What is the physical significance of the rate of change of a body's momentum with respect to the displacement it travels through, $dp/dx$?

Question

This is something I've been trying to wrap my head around as I review mechanics, particularly conservation laws, in depth.
I see the conservation of momentum and conservation of energy as sort of two parallel systems for investigating the motion/interactions of several bodies:

One is imbued to a body by having work done, i.e., a force applied over a displacement ($int F ,ds = W = Delta E$), whereas the other is imbued to a body by transferring an impulse, i.e., a force applied over a time ($int F ,dt = J = Delta p$).

Despite the fact that the "conservation of energy" perspective on a system is most important when dealing with forces that vary through space (e.g. simple harmonic motion, a mass-spring system, etc.), it is still common convention to have a measurement of the rate at which energy is transferred or work is done with respect to time, i.e., power ($dE/dt$).

However, it seems to me that there's no such parallel measurement for momentum - some kind of differential with respect to position ($dp/dx$), which I understand would have units $mathrm{kg,s^{-1}}$, which suggests it might be a rate of change of mass; yet I don't really get why.

Does that differential of momentum with respect to position/displacement have any physical significance? Am I missing something here? Or is a differential with respect to time as with $P = dE/dt = dW/dt$ just inherently more meaningful and useful than any rate of change with position?

ZeroTheHero · Accepted Answer

$dp/dx$ doesn’t have any immediate interpretation.  This is because conserved quantities are conserved in time so their rate of change is $0$.
In 1d it is possible to manipulate
$$
frac{dp}{dx}=frac{dp/dt}{dx/dt}=frac{F}{v}
$$
and in $3d$ you might consider the ratio $dvec p/{ds}=vec F/vert vec vvert$ but there is no special meaning to this ratio.  Moreover, if the speed $v$ instantaneously $0$ the ratio hardly makes sense.
Finally unitwise $dp/dx$ has units of kg/s, and it is difficult to thing of something that would be expressed in such units except a rate of mass loss or gain.

Umaxo · Answer

You usually parametrize evolution of system with time, not position. So if you have quantity defined on some path, it is natural to look at time-dependance not space-dependance. You can still do the later, and it probably is being done if the particular system in question is better expressed in space-dependent view. But in general it seems like an unnatural idea. The evolution with time is on the other hand very fundamental.
This changes when you have quantity defined throughout space. Then gradient has quite natural meaning and you can naturally form quantities like momentum density of the fluid and look at its gradient.

What is the physical significance of the rate of change of a body's momentum with respect to the displacement it travels through, $dp/dx$?

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