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?



Classical mechanics motivation for poisson manifolds?
For many reasons and purposes, it is the Poisson bracket, not the symplectic form, that plays a primary role.


* Equations of motion and, more generally, the evolution of observables have an easy form:
$$ \frac{\partial f}{\partial t}=\{H,f\}.$$
* Conserved quantities form a Poisson subalgebra:
$$\{H,F\}=\{H,G\}=0 \implies \{H,\{F,G\}\}=0. $$
* Since symmetries (Hamiltonian group actions) are Poisson by nature, the moment map is defined in the Poisson setting:
$$ M\ni P\mapsto (X \mapsto H\_X(P)).$$
Unlike in the symplectic case, *both* steps in the Hamiltonian reduction, restriction to the level set and factorization by the action of the stabilizer, are naturally carried out in the Poisson category, even for singular reduction.
* Quasiclassical approximation in quantum mechanics (and conversely, quantization of classical mechanical systems) is expressed via the Poisson bracket:
$$[\hat{F},\hat{G}]=ih\widehat{\{F,G\}} +O(h^2). $$


On the other hand, many natural systems have a degenerate Poisson bracket and/or are infinite-dimensional.


* The phase space of various tops is the dual space $\mathfrak{g}^\*$ of a Lie algebra. This is a universal example of a linear Poisson structure. Symplectic leaves are the coadjoint orbits and the Poisson center is given by the (classical) "Casimir functions". 
* Classical integrable systems such as KdV admit a bi-Hamiltonian structure (i.e. a pair of compatible Poisson brackets). This has no analogues in symplectic theory. 
* Some of these structures are obtained by a reduction from a linear Poisson structure on a suitable infinite-dimensional Lie algebra (loop algebra, algebra of matrix differential or pseudo-differential operators, etc).


Finally, a related practical consideration: even if you are interested in studying a symplectic manifold, frequently this is best accomplished by embedding it as a symplectic leaf into a simpler Poisson manifold, *whether or not it has direct physical meaning.* In particular, this applies to flag manifolds of semisimple Lie groups, which are topologically complicated objects, but can be identified with coadjoint orbits, thus embedded into a vector space with a linear Poisson structure.


