Thread by @robertghrist

what is this?
it has something to do with a spinning tennis racket...
1/5

spinning a tennis racket about the longest or shortest principal axes is stable, but when you spin about the "middle" axis, there's a curious instability... 2/5

on the left is rotation about the three principal axes, assuming three distinct principal moments of inertia. to the right are more perturbed initial conditions. note the stability about the long and short axes, as well as the periodic "flips" with middle-axis rotation... 3/5

the dynamics are 3-dimensional, but conservative; restricting to an invariant sphere of constant angular momentum reveals three pairs of equilibria (+/- rotation about each axis). two pairs are centers; the last pair -- the intermediate axis -- are (unstable) saddles! 4/5

so, the real reason why the tennis racket behaves as it does is, as usual, eigenvalues: pure imaginary along the major & minor axes; real (+/-) along the middle axis. the occasional flips are due to shadowing a heteroclinic cycle. this is the intermediate axis theorem. 5/5