Center Of Mass Coordinates

In kinematics, both classical or relativistic, the center of mass coordinate system is a coordinate system in which the center of mass of the system remains still (it has no velocity). It is especially useful for the kinematic study of an assembly of particles. We shall note a variable in the center of mass system with a bar over it, so that, e.g., \bar\mathbf{v}_{\mathrm{cm}}=\mathbf{0} by definition. In any coordinate system,
\mathbf{v}_\mathrm{cm}={\sum_i m_i\mathbf{v}_i\over\sum_i m_i}
where the sum extends over i the index of particles with mass m_i and velocity \mathbf{v}_i. From the previous equation we see that in the center of mass system, the total linear momentum p is therefore zero:
\sum_i \bar\mathbf{p}_i=\mathbf{0}
where \mathbf{p}_i=m_i\mathbf{v_i}.

 

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