1 Some Examples

In this section we present some elementary examples exhibiting the general geometric

features that will be discussed in the body of the paper. They focus on the ideas of reconstruction

of dynamics and the "phases" obtained when reconstruction is performed on a closed loop. In this

case, we shall distinguish between a geometric and a dynamic phase. Such phenomena naturally

occur in Hamiltonian systems depending on a parameter, for example, in moving systems or in

integrable systems depending on a "slow" parameter. The reader will find additional examples in

§5. In particular, the rotating top in a gravitational field (the heavy top) and the dynamics of a

system of planar coupled rigid bodies are treated there. The formula for the phase of the system of

coupled planar rigid bodies is first computed by hand in §5E, so this can be read as part of the

present section if desired.

Before beginning any serious examples, we will give an elementary example—Elroy's

beanie—which still illustrates many of the interesting features of more complicated examples. In

general, the theory and examples in this work can be divided into two types—those involving

adiabatic phenomena and those that are "pure mechanical" or "pure reconstruction". Our first main

example on moving systems in §1A is of the adiabatic type, while Elroy's beanie is purely

mechanical.

Example—Elroy's Beanie Consider two planar rigid bodies joined by a pin joint at their

center of masses. Let I2 and I2 be their moments of inertia, and 0

p

and 02 be the angle they

make with a fixed inertial direction, as in the figure.

Elroy's Beanie

inertial frame

3