346 J. E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRYwhere the sum is a Whitney sum of vector bundles over Q/G (fiberwise a direct
sum) and fj is the associated vector bundle to the co-adjoint action of G on g.
The description of the Poisson structure on this bundle (a synthesis of the canonical
bracket, the Lie-Poisson bracket and curvature) may be found in Cendra, Marsden
and Ratiu [1998].
Lagrangian reduction. The Lagrangian analogue of the bundle picture is the
dual isomorphism
TQ/G ~ T(Q/G) EB gwhose geometry is developed in Cendra, Marsden and Ratiu [1998]. In particular,
the equations and variational principles are developed on this space. For Q = G
this reduces to the Euler-Poincare picture we had previously. For G abelian, it
reduces to the Routh procedure.
Ifwe have an invariant Lagrangian on TQ it induces a Lagrangian l on (TQ)/G
and hence on T( Q/G)EBg. Calling the variables r°', r°' and D°', the resulting reduced
Euler-Lagrange equations, also called the Lagrange-Poincare equations, (im-
plicitly contained in Cendra, Ibort and Marsden [1987] and explicitly in Marsden
and Scheurle [1993b]) are
d 8l
dt or°'
az
or°'
d 8l
dt ()[lboDP 8l ( -B°' a/3r ·/3 + '>ca ad" n d) G
()[la 8l ( -C,ca af3r. a + c a db" nd) G
where B~/3 is the curvature of the connection A~, cgd are the structure constants
of the Lie algebra g and where ~~d = cgdA~.
Using the geometry of the bundle TQ/G = T(Q/G) EB g, one obtains a nice
interpretation of these equations in terms of covariant derivatives. One easily gets
the dynamics of particles in a Yang-Mills field (these are called Wong's equation)
as a special case; see Cendra, Holm, Marsden and Ratiu [1998] for this example.
Wong's equations will come up again in Lecture 4 on optimal control.
We also mention that methods of Lagrangian reduction have proven very useful
in optimal control problems. It was used in Koon and Marsden [1997] to extend
the falling cat theorem of Montgomery [1990] to the case of nonholonomic systems.
Cotangent bundle reduction is very interesting for group extensions, such as
the Bott-Virasoro group described earlier, where the Gelfand-Fuchs cocycle may be
interpreted as the curvature of a mechanical connection. This is closely related to
work of Marsden, Misiolek, Perlmutter and Ratiu [1998a,b] on reduction by stages.
This work in turn is an outgrowth of earlier work of Guillemin and Sternberg
[1980], Marsden, Ratiu and Weinstein [1984a,b] and many others on systems such
as the heavy top, compressible flow and MHD. It also applies to underwater vehicle
dynamics as shown in Leonard [1996] and Leonard and Marsden [1997].
In semidirect product reduction, one supposes that G acts on a vector space V
(and hence on its dual V*). From G and V we form the semidirect product Lie
group S = G@V, the set G x V with multiplication
(91, vi) · (92, V2) = (9192, V1 + 91 V2) ·
The Euclidean group SE(3) = S0(3)@ffi.^3 , the semidirect product of rotations
and translations is a basic example. Now suppose we have a Hamiltonian on TG
that is invariant under the isotropy Ga 0 for ao E V. The semidirect product