362 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
0.1 0 .1cf^0 >^0
- 1 0.^1
0 10 20 30 40 0 10 20 30 40 - 1 0. 1
N 0 N 0
a >- 1 0 .1
0 10 20 30 40 0 10 20 30 40
2
2
d' (^1) > "'
0 0
1 1
0 10 20 30 40 0 10 20 30 40
F igure 2.5. The velocities are stable.
Cor o llary 2.2. (Semidirect product reduction.) The reduction ofT*G by G a
at values μa = μ jga gives a space that is isomorphic to the coadjoint orbit through
the point()= (μ, a) E .s* = g* x V*, the dual of the Lie algebra .s of S.Of course if one has an S -invariant Hamiltonian H , one can reduce the dynamics
of a given H in two stages.
This theorem explains why one gets Lie-Poisson dynamics for the underwater
vehicle system on the dual of the Lie algebra of the semidirect product S0(3)@(IR^3 x
IR^3 ), starting with the physical configuration space Q = SE(3) and reducing it by
the symmetry group SE(2) x IR^2. This remarkable fact can also be checked directly
from the dynamical equations which we write down shortly.
Generalized stability theorem. To deal with the non-compactness, we make
some special assumptions. The action of (Ga)μa on g~ is proper and there is an
inner product on g~ that is invariant under this action; for example, both of these
conditions hold if (Ga)μa is compact (and this holds if G is compact).
Choose a vector subspace E[ze] C kerDJa ([ze]) that complements the tangent
space to the (Ga)μa-orbit of [ze]· Let the Hamiltonian reduced to the first stage
space Pa be denoted H a and assume that the second derivative of H a - (Ja)t.e at
[ze] E Pa restricted to E[ze] is definite.
Theor e m 2.3. In addition to the above, assume μa E g~ is generic. Then the
point [ze] E Pa is Liapunov stable modulo the action of (Ga)μa for the dynamics of