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I've heard that

"First, unlike energy, momentum cannot be stored. There's no such thing as potential momentum. Potential energy exists because energy isn't about moving, it's about doing."

However, there is an entire chapter on potential momentum on Wikibooks. Am I misunderstanding the chapter, or do the two sources conflict with each other? If so, who is correct?

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    $\begingroup$ That’s a rather confusing quote… where did you hear that? $\endgroup$
    – knzhou
    Commented Oct 7, 2021 at 6:57
  • $\begingroup$ Kinetic energy is by definition about moving. $\endgroup$
    – my2cts
    Commented Oct 7, 2021 at 9:35
  • $\begingroup$ I'm sorry for causing any trouble; I've expanded the quote so as to be more clear as to what I meant. $\endgroup$ Commented Oct 7, 2021 at 10:16
  • $\begingroup$ The quote is still confusing. Of course momentum can be 'stored'. That's what particles do, 'store' energy and momentum. Without a reference this question cannot be answered in its present form. $\endgroup$
    – my2cts
    Commented Oct 7, 2021 at 11:41

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A free particle has constant potential energy, without loss of generality $0$, so the Minkowski square norm of its $4$-momentum is$$E^2/c^2-\vec{p}^2=m^2c^2,$$with $\vec{p}$ the $3$-momentum an $m$ the rest mass. The RHS is invariant, meaning observer-independent, do not confuse this with conserved, meaning time-independent. (I'll discuss conserved quantities later.) Now to generalize: when the particle isn't free, we generalize this equation while preserving its RHS, say as$$(E-U)^2/c^2-(\vec{p}-\vec{Q})^2=m^2c^2.$$This is the Minkowski square norm of the $4$-momentum minus a $4$-vector whose energy part is the potential energy $U$. It therefore makes sense to call the $3$-vector $\vec{Q}$ the potential momentum.

The source you quote uses that term with a very different meaning, then explains why that one doesn't make sense: in other words, it explains why a different "energy is like this, therefore momentum is like that" analogy doesn't work. The familiar kinetic momentum is conserved, rather than being one of two terms whose sum is conserved while the terms themselves separately aren't. By contrast, energy conservation is only the conservation of a sum of two non-conserved energy terms, one kinetic, one potential. So while one can store a non-conserved amount of energy as what we call potential, momentum doesn't work the same way.

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  • $\begingroup$ If that was the Electromagnetic filed, ${\bf Q}$ would be what, the vector potential? $\endgroup$ Commented Dec 2, 2021 at 10:19
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    $\begingroup$ @AlexanderCska That makes sense in light of $\vec{\pi}=\vec{p}-q\vec{A}$ with $\vec{\pi}$ ($\vec{p}$) the canonical (kinetic) momentum. $\endgroup$
    – J.G.
    Commented Dec 2, 2021 at 12:26

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