motivating the conservation of symplectic area by way of general (coordinate) covariance I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for turning a Hamiltonian into a vector field, the integral curves of which will foliate phase space into orbits.
Of course, this criterion alone isn't enough; there are many structures that can fulfill this sort of need -- any old rank-2 tensor field might do. Most people who have read about symplectic geometry on the internet probably know how Henry Cohn fills in the extra details, here:
http://research.microsoft.com/en-us/um/people/cohn/Thoughts/symplectic.html
The hardest part of the motivation, I think, is explaining why the tensor field that converts the Hamiltonian differential dH into a vector field needs to be alternating. Cohn tells a nice story about how conservation of energy along the flows of a Hamiltonian vector field forces the tensor to be alternating.
But I'm wondering whether there isn't an alternative -- perhaps more general -- story to tell, by way of general covariance. Couldn't we just say that if we want Hamilton's equations to be formally invariant under a smooth coordinate change, we need the symplectic area form to be conserved? As I understand it, a "canonical transformation" is just a diffeomorphism of phase space coordinates that leaves Hamilton's equations invariant, and Poincare showed that a diffeomorphism is a canonical transformation if and only if it preserves the symplectic area form.
My question is -- is all of this right? Is the definition of "canonical transformation" really just "diffeomorphism that leaves Hamilton's equations invariant," and is it true that a diffeomorphism on phase space is canonical if and only if it preserves the symplectic area form?
It seems to me that this is a much more straightforward way to motivate the at first bizarre-seeming symplectic formalism than is usually given -- because it falls right out of a more general physical principle, diffeomorphism invariance. In other words, it's a top-down motivation (we need symplectic area to be conserved if we're going to be covariant), instead of an unilluminating bottom-up motivation (look, symplectic area happens to be conserved, so we should do everything with a symplectic 2-form!).
 A: As a physics undergraduate a canonical transformation has been introduced as a diffeomorphism that leaves the canonical equations invariant. There was, however, no differential geometry involved at all. 
A semester later in a course on differential geometry I stumbled upon Arnold's Mathematical Methods of Classical mechanics, where a canonical transformation is a diffeomorphism of phase space (as a $2n$-dimensional symplectic manifold) $f:P\to P$ that preserves the symplectic 2-form $\omega$, i.e. $\omega=f^*\omega$.
Understanding the relation between the two definitions of canonical transformations requires the construction of the Hamiltonian phase flow via the Isomorphism $I^{-1}:T_xP\to T_x^*P$ obtained by the symplectic 2-form $\omega$. Thus, to the 1-form $\mathrm{d}H$, given by the exterior derivative of the Hamiltonian, we can define the Hamiltonian vector field $I\mathrm{d}H$, which yields for a curve $\mathbf{x}:\mathbb{R}\to P$ the well known canonical equations $$\dot{\mathbf{x}}=(\dot{\mathbf{q}},\dot{\mathbf{p}})=I\mathrm{d}H\qquad\Leftrightarrow\qquad\dot{\mathbf{p}}=-\frac{\partial H}{\partial\mathbf{q}},~\dot{\mathbf{q}}=\frac{\partial H}{\partial\mathbf{p}}.$$ 
In this manner one finds starting from the canonical equations in its simplest derivation (from Legendre transformation of the Lagrangian), that a symplectic structure on the phase space is necessary to reformulate Hamiltonian mechanics in terms of vector fields and flows. Only with this explicit use of the symplectic 2-form it seems meaningful to postulate invariance of canonical equations w.r.t. symplectomorphisms. So this postulate does not really explain the need for symplectic geometry in the Hamiltonian formalism.
I agree with you that postulating general invariance under symplectomorphisms in Hamiltonatian mechanics seems more satisfying, but my perception is that this would be the second step before the first, since most physicists learn canonical mechanics before differential geometry. (As a side note: It seems that, opposed to (modern) mathematicians, physicists sometimes tend to keep things in the way they were constructed at the cost of simplicity, especially in differential geometry where one wants to define coordinate free global expressions. (E.g. many physicists define the tangent space via coordinate vectors of Euclidean space and then glue them together by an equivalence relation to forget those coordinates and obtain a well defined tangent space.)
