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The intuitive way to handle this is to think about it in
angular momentum space. The contours of constant energy
are ellipsoids in this space. Let's choose one such
contour, i.e. let's choose a definite energy.
If the norm of the angular momentum is as small as possible
(consistent with the chosen energy) the angular momentum must
be at the short axis of the ellipsoid. If the norm is merely
/near/ the smallest possible value, the angular momentum must
be near the short axis. The geometry of spheres intersecting
with an ellipsoid guarantees it.
Similarly, if the norm of the angular momentum is at or near
the largest possible value, the angular momentum vector must
be at or near the long axis of the ellipsoid.
Now (!) consider the intermediate case. There are lots of
places the angular momentum vector could wander relative to
the ellipsoid. (In any inertial frame the angular momentum
vector doesn't wander at all, but the ellipsoid does, as
we re-orient the object. Alternatively, in a frame comoving
with the body, the ellipsoid stays fixed but the angular