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Concept# Center-of-momentum frame

Summary

In physics, the center-of-momentum frame (COM frame), also known as zero-momentum frame, is the inertial frame in which the total momentum of the system vanishes. It is unique up to velocity, but not origin.
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" is a short for "center-of-momentum ".
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 single point) remains at the origin. In all center-of-momentum 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 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.
In relativity, the COM frame exists for an isolated massive system. This is a consequence of Noether's theorem. In the COM frame the total energy of the system is the rest energy, and this quantity (when divided by the factor c2, 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.

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