The Serret-Andoyer transformation is a classical method for reducing the fr
ee rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimension
al Hamiltonian flow. First, we show that this transformation is the computa
tion, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein)
reduction associated with the lifted left-action of SO(3) on T*SO(3)-a gene
ralization and extension of Noether's theorem far Hamiltonian systems with
symmetry. In fact, we go on to generalize the Serret-Andoyer transformation
to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperrg
ular Hamiltonian functions. Interpretations of the Serret-Andoyer variables
, both as Eulerian coordinates and as canonical coordinates of the co-adjoi
nt orbit, are given. Next, we apply the result obtained to the controlled r
igid body with momentum wheels. For the class of Hamiltonian controls that
preserve the symmetry on T*SO(3), the closed-loop motion of the main body c
an again be reduced to canonical form. This simplifies the stability proof
for relative equilibria, which then amounts to verifying the classical Lagr
ange-Dirichlet criterion. Additionally, issues regarding numerical integrat
ion of closed-loop dynamics are also discussed. Part of this work has been
presented in [16].