Momentum conservation

Momentum conservation states that total momentum stays constant when the net force exerted from outside the chosen collection is zero. Use a closed collection, meaning the same fixed-mass point masses are followed throughout, and an inertial frame. Then

P = \sum_{i} p_{i}, \frac{d P}{d t} = \sum_{i} F_{i}^{ext} .

To see why, sum Newton's second law over the members. If

F_{ij}

is the force on member

i

from member

j

F_{ij} = - F_{ji}

. Each pair cancels in the total:

\frac{d P}{d t} = \sum_{i} F_{i}^{ext} + \sum_{i < j} (F_{ij} + F_{ji}) = \sum_{i} F_{i}^{ext} .

The internal forces act on different bodies and can change their individual momenta; their contributions cancel only in the collection's sum. An isolated system has no external interaction, but balanced external forces also suffice for momentum conservation. A closed system merely excluding matter exchange need not have conserved momentum. ^[ref]

The usefulness is that the detailed internal forces during a collision need not be known. Both an elastic collision and a perfectly inelastic collision obey the same isolated momentum balance; kinetic energy conservation is an additional condition for the elastic collision. If the external impulse is small but nonzero, the total momentum change is correspondingly small. For a variable-mass system exchanging matter, the entering or leaving momentum must also be included.

In a translation-invariant field theory, integrating over the spacetime zero mode enforces momentum conservation at interactions and scattering amplitudes.