I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary cotangent bundle of configuration space -- because there are natural problems where the configuration space is not trivial. **But what motivation is there, from a physical standpoint, for passing further still into the realm of non-exact symplectic manifolds or those that can't be realized as a cotangent bundle?** By an exact symplectic manifold, I mean one where the symplectic form is exact, the differential of a 1-form. 

(I also don't understand whether those two ideas -- non-exact symplectic manifolds and symplectic manifolds that can't be realized as a cotangent bundle -- are equivalent. The question was asked [here][1], but I don't see a simple "yes" or "no." That could be because the answer isn't known, or it could be because I don't have a formal enough definition for 'sympletic manfiold that can be realized as a cotangent bundle.')

José Figueroa-O'Farrill gave a partial answer to this question in his answer [here][2]. He writes: 

> Not every space of states is a cotangent bundle, of course. One can
> obtain examples by hamiltonian reduction from cotangent bundles by
> symmetries which are induced from diffeomorphisms of the configuration
> space, for instance. Or you could consider systems whose physical
> trajectories satisfy an ODE of order higher than 2, in which case the
> cotangent bundle is not the space of states, since you need to know
> more than just the position and the velocity at a point in order to
> determine the physical trajectory.

I don't know much about symplectic reduction, but I don't see it as a very natural example, since there must have been a more fundamental problem that didn't demand a general symplectic manifold instead of a cotangent bundle. The example about ODEs of order greater than 2 is interesting. But I'm wondering if anyone can offer a fuller explanation about the role arbitrary symplectic manifolds play in *physics* rather than math. I suspect part of the story will be about quantization.


  [1]: http://mathoverflow.net/questions/35900/when-is-a-symplectic-manifold-equivalent-to-a-cotangent-bundle
  [2]: http://mathoverflow.net/questions/16460/how-to-see-the-phase-space-of-a-physical-system-as-the-cotangent-bundle