Unit-direction kinetic-energy identity ($v^2=(\mathbf{u}\cdot \mathbf{v}{)}^2-r^2\mathbf{u}\cdot \ddot{\mathbf{u}}$)

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.