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 a£ 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