Kinetic energy

The nonrelativistic kinetic energy of a particle is $T=m{v}^2/2$.

Writing particle positions relative to the center of mass as ${\rho }_i=r_i-R$ makes ${\sum }_i{m}_i{\rho }_i=0$, so the cross term in the squared velocities vanishes. The total kinetic energy splits into center-of-mass translation and relative motion as displayed. For a rigid body, the latter is $\frac{1}{2}I{\omega }^2$, where $I$ is the moment of inertia about the instantaneous rotation axis through the center of mass.

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.

The kinetic energy density of a continuum with mass density $\rho$ and velocity $\mathbf{v}$ is $\rho |\mathbf{v}{|}^2/2$. For a displacement field, the material velocity is its time derivative.

A rigid body rotating about a fixed axis with moment of inertia $I$ and angular speed $\omega$ has kinetic energy $K=I{\omega }^2/2$.