In this chapter we discuss the cotangent bundle reduction theorem. Versions of this are already given in Smale [1970], but primarily for the abelian case. This was amplified in the work of Satzer [1977] and motivated by this, was extended to the nonabelian case in Abraham and Marsden [1978]. An important formulation of this was given by Kummer [1981] in terms of connections. Building on this, the “bundle picture” was developed by Montgomery, Marsden and Ratiu [1984] and Montgomery [1986].

From the symplectic viewpoint, the principal result is that the reduction of a cotangent bundle T*Q at µ ∈ g* is a bundle over T*(Q/G) with fiber the coadjoint orbit through µ. Here, S = Q/G is called shape space. From the Poisson viewpoint, this reads: (T*Q)/G is a g*-bundle over T*(Q/G), or a Lie-Poisson bundle over the cotangent bundle of shape space. We describe the geometry of this reduction using the mechanical connection and explicate the reduced symplectic structure and the reduced Hamiltonian for simple mechanical systems.

Mechanical G-systems

By a symplectic (resp. Poisson) G-system we mean a symplectic (resp. Poisson) manifold (P, Ω) together with the symplectic action of a Lie group G on P, an equivariant momentum map J : P → g* and a G-invariant Hamiltonian H : P → ℝ.

Following terminology of Smale [1970], we refer to the following special case of a symplectic G-system as a simple mechanical G-system.