16 Marsden, Montgomery, and Ratiu
In this paper we will put this geometry into a more general context and will synthesise it with our
work on connections associated with moving systems.
§1G The Rigid Body
The motion of a rigid body is a geodesic with respect to a left-invariant Riemannian metric
(the inertia tensor) on S0(3). The corresponding phase space is P = T*S0(3) and the momentum
map J : P
R3
for the left S0(3) action is right translation to the identity. We identify so(3)*
with so(3) via the Killing form and identify
R3
with so(3) via the map v h ) v where v(w) =
v x w, x being the standard cross product. Points in so (3)* are regarded as the left reduction of
T SO (3) by SO (3) and are the angular momenta as seen from a body-fixed frame. The reduced
spaces
J~l(\i)/G
are identified with spheres in
R3
of Euclidean radius ilM-ll with their
symplectic form co = - dS /|| |i || where dS is the standard area form on a sphere of radius || |i ||
and where G consists of rotations about the |i.-axis. The trajectories of the reduced dynamics
are obtained by intersecting a family of homothetic ellipsoids (the energy ellipsoids) with the
angular momentum spheres. In particular, all but at most four of the reduced trajectories are
periodic. These four exceptional trajectories are the well known homoclinic trajectories.
Suppose a reduced trajectory FI(t) is given on P , with period T. After time T, by how
much has the rigid body rotated in space? The spatial angular momentum is n = |i = gll, which is
the conserved value of J . Here g e SO (3) is the attitude of the rigid body and E L is the body
angular momentum. If 11(0) = II(T) then p. = g(0)n(0) = g(T)II(T) and so gCO"1^ = g(O)-1}!
i.e.,
g(T)g(0)-1
is a rotation about the axis \i. We want to compute the angle of this rotation.
To answer this question, let c(t) be the corresponding trajectory in J_1(M) c P. Identify
T*SO(3) with 80(3) x
R3
by left trivialization, so c(t) gets identified with (g(t), Il(t)). Since
the reduced trajectory TI(t) closes after time T, we recover the fact that c(T) = g c(0) for some g
G G . Here, g = g(T)g(0)_1 in the preceding notation. Thus, we can write
g = exp[(Ae)Q (1)
where £ = |i/|||i|| identifies Q with R by h^a, for a e R. Let D be one of the two
spherical caps on
S2
enclosed by the reduced trajectory, A be the corresponding oriented solid
angle, i.e., | A| = (area
D)/|||i||2,
and let H be the energy of the reduced trajectory. All norms
are taken relative to the Euclidean metric of
R3.
We shall prove below that modulo 2TC, we have
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