The Orbit Bundle Picture of Cotangent Bundle Reduction

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T ∗Q/G, decomposed as a Whitney sum bundle, T ∗(Q/G) ⊕ g̃∗ over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found.