Center-of-momentum frame

In physics, the center-of-momentum frame (also 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: a reference frame). 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.
Properties.
General.
The center of momentum frame is defined as the inertial frame in which the sum of the linear momenta of all particles is equal to 0. 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
where 
is the velocity of the mass center. The total momentum in the center-of-momentum system then vanishes:
Also, the total energy of the system is the "minimal energy" as seen from all inertial reference frames.
Special relativity.
In relativity, the COM frame exists for an isolated massive system. This is a consequence of . In the COM frame the total energy of the system is the "rest energy", and this quantity (when divided by the factor "c"2, where "c" is the speed of light) gives the rest mass (invariant mass) of the system:
The invariant mass of the system is given in any inertial frame by the relativistic invariant relation
but for zero momentum the momentum term ("p"/"c")2 vanishes and thus the total energy coincides with the rest energy.
Systems that have nonzero energy but zero rest mass (such as photons moving in a single direction, or equivalently, plane electromagnetic waves) do not have COM frames, because there is no frame in which they have zero net momentum. Due to the invariance of the speed of light, a massless system must travel at the speed of light in any frame, and always possesses a net momentum. Its energy is—for each reference frame—equal to the magnitude of momentum multiplied by the speed of light:
Two-body problem.
An example of the usage of this frame is given below – in a two-body collision, not necessarily elastic (where "kinetic energy" is conserved). The COM frame can be used to find the momentum of the particles much easier than in a lab frame: the frame where the measurement or calculation is done. The situation is analyzed using Galilean transformations and conservation of momentum (for generality, rather than kinetic energies alone), for two particles of mass "m"1 and "m"2, moving at initial velocities (before collision) u1 and u2 respectively. The transformations are applied to take the velocity of the frame from the velocity of each particle from the lab frame (unprimed quantities) to the COM frame (primed quantities):
where V is the velocity of the COM frame. Since V is the velocity of the COM, i.e. the time derivative of the COM location R (position of the center of mass of the system):
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