Why isn't there such a thing as "internal momentum"?

Question

The three most well-known conserved quantities in classical physics are energy, momentum, and angular momentum.

Suppose we have a system with no external forces acting on it. We can talk about the system's internal energy as the sum of all kinetic and potential energies of its particles and interactions between particles. If you think about subparts of the system, the internal energy is just what remains of the kinetic and potential energies when you consider the total system's center of mass reference frame (which is inertial because there are no external forces).

We can also talk about an analogue of this when it comes to angular momentum. The so-called spin angular momentum is what remains of the angular momentum of the when you consider the total system's center of mass reference frame.

Why isn't there an analogue of this for linear momentum?

Answer 1

Very simply, if we try to generalize the notion of an "internal $x$" of a system $S$ given any quantity $x$ by defining it to be the value of $x$ in the center of mass reference frame of system $S$, then "internal momentum" is always going to be zero, because the total momentum of a system $S$ in the center of mass reference frame of $S$ is always zero. This renders the concept of "internal momentum" trivial and uninteresting. As far as I can tell, there isn't anything deeper than that (although others can provide their own answers if they disagree).

Answer 2

It boils down to our choice of the reference frame which we use to define "internal energy" or "internal momentum".

For example, consider a gas in a mass-less container that does not deform, and let the container experience sudden change of motion, e.g. its motion is quickly stopped by impacting on some heavy body with a locking mechanism or a velcro patch. The gas will not stop immediately, it will continue to move with respect to the stopped container, will compress somewhere and expand elsewhere, complicated flow can develop, and only later the gas will settle to equilibrium state in which it is at rest with respect to the container.

Now, right after the container stops, what is internal energy and internal momentum of the gas?

If we define internal energy/momentum with respect to the container, which is quite sensible thing to do, we get higher value of energy than if we define it with respect to the center of mass of the gas, and we get non-zero internal momentum.

If we define internal energy/momentum with respect to center of mass of the gas, then internal momentum stays zero. But this frame can be cumbersome for describing state of the gas, as in this frame, not only the gas elements, but also the container itself moves with acceleration, and does work on the gas.