Center-of-mass frame

In physics, the center-of-momentum frame (zero-momentum frame, or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The center of momentum of a system is not a location (but a collection of relative momenta/velocities). Thus "center of momentum" means "center-of-momentum frame" and is a short form of this phrase.

A special case of the center-of-momentum frame is the center-of-mass frame: an inertial frame in which the center of mass (which is a physical point) remains at the origin. In all COM frames, the center of mass is at rest, but it is not necessarily at the origin of the coordinate system.

In special relativity, the COM frame is necessarily unique only when the system is isolated.

The center of momentum frame is defined as the inertial frame in which the sum over the linear momentum of each particle vanishes. Let S denote the laboratory reference system and S′ denote the center-of-momentum reference frame. Using a galilean transformation, the particle velocity in S′ is

${\displaystyle v'=v-V_{c},}$

where ${\displaystyle V_{c}={\frac {\sum _{i}m_{i}v_{i}}{\sum _{i}m_{i}}}}$

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