Center-of-mass motion

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.

In the two-body problem, these coordinates separate the common translation from the internal separation. For total mass $M$ and reduced mass $\mu$, the kinetic energy is $M{\dot{R}}^2/2+\mu {\dot{r}}^2/2$. The conjugate momenta are $P=M\dot{R}$ and $p=\mu \dot{r}$ for the ordinary Newtonian kinetic term.

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 particles with masses $m_j$ and positions $r_j$, Jacobi coordinates consist of the total center of mass and successive relative coordinates ${\rho }_j=r_{j+1}-({\sum }_{l=1}^j{m}_l{r}_l)/({\sum }_{l=1}^j{m}_l)$. They separate translation from internal motion and diagonalize the kinetic energy with the corresponding reduced masses.

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.