Regid Body
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Center of mass $\vec R$ is defined as

.. math::
   \int \rho(\vec r) \vec r d^3 \vec r = \int \rho(\vec r) \vec R d^3 \vec r .

Equivalently,

.. math::
   \int \rho(\vec r) \left( \vec r - \vec R \right) d^3 \vec r = 0.