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, 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. 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 general symplectic manifolds play in physics rather than math. I suspect part of the story will be about quantization.
EDIT: I just saw this question on whether symplectic reduction can be considered "interesting from a physical point of view." Figured it was appropriate to link to here, although I'm still interested in any bigger-picture insights that don't have to do with reduction.
Best Answer
Actually the first case in history of a symplectic manifold wasn't a cotangent space. It was the space of Keplerian motions of a planet, represented locally by its Keplerian elements. The Lagrange symplectic structure on this space is defined by the so-called Lagrange parenthesis he introduced at this time (three papers in 1808/09/10)(*). That manifold is actually even non Hausdorff, but its greatest hausdorff quotient is a still a manifold (this is known as the "regularization theorem"). This manifold is symplectic but not a cotangent (but however contractible to the sphere $S^3$). Extended with the repulsive motions, it is an algebraic manifold defined by the following equations [Sou]. $$ \left\{ \begin{array}{rcl} ||{A}||^2 -fx^2 & = & 1 \\ y^2 - f ||{B}||^2 & = & 1 \end{array} \right. \quad \& \quad \left\{ \begin{array}{rcl} A \cdot B - xy &=& 0 \\ xy - f\tau &=& 0, \end{array} \right. $$ where $A,B \in {\bf R}^3$ and $x,y,f,\tau$ belong to $\bf R$. Actually this manifold is the result of the gluing of $TS^3$ and $TH^3$ along $TS^2\times {\bf R}$ (where $H^3$ is the 3 dimensional pseudo-sphere). I made the following picture, for $A$ and $B$ in $\bf R$, to get a visual idea of the manifold. The bottom represents the $TS^3$ part, the top represents $TH^3$ and it is glued along two lines representing $TS^2\times {\bf R} \simeq S^2 \times {\bf R}^3$.
Remark. There exists also the examples of compact symplectic manifolds representing internal degrees of symmetries, as mentioned in Tobias answer. In the same spirit there is the Grassmannian manifolds ${\rm Gr}(2,n+1)$ of $2$-planes in ${\bf R}^{n+1}$, representing the space of un-parametrized geodesics on the sphere $S^n$. We can regard this space of geodesics as the space of light rays on the Euclidean sphere where the speed of light would be infinite.
---------- Edit March 28, 2017
On a conceptual point of view, I just finished to write a paper:
Universal Structure Of Symplectic Manifolds
http://math.huji.ac.il/~piz/documents/ESMIACO.pdf,
---------- Edit November 15, 2019
http://math.huji.ac.il/~piz/documents/ESMIALCO.pdf,
This paper has been enhanced to make the symplectic manifold an orbit of the linear coadjoint action of a central extension of the group of Hamiltonian diffeomorphisms, independently of the group of periods. That is, even if the symplectic form is not integral.
that proposes a way, based on diffeology, to understand the statement: "Every symplectic manifold is a coadjoint orbit".
---------- Notes
(*) I published this paper on the origins of symplectic geometry, but in french, where Lagrange's construction is explained.
---------- Reference
[Sou] Jean-Marie Souriau. Géométrie globale du problème à deux corps. In Modern Developments in Analytical Mechanics, pp. 369-418. Accademia della Scienza di Torino, 1983.