344 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
maps the Euler-Poincare equations to the Lie-Poisson equations and vice-versa.^3
Lie-Poisson systems have a remarkable property; they leave the coadjoint orbits
in g invariant. In fact the coadjoint orbits are the symplectic leaves of g. For
each of examples 1 and 3, the reader may check directly that the equations are Lie-
Poisson and that the coadjoint orbits are preserved. For example 2, the preservation
of coadjoint orbits is essentially Kelvin's circulation theorem. See Marsden and
Weinstein [1983] for details. For example 1, the coadjoint orbits are the familiar
momentum spheres, shown in figure 1.1.
Figure 1.1. The rigid body momentum sphere.History and literature. Lie-Poisson brackets were known to Lie around 1890 ,
but apparently this aspect of the theory was not picked up by Poincare. The coad-
joint orbit symplectic structure was discovered by Kirillov, Kostant and Souriau
in the 1960's. They were shown to be symplectic reduced spaces by Marsden and
Weinstein [1974]. It is not clear who first observed explicitly that g* inherits the
Lie-Poisson structure by reduction as in the preceding Lie-Poisson reduction theo-
rem. It is implicit in many works such as Lie [1890], Kirillov [1962], Guillemin and
Sternberg [1980] and Marsden and Weinstein [1982, 1983], but is explicit in Holmes
and Marsden [1983] and Marsden, Weinstein, Ratiu, Schmid and Spencer [1983].
Symplectic and Poisson reduction. The ways in which reduction has been
generalized and applied has b een nothing short of phenomenal. Let me sketch just
a few of the highlights (eliminating many important references).
First of all, ill an effort to synthesize coadjoint orbit reduction (suggested by
work of Arnold, Kirillov, Kostant and Souriau) with techniques for the reduction of
cotangent bundles by Abelian groups of Smale [1970], Marsden and Weinstein [1974]
(^3) A cautionary note. The heavy top is a n example of a Lie-Poisson system on se(3)*. However, its
inverse Legendre transformation (using the standard h) is degenerate! This is an indication that
something is missing on the Lagra ngian side and this is indeed the case. The resolution is found
in Holm, Marsden and Ratiu [1998a].