Symmetry, Reduction, and Phases in Mechanics
^ M x ^ a H B i
Figure 1D-1
There are interesting answers to these questions that are related to the dynamics of Yang-
Mills particles moving in the associated gauge field of the problem. See Montgomery [1989] and
references therein. This bundle approach to mechanics will be a theme developed in this work as
We give two simple examples of how this works. Additional details will be given for this
type of example in §5. First, consider three coupled bars (or coupled planar rigid bodies) linked
together with pivot (or pin) joints, so the bars are free to rotate relative to each other. Assume the
bars are moving freely on the plane with no external forces and that the angular momentum is zero.
However, assume that the joint angles can be controlled with, say, motors in the joints. Figure
1D-1 shows how the joints can be manipulated, each one going through an angle of 2% and yet
the overall assemblage rotates through and angle n. A formula for the reconstruction phase
applicable to examples of this type is given in Krishnaprasad [1989].
A second example is the dynamics of linkages; see §5E for more details. This type of
example is consideres in Krishnaprasad [1990], Yang and Krishnaprasad [1989], and
Krishnaprasad and Yang [1990], including comments on the relation with the three manifold
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