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Momentum is the Noether charge of translational
invariance. As such, even fields as well as other things can have momentum, not just particles. However, in curved spacetime which isn't
asymptotically Minkowski, momentum isn't defined at all.
In physics, momentum is a physical quantity related to the
velocity and mass of an object.
Momentum in classical mechanics
In classical mechanics, momentum (traditionally written as
p) is defined as the product of mass and velocity. It is thus a vector
quantity.
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Impulse
The change in momentum, called the impulse, is equal to force times the change in
time.
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The SI unit of momentum is newton-seconds, which can alternatively be expressed with the
units kg.m/s.
An impulse changes the momentum of an object. An impulse is calculated as the
integral of force
with respect to duration.
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using the definition of force yields:
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See also angular momentum.
Momentum in relativistic mechanics
It is commonly believed that the physical laws should be invariant under
translations. Thus, the definition of momentum was
changed when Einstein formulated Special relativity so that its magnitude would remain invariant under relativistic
transformations. See physical conservation law.
We now define a vector, called the 4-momentum thus:
- [E/c p]
where E is the total energy of the system, and p is called the "relativistic momentum" defined thus:
-
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where
- .
Setting velocity to zero, one derives the result that objects have a rest mass which is related by the expression E = mc2
The "length" of the vector that remains constant is defined thus:
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Massless objects such as photons also carry momentum; the formula is
p=E/c, where E is the energy the photon
carries and c is the speed of light.
Momentum in quantum mechanics
In quantum mechanics momentum is defined as an operator on the wave function. The
Heisenberg uncertainty principle
defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum
mechanics position and momentum are interchangeble.
Origin of Momentum
Momentum arises from the condition that an experiment must give the same results regardles of the position or velocity of the
observer. More formally it is the requirement of invariance under translation. Classical momentum is the result of the invariance
of translation in three dimensions. Relativistic momentum as proposed by Albert Einstein arises from the invariance of Four-vectors under lorentzian translation. These Four-vectors appear spontaneously in the Green's
function from Quantum field theory.
Figurative use
A process may be said to gain momentum. The terminology implies that it requires effort to start such a
process, but that it is relatively easy to keep it going.
See also
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