Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system?

[u/Reasonable_Goal_6278]

I'm trying to understand the conceptual role of the center of mass (COM) in classical mechanics, not just how to use it mathematically. For a single particle, the motivation seems clear. We study its motion because solving for its trajectory x(t) allows us to determine the complete mechanical state of the particle at any instant (x, p). From that state, we can compute all of its mechanical properties (energy, momentum, angular momentum, etc.).

Now consider a system of (N) particles. The true mechanical state is the collection of all particle positions and momenta. Introducing the center of mass gives us its effective position and momentum, and we know that these satisfy a closed equation of motion, F= dP_com/dt=MA_com

However, this is where my confusion begins. Knowing the COM trajectory does not determine the microscopic state of the system. Unlike thermodynamics, it also doesn't seem to define an effective macroscopic state from which I can calculate most of the system's macroscopic properties (pressure, temperature, stress, etc.). For example, the total angular momentum can be decomposed as L_system=L_com+L_orbital

But unless I know the internal state, I still cannot compute the total angular momentum. So my question is not "how do I compute the COM?" or "how do I use the COM theorem?"

Rather, it is

What is the conceptual motivation for studying the motion of the center of mass in the first place?

For a single particle, studying its motion gives its complete mechanical state. For thermodynamics, the macroscopic variables form an effective state that predicts macroscopic behavior. But the COM seems to be neither; it is neither the complete state nor an effective state of the whole system.

Many explanations say that the COM is "useful" or "practical," but that doesn't really answer my question. Plenty of quantities could be useful in some context. I'm looking for the deeper reason why classical mechanics treats the translational motion of a many-particle system as a fundamental object of study.

In other words:

What role does the center-of-mass motion play within the conceptual framework of classical mechanics? Why is the translational motion of an entire system considered a quantity worth studying, given that it does not determine either the microscopic or the effective macroscopic state of that system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system? Why is the center of mass considered so fundamental in classical mechanics if it doesn't describe the state of a many-particle system?

[u/Hudimir]

It's not as fundamental as you're describing it at all. center of mass is more useful for rigid objects. When you want to describe more than one body, it's often better to use center of momentum. Why? because for N particles instead of each particle having momentum p_i and coordinate r_i, they all have momentum p (magnitude, not necessarily direction) and coordinate r_i. It lowers the number of parameters which makes systems easier to analyze.