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
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