LECTURE 2. STABILITY AND UNDERWATER VEHICLE DYNAMICS 367More realistic models of the fluid dynamics, especially in cases when vorticity is
generated by the body-fluid interaction as well as the inclusion of elastic and flexible
properties of the body would also be interesting, so would the effect of dissipation
after Bloch, Krishnaprasad, Marsden and Ratiu [1994, 1996].
The underwater vehicle problem should continue to provide interesting addi-
tional motivations for the continued development of the basic theory.
The planar skater and phases. We now turn our attention to the topic of geo-
metric phases, which we have already seen is important in the underwater vehicle.
We begin with a closer look at the basic theory.
For a mechanical system with Hamiltonian of the form kinetic plus potential
and with a symmetry, one can think of the zero momentum map level set as a
constraint. Let g denote the group position and r the internal shape: then
we can write the equation J = 0 as
g-^111 = -A(r)r,
which defines A , the mechanical connection; this definition agrees with the one
we had earlier. For nonzero momentum valuesμ of J, then the equation J =μcan
be written as
g-^1 !J = -A(r)r + (I(r,g))-^1 μ,
which also defines I , the locked inertia tensor.
A good example illustrating these ideas is the planar skater, shown in Figure
2.9. It consists of three linked rigid bodies in the plane (imagine them sliding on
an ice rink).
Figure 2.9. The planar skater consists of three interconnected bodies that
are free to rotate about their joints.Suppose that the angles 'l/J 1 and 'ljJ 2 perform a cyclic or periodic motion (imagine
motors in the joints). Note that if the angular momentum starts out zero, it remains
so since no external torques act on the system. The cyclic inputs to this system
are shown as a base input curve, while the actual trajectory of the motion is shown
lifted above the input curve, as in Figure 2.10.
After completing one cycle of internal shape changes, the skater has undergone
a net rotation (change in B). The area enclosed by the base inputs is proportional to
the net rotation. This is a simple example of the geometric phase, or holonomy,
associated with the cyclic shape inputs. The planar skater is a simple enough