We calculated the rotational energy for a rigid body rotating about an
axis through a fixed point O in the body. While generally this looks
complicated due to a non-trivial inertia tensor, we showed that it
reduces to the familiar freshman physics result for rotation of a body
about a single axis 
.
We then introduced the principal axes, noting that the inertia tensor
can always be diagonalized by a change of basis (choice of coordinate
axes). The principal axes diagonalize the inertia tensor, with the
diagonal entries 
and 
known as the principal moments
of inertia; they give the familiar moment of inertia for rotation
about the 
axes respectively, in the basis
of principal axes. Discussed nomenclature of the degeneracies of these
principle moments of inertia: the asymmetric top with 
the symmetric top with 
, and the spherical
top with 
.
We noted that our derivation of 
and 
relied on
measuring them relative to a fixed point in the body, and that one can
show our results are still valid for and
relative to 1) any constant velocity point in the body; and 2) the
center of mass. We also noted that Euler's equation, 
, is valid in all these frames; although 
must be measured in a space-fixed (inertial) frame.
We related as measured in a space-fixed (inertial)
frame to the apparent perceived by an observer in the
body-fixed (rotating) frame. We then picked our body axes as the
principal axes, to write 3 one-dimensional Euler's equations for the
motion in the body-fixed frame.
We examined Euler's equations to see under what conditions the uniform
rotations we studied in freshman physics (torque-free rotations with
constant 
) can occur. We found that uniform rotation
could oncly occur about a principal axis of the rigid body. For bodies
with symmetries, more axes are principal axes (that is, diagonalize
the inertia tensor). For symmetric tops (cylindrical symmetry) any
choices of axes for the xy-plane are principal axes, while for
spherical tops (spherical symmetry), any axes at all are principal
axes. We commented on stability of such uniform rotations, quoting the
result that rotations about axes with the smallest and largest I are
stable, while rotations about the axis with intermediate I is
unstable. Thus for a book whose width, height, and depth are all
different, uniform rotations about the longest and shortest axes are
stable, while those about the middle axis are not (if nudged away from
the middle axis, a book rotating initially about the middle axis will
jump to rotation about one of the other two axes).
Finally, we surveyed the classic physics problem of the torque-free
rotation of a symmetric top (like a gyroscope or our slightly
nonspherical spinning earth). We noted that the motion looks different
in the space frame and body frame. In the space frame, is
constant, and the top's motion is the sum of two distinct motions: (1)
the top spins about its own symmetry axis with a frequency 
; and (2) the spinning top precesses about the fixed
with frequency 
. This is the motion we see when
looking at an ideal massless gyroscope from our own external
perspective.
Motion in the body frame is derived in your Marion and Thornton
readings. Here the body's symmetry axis is fixed, and both
and appear to precess about it with frequency
. Thus for an asymmetric earth (which
rotates about the pole axis), if the pole axis is not the symmetry
axis, earth-based observers see the poles precess about the earth's
symmetry axis.
--
KB