In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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 classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
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.
[Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ]
A special case of the center-of-momentum frame is the center-of-mass frame: an inertial frame in which the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
(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
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
, 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
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
, 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
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the system is the ''minimal energy'' as seen from all
inertial reference frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
s.
Special relativity
In
relativity, the COM frame exists for an isolated massive system. This is a consequence of
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
. In the COM frame the total energy of the system is the ''
rest energy
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
'', and this quantity (when divided by the factor ''c''
2, where ''c'' is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
) gives the
rest mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
(
invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
) of the system:
:
The
invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
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
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
(such as
photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alway ...
moving in a single direction, or, equivalently,
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
s) 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
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, 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
In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime.
The term is most often used in t ...
: the frame where the measurement or calculation is done. The situation is analyzed using
Galilean transformations
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
and
conservation of momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
(for generality, rather than kinetic energies alone), for two particles of mass ''m''
1 and ''m''
2, moving at initial velocities (before collision) u
1 and u
2 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):
:
so at the origin of the COM frame, R' = 0, this implies
:
The same results can be obtained by applying momentum conservation in the lab frame, where the momenta are p
1 and p
2:
:
and in the COM frame, where it is asserted definitively that the total momenta of the particles, p
1' and p
2', vanishes:
:
Using the COM frame equation to solve for V returns the lab frame equation above, demonstrating any frame (including the COM frame) may be used to calculate the momenta of the particles. It has been established that the velocity of the COM frame can be removed from the calculation using the above frame, so the momenta of the particles in the COM frame can be
expressed in terms of the quantities in the lab frame (i.e. the given initial values):
:
notice the
relative velocity
The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.
Classical mechanics
In one dimension (non-relativistic)
We begin with relative motion in the classi ...
in the lab frame of particle 1 to 2 is
:
and the 2-body
reduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...
is
:
so the momenta of the particles compactly reduce to
:
This is a substantially simpler calculation of the momenta of both particles; the reduced mass and relative velocity can be calculated from the initial velocities in the lab frame and the masses, and the momentum of one particle is simply the negative of the other. The calculation can be repeated for final velocities v
1 and v
2 in place of the initial velocities u
1 and u
2, since after the collision the velocities still satisfy the above equations:
[''An Introduction to Mechanics'', D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, ]
:
so at the origin of the COM frame, R = 0, this implies after the collision
:
In the lab frame, the conservation of momentum fully reads:
:
This equation does ''not'' imply that
:
instead, it simply indicates the total mass ''M'' multiplied by the velocity of the centre of mass V is the total momentum P of the system:
:
Similar analysis to the above obtains
:
where the final
relative velocity
The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.
Classical mechanics
In one dimension (non-relativistic)
We begin with relative motion in the classi ...
in the lab frame of particle 1 to 2 is
:
See also
*
Laboratory frame of reference
In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime.
The term is most often used in t ...
*
Breit frame In particle physics, the Breit frame (also known as infinite-momentum frame or IMF) is a frame of reference used to describe scattering experiments of the form A + B \rightarrow A + \sum C_i, that is experiments in which particle A scatters off pa ...
References
Classical mechanics
Coordinate systems
Frames of reference
Geometric centers
Kinematics
Momentum
{{classicalmechanics-stub