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?"