Symmetry, Reduction, and Phases in Mechanics 5

A6

= 2 _

d ¥ =

Ii + I^O

2% (

T

\

l2

V1!

+ h J

2% . (2)

This is the amount by which Elroy rotates, each time his beanie goes around once.

Notice that the result (2) is independent of the detailed dynamics and only depends on the

fact that angular momentum is conserved and the beanie goes around once. In particular, we get the

same answer even if there is a "hinge potential" hindering the motion or if there is a control present

L

in the joint. Also note that if Elroy wants to rotate by - 27ck radians, where k is an

integer, all he needs to do is spin his beanie around k times, then reach up and stop it. By

conservation of angular momentum, he will stay in that orientation after stopping the beanie.

Here is a geometric interpretation of this calculation. The connection we used is Amech =

d9 + j d\|/. This is a flat connection for the trivial principal S1 -bundle % : S1 x S1 - S1

1 + *2

given byTC(9,y) = \\f . Formula (2) is the holonomy of this connection, when we traverse the

base circle, 0 \j/ 2K . (We note that this is the same connection that appears in the Aharonov-

Bohm effect.)

§1A Moving systems

Begin with a reference configuration Q and a Riemannian manifold 5- Let M be a space

of embeddings of Q into S and let mt be a curve in M . If a particle in Q is following a curve

q(t), and if we imagine the configuration space Q moving by the motion mt, then the path of the

particle in S is given by mt(q(t)). Thus, its velocity in S is given by the time derivative:

Tq(i)m{q(t) + Zt(mt(q(t))) (1)

where Zt, defined by Zt(mt(q)) = -rr mt(q), is the time dependent vector field (on S with

domain mt(Q)) generated by the motion mt and T (t)mtw is the derivative (tangent) of the map

n^ at the point q(t) in the direction w. To simplify the notation, we write

T^ = Tq(t)mt and g(t) = mt(q(t)).

Consider a Lagrangian on TQ of the form kinetic minus potential energy. Using (1), we thus

choose

L

m

t

^'

v ) =

2 «

mt'v

+ W ) )

II2

- V(q) - U(#t)) (2)

where V is a given potential on Q and U is a given potential on S.