Center of mass

For masses $m_i$ at positions ${\mathbf{r}}_i$, the center of mass is $\mathbf{R}=({\sum }_i{m}_i{\mathbf{r}}_i)/({\sum }_i{m}_i)$, with the analogous mass-density integral for a continuous body.

Suppose a body of mass $M_0$ has centre of mass ${\mathbf{R}}_0$, and material of mass $M_c<M_0$ with centre of mass ${\mathbf{R}}_c$ is removed. Additivity of the first mass moment gives the remaining centre of massWhen the original centre of mass is at the origin, the remaining centre of mass points opposite the removed material's first mass moment. For a constrained rigid body rotating about a fixed axis, this displaced centre of mass undergoes centripetal acceleration even if the applied axial torque is zero.

The center of mass moves as if the total external force acted on the total mass; internal forces cancel in its equation.

Center-of-mass reduction separates translational motion from internal coordinates by using the conserved or externally forced center coordinate.

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.

For three particles, choose the relative position of particles one and two, the position of their center of mass relative to particle three, and the total center of mass. The kinetic energy diagonalizes with masses

For Jacobi coordinates $\mathbf{a},\mathbf{b},\mathbf{c}$ and their corresponding reduced masses $\alpha ,\beta ,\gamma$, total angular momentum is

For three ordered angular positions $q_1,q_2,q_3$, define consecutive gapsand the collective angle $\Phi =(q_1+q_2+q_3)/3$. Then $\alpha +\beta +\gamma =2\pi$. For three equal masses on a circle of radius $r$,Thus the ignorable coordinate $\Phi$ decouples from the relative gap dynamics whenever the potential depends only on the gaps.

For masses $m_1,m_2$, the relative coordinate obeys a one-body equation with reduced mass