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In the remaining two chapters of this part we are going to take a different technical approach to the reduction by stages problem. It will be mainly based on thinking of the (connected components of the) level sets of the various momentum maps in the setup as the accessible sets of a distribution that we will introduce in the following paragraphs. This point of view has been exploited in Ortega [2002] and Ortega and Ratiu [2002, 2004a] in the context of the so-called optimal momentum map .

In this chapter we will use the distribution theoretical approach to formulate a reduction by stages theorem that only requires an easily verifiable point set topological condition. This condition is satisfied by a large class of Lie groups, for example, compact ones. Notice that this statement could not have been made had we followed exclusively the purely algebraic approach in §5.2. Having said that, we will analyze the relation between the stages theorem in this chapter and that in the previous one.

As we already pointed out the main difference between the point and orbit reduced spaces is the invariance properties of the submanifolds out of which they are constructed. More specifically, if we mimic in the optimal context the standard orbit reduction procedure, the optimal orbit reduced space that we should study is G · J ^−1( ρ )/ G  =  J ^−1( O _ ρ )/ G , where O _ ρ  :=  G · ⊂  M / A ʹ_ G . The following pages constitute an in-depth study of this quotient and its relation with new (pre)-symplectic manifolds that can be used to reproduce the classical orbit reduction program and expressions.

As we already saw in Part II, the reduction by stages procedure consists of carrying out reduction in two shots using the normal subgroups of the symmetry group. To be more specific, suppose that we are in the same setup as Theorem 13.5.1 and that the symmetry group G has a closed normal subgroup N . In this chapter we will spell out the conditions under which optimal reduction by G renders the same result as reduction in the following two stages: we first reduce by N ; the resulting space inherits symmetry properties coming from the quotient Lie group G/N that can be used to reduce one more time.