12 Marsden, Montgomery, and Ratiu
through (cp0, \y0, |i, p^ ) . Note that this connection on J~l([l) T*S* is the pull-back of the
connection on T2 » S1 , (9, \\f) h-» \\f , whose horizontal space at any point is generated by ,
by the map which is the restriction of the cotangent bundle projection T*T2 - T2 to J - 1 ^) -
Relative to this connection and identifying
T*T2
with
TT2
using the kinetic energy metric
ds2
=
d(p2
4-
d\i/2,
u is the generator of the vertical part of this curve; note II7— II = II 7—II = 1.
Ildp II II dv II
Thus the differential equation in Step 2 is on the group S1 and has right hand side given by the
generator of the vertical part of the horizontally lifted curve in Step 1. Roughly, this describes
the method of reconstruction of dynamics. We shall explain this in §2 and address the
specific case of Lagrangian systems in §3, circumventing the use of the connection in Step 1.
§1D Coupled bodies, linkages and optimal control
The above example can be generalized to the case of coupled rigid bodies. Already the case
of a single rigid body in space is an interesting example that will be discussed in §1G below. For
several coupled rigid bodies, the dynamics is quite complex. For instance for bodies in the plane,
the dynamics is known to be chaotic, despite the presence of stable relative equilibria. See Oh,
Sreenath, Krishnaprasad, and Marsden [1989]. Berry phase phenomena for this type of example
are quite interesting and are related to some of the work of Wilczek and Shapere on locomotion in
micro-organisms. (See, for example, Shapere and Wilczek [1987]). In this problem, control of the
system's internal variables can lead to phase changes in the external variables. These choices of
variables are related to the variables in the reduced and the unreduced phase spaces, as we shall
see. In this setting one can formulate interesting questions of optimal control such as "when a cat
falls and turns itself over in mid flight (all the time with zero angular momentum!) does it do so
with optimal efficiency in terms of say energy expended?"
Previous Page Next Page