= Kinetic energy decomposition about the center of mass
{title2=$T=\frac12M|\dot R|^2+\frac12\sum_i m_i|\dot\rho_i|^2$}
= König theorem for kinetic energy
{c}
{synonym}
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 $\tfrac12 I\omega^2$, where $I$ is the <moment of inertia> about the instantaneous rotation axis through the <center of mass>.
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