TITLE: Classical mechanics motivation for poisson manifolds?
QUESTION [24 upvotes]: Suppose I want to understand classical mechanics. 
Why should I be interested in arbitrary poisson manifolds and not just in symplectic ones?
What are examples of systems best described by non symplectic poisson manifolds?

REPLY [7 votes]: The following  article by J. Butterfield suggests 3 reasons motivating Poisson manifolds (over symplectic manifolds)(page 81)

Parameters and stability. Sometimes it is easier to analyze stability of the dynamics on a Poisson manifold than on its symplectic leaves.
Naturality: The rigid rotator problem is more natural to analyze on SO(3) than on its coadjoint orbit.

3.Reduction: When the configuration space is a Lie group, it is natural to use T*G/G as the reduced phase space. This is a Poisson manifold isomorphic to the dual of the Lie algebra and having a Lie Poisson structure.