4 Marsden, Montgomery, and Ratiu

Elroy and his beanie

Conservation of angular momentum states that l1Ql + I202 = M - = constant in time, where

the overdot means time derivative. The shape space of a system is the space whose points give

the shape of the system. In this case, shape space is the circle

S1

parametrized by the hinge

angle \\f = 02 - 0X . We parametrize the configuration space of the system not by 0X and 02

but by 0 = 0X and \|/ . Conservation of angular momentum reads

L0 + L(0 + w) = a ; thatis, d0 + z—dy = — - — d t . (1)

I1

+

I2 I1

+

I2

The left hand side of (1) is the mechanical connection discussed in detail in §2.4. Suppose that the

beanie (body #2) goes through one full revolution so that \\f increases from 0 to 2n . Suppose,

moreover, that the total angular momentum is zero: [I = 0. From (1) we see that the entire

configuration undergoes a net rotation of