364 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRYreduction y S l
""'"'" l n by V
Figure 2. 7. Reduction by stages.given in Lecture 1, we get
l1 JD+ Dv
P Mv+DTD.Here, J is the sum of the body inertia matrix plus the added inertia matrix and M
is the mass matrix for the body alone, plus the added mass matrix associated with
the fluid.
For an ellipsoidal body, with appropriate choice of body-fixed coordinate frame,
M and J are diagonal and D = mfa, where ra is the vector from the center of
buoyancy to the center of gravity. This simplifies matters somewhat.
The underwater vehicle dynamics has Lie-Poisson form on tu . This fact follows
by constructing the physical phase space of T SE(3) reduced by the symmetry
group SE(2) x IR with the aid of th<) semidirect product reduction theory. The
Hamiltonian comes from the Lagrangian in Lecture 1 via the Legendre transform