On Fri, Sep 21, 2012 at 8:02 PM, Bill Gosper <billgosper@gmail.com> wrote:
DanA> I presume you mean why you can't flip a rectangular solid (a x b x c with a < b < c) around its middle axis. Ignoring both gravity and air resistance.
Well, if you look at the flow on the phase space (a scalene ellipsoid) that shows how physics evolves, it has one pair of saddle points (at +-the middle axis) and two pair of centers (at +- the other two axes).
If you flip the rectangular solid around its smallest or largest axis, any small perturbation will just shift it to a nearby closed orbit, so it will stay close to rotating around that axis.
But if you flip it about the middle axis, the local saddle structure has all its nearby (but unequal) orbits going far away from rotation about that axis.
I was not able to access Gene's writeup since I got a 404 error from the URL below, so can't compare what he wrote to the above. Gene?
--Dan
GMail ior your mail reader got faked out by the blanks in the URL. If the following fails, try "physically" pasting into the top browser pane.
http://gosper.org/Rigid <http://gosper.org/Rigid%20Body%20(1).pdf> http://gosper.org/Rigid Body (1).pdf
--rwg
AAGH! I'm typing this into the GMail Reply window, where the URL looks perfect. Before sending, I tested it with the Link feature. Yet it's wildly garbled in my copy of the math-fun mail.
I just "cleansed" the string by pasting into Mathematica, which introduced new blanks for implicit multiplication! Try pasting this into your browser: http://gosper.org/Rigid%20Body%20(1).pdf --rwg