368 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
0.45
0.4
0.35
0.3
e 0.25
0.2
0.15
0.1
0.05
00.3 0.2 0.1----
0.7
0.6
0.5
0.4
0 .3 <pl
0.2
0.1
0 -0.1 -0.2 -0.3 -0.4
0
<pz resulting motion
base inputsFigure 2.10. Input and output motions for the planar skater.example that it can be dealt with by hand in a simple way, using the definition of
angular momentum and its conservation. More sophisticated examples, such as the
falling cat or the reorientation of spacecraft requires more elaborate mathematics.
One of the keys to a deeper understanding is to get a better vision of holonomy.
Holonomy is also a basic notion in geometry per se and can be illustrated
using parallel transport on the sphere (the same geometry is used to understand
the Foucault pendulum). See Figure 2.11. In this figure, one parallel transports
an arrow around a triangle on the sphere whose sides are made up of portions of
great circles. In this case, parallel transport along the arcs is parallel transport
in the naive sense. Notice that when the arrow returns to its original location, it
has rotated relative to its original position. This rotation is another example of a
phase shift.
The general notion of holonomy in geometry involves the splitting of the tan-
gent space of a bundle with the use of a connection (more generally a distribution),
as shown in Figure 2.12.
Interestingly, geometric phases or Berry phases in quantum mechanics ·can
be viewed as a special case of phases for classical systems by using the (now well
known) fact that with Hilbert space as our phase space (the symplectic structure
being the imaginary part of the Hermitian inner product), we get the Schrodinger
equation as a special case of Hamilton's equations. The symmetry group is 51 phase
shift and the reduction is the classical process of relating quantum mechanics on
Hilbert space to that on projective Hilbert space studied by Bargmann and Wigner.
(See also Marsden, Montgomery and Ratiu [1990] and Marsden and Ratiu [1998]
for a discussion.)
Phases for a relative equilibrium for the underwater vehicle were shown in
earlier figures. The body angular and linear velocities remain stable while
the group variables (rotation around one axis and translations) can drift, as we
have explained.
Phases for the three-wave interaction. We return now to the three wave
interaction discussed in the first lecture and briefly discuss phases for the problem.