot\ddot{\mathbf u}$}
For a twice differentiable position away from the origin, write $\mathbf r=r\mathbf u$ with radial <unit vector> $\mathbf u$. Its unit-length constraint gives $\mathbf u\cdot\dot{\mathbf u}=0$ and $\mathbf u\cdot\ddot{\mathbf u}=-|\dot{\mathbf u}|^2$, separating radial and directional contributions to speed and <kinetic energy>. The identity is kinematic and does not require a particular force law.
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