Properties of the Center of Mass

Question

My students are currently going through the rigid rotor and hydrogen atom unit in their quantum physical chemistry course and I found myself at a loss on how to justify what seems a natural way to motivate the concept of the center of mass. As this answer nicely shows, the center of mass can be motivated as the unique point that a force can be applied to for a rigid body such that the body can be treated as a point mass for linear momentum kinematics. But since we are studying rotations my mind is preoccupied with the ways the center of mass is special for rotational kinematics.

My question is whether it is conceptually valid to say that the center of mass is also the point about which a rigid body can continue to rotate with a constant angular momentum (or rotational kinetic energy) without the addition of any additional external forces. In other words, it needs no "help" to continue rotating while a body rotating about a point other than its center of mass clearly would need additional help lest it fly off in a tangential trajectory. Moreover, I am not sure how I could prove such a statement. I am somewhat familiar with the notion that any canonical transformation that preserves Hamilton's equations (such as a coordinate rotation) gives a corresponding conservation law, but that just shifts the problem to proving that the Hamilton's equations for a rotating rigid body are unchanged by a rotation of the coordinate frame. If that were simple to show, then it be clear that the angular momentum is conserved and a rotating rigid body would have constant angular momentum and therefore rotational kinetic energy.

If this has been asked or answered in some way before I preemptively apologize for failing to find the relevant question(s) or answer(s).

Answer 1

If there are no external forces acting on a body (or even if there are external forces but they net to zero) then its centre of mass will move in a straight line with constant speed.

Parts of the body may still move and indeed accelerate relative to its centre of mass, but these motions will be caused or constrained by internal forces. These internal forces net to zero because of Newton's 3rd Law, and hence the internal forces do not affect the motion of the centre of mass.

Wikipedia's article on rigid body dynamics provides outline proofs, and all of this will be covered in any text on classical mechanics

Answer 2

I think I have the argument you're looking for. The answer to your question is yes, although a more intuitive, but equivalent way of stating it would be that the center of mass is the only point about which objects unconstrained to rotate about any axis can rotate. Your wording is equivalent and the argument you linked is a corollary to demonstrating what you're looking for. Here's the complete justification.

Suppose you have some rigid body, say a chair. Initially you let the chair go and because this is outer space it just floats in front of you. As it's floating, you hit the chair suddenly and forcefully. Regardless of where you hit it, the chair will (1) start moving away from you and (2) (in general) spin. Because it's spinning, it must be spinning about some axis. But we know that \begin{equation} m\vec{a}_\text{cm} = \sum_\text{ext}\vec{F} = 0 \Rightarrow \vec{v}_\text{cm} = \text{constant} \end{equation} So its center of mass must be moving with constant velocity. And the only way it's possible for the center of mass to be moving with constant velocity is if it's not also rotating about some axis (since if it was rotating about some other axis, it would have some centripetal acceleration and thereby net force on it). Hence the only way the chair's center of mass can be moving at constant velocity and spinning is if its rotation is around the center of mass. So unconstrained bodies can only rotate about axes passing through their center of mass.