= Momentum conservation
= Conservation of momentum
{synonym}
<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
$$
\mathbf P=\sum_i\mathbf p_i,\qquad
\frac{d\mathbf P}{dt}=\sum_i\mathbf F_i^{\mathrm{ext}}.
$$
To see why, sum <Newton's second law> over the members. If $\mathbf F_{ij}$ is the <force> on member $i$ from member $j$, <Newton's third law> assumes $\mathbf F_{ij}=-\mathbf F_{ji}$. Each pair cancels in the total:
$$
\frac{d\mathbf P}{dt}
=\sum_i\mathbf F_i^{\mathrm{ext}}
+\sum_{i<j}(\mathbf F_{ij}+\mathbf F_{ji})
=\sum_i\mathbf F_i^{\mathrm{ext}}.
$$
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>. https://openstax.org/books/university-physics-volume-1/pages/9-3-conservation-of-linear-momentum[OpenStax University Physics Volume 1, section 9.3: Conservation of Linear 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.
Back to article page
