1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. REDUCTION THEORY 347

reduction theorem states that reduction of T * G by Gao gives reduced spaces that

are symplectically diffeomorphic to coadjoint orbits in the dual of the Lie algebra of

the semi-direct product: (g@ V)*.

This is a very important construction in applications where one has "advected

quantities" (such as density in compressible flow). Its Lagrangian counterpart,

which is not simply the Euler-Poincare equations on g @ V , is developed in Holm,

Marsden and Ratiu [1998a] along with applications to continuum mechanics. Cen-

dra, Holm, Hoyle and Marsden [1998] have applied this idea to the Maxwell-Vlasov

equations of plasma physics.

If one reduces the semidirect product group S = G@ V in two stages, first by

V and then by G, one recovers the semidirect product reduction theorem mentioned

above.

A far reaching generalization of this semidirect product t heory is given in Mars-

den, Misiolek, Perlmutter and Ratiu [1998a,b] in which one has a group M with a

normal subgroup N C M and M acts on a symplectic manifold P. One wants to

reduce P in two stages, first by N and then by M / N. On the Poisson level this is

easy: P/M ~ (P/N)/(M/N) but on the symplectic level it is quite subtle. Cendra,

Marsden and Ratiu [1998] have developed a Lagrangian counterpart to reduction

by stages.

Singular reduction. Singular reduction starts with the observation of Smale

[1970] that z E Pis a regular point of J iff z has no continuous isotropy. Motivated

by this, Arms, Marsden and Moncrief [1981] showed that the level sets J-^1 (0)

of an equivariant momentum map J have quadratic singularities at points with

continuous symmetry. While easy for compact group actions, their main examples

were infinite dimensional! The structure of 1 -^1 (0)/G for compact groups was

developed in Sjamaar and Lerman [1990], and extended to 1-^1 (μ)/G,, by Bates and

Lerman [1997] and Ortega and Ratiu [1997a]. Many specific examples of singular

reduction and further references may be found in Bates and Cushman [1997].

The method of invariants. An important method for the reduction construction

is called the method of invariants. This method seeks to parameterize quotient

spaces by functions that are invariant under the group action. The method has a
rich history going back to Hilbert's invariant theory and it has much deep math-
ematics associated with it. It has been of great use in bifurcation with symmetry
(see Golubitsky, Stewart and Schaeffer [1988] for instance).
In mechanics, the method was developed by Kummer, Cushman, Rod and
coworkers in the 1980 's. We will not attempt to give a literature survey here, other
than to refer to Kummer [1990], Kirk, Marsden and Silber [1996] and the book of
Bates and Cushman [1997] for more details and references. We shall illustrate the
method with a famous system, the three wave interaction, based on Alber, Luther,
Marsden and Robbins [1998b].