Phase space reduction and Poisson structure

Let (P,π,B,G) be a G-principal fiber bundle. The action of G on the cotangent bundle T*P is free and Hamiltonian. By Liberman and Marle [Symplectic Geometry and Analytical Mechanics (Reidel, Dortrecht, 1987)] and Marsden and Ratiu [Lett. Math. Phys. 11, 161 (1981)] the quotient space T*P/G is a Poisson manifold. We will determine the Poisson bracket on the reduced Poisson manifold T*P/G, and its symplectic leaves.