  Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The
G-action lifts to an action on the total space T of the cotangent bundle of Q
and hence on the standard symplectic Poisson algebra of smooth functions on T.
The Poisson algebra of G-invariant functions on T yields a Poisson structure on
the space T/G of G-orbits. We relate this Poisson algebra with extensions of
Lie-Rinehart algebras and derive an explicit formula for this Poisson structure
in terms of differentials. We then show, for the particular case where the
G-action on Q is principal, how an explicit description of the Poisson algebra
derived in the literature by an ad hoc construction is essentially a special
case of the formula for the corresponding extension of Lie-Rinehart algebras.
By means of various examples, we also show that this kind of description breaks
down when the G-action does not define a principal bundle.
