= Kinetic-energy decomposition about the centre of mass
{title2=$T=\tfrac12M|\dot{\mathbf R}|^2+\tfrac12\sum_i m_i|\dot{\mathbf y}_i|^2$}
For fixed particle <masses> and positive total <mass>, write $\mathbf x_i=\mathbf R+\mathbf y_i$ with $\mathbf R$ the <centre of mass>. The weighted relative <velocities> sum to zero, so expanding the <kinetic energy> eliminates the mixed term and yields the displayed decomposition. For a <rigid body>, the relative contribution is the <rotational kinetic energy> $I\omega^2/2$ about the chosen centre-of-mass axis.
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