My question is related to this recent post, which deals with the tennis racket theorem
There are surely a thousand ways to prove the result described in that question, i.e. that rotation about the principal axis of a rigid body with the intermediate inertial moment is unstable. The most popular answer there explains it as a basically a result of the conservation laws of the system: since the dynamics of the angular momentum vector are such that it is simultaneously constrained to the surface of a sphere and an ellipsoid, the trajectories through angular momentum space must take certain geometric forms, and one is lead very naturally to the conclusion that rotation about one axis must be unstable.
This argument has always been very appealing to me, partially because it is fairly minimal. I've wondered, as a result, how minimal this argument can be made. For example, suppose we didn't know about conservation of energy, but only knew about conservation of angular momentum. Then, we're interested in the dynamics of a vector with fixed magnitude, i.e. dynamics on the 2-sphere. From the hairy ball theorem, one knows that every vector field on the 2-sphere must have topological charge 2. Then using our knowledge that there are in fact six fixed points, which come in identical pairs, we immediately see that some of them must have negative topological charge, and therefore be unstable. I've obviously added a few assumptions on top of conservation of angular momentum, but is there a way to make this rigorous, or do it in fewer assumptions? How general is this result? Given a few discrete symmetries and the basic topology of the problem, is this more or less inevitable?
Edit: Thinking a little more about this, I'm not sure we even need the conservation of angular momentum assumption; we can just think about the direction of the angular momentum vector, and ignore the magnitude.
