TITLE: What is so geometric about symplectic geometry?
QUESTION [37 upvotes]: Symplectic geometry is often motivated by the Hamilton's equation which in turn are a reformulation of Newton's third law. But the subject itself is of independent mathematical interest. What I don't understand is why symplectic geometry is, in fact, geometry. Compared with Riemannian geometry where angles, lengths and geodesics can be defined and which can be used to provide models for classical geometry, the notion of a smooth manifold equipped with some two-form doesn't really seem geometric to me at all.
In fact I would argue that the aspect of providing a model for classical geometries is the defining feature of Riemannian geometry making it "geometry".
The only "geometric interpretion" of symplectic geometry I always see is "symplectic geometry is the geometry of phase spaces" with no mention what the geometric content of phase spaces actually is. Notions like a two-form or a Liouville form seem to me much more algebraic than geometric in nature.
Is there a more conceptual way of understanding symplectic structures by motivating them through geometric properties? Or perhaps even some axioms of some geometric space for which symplectic geometry provides a concrete framework to construct a model for?
A remark: One could answer this by defining geometry as the study of some space with an associated symmetry group (in the Riemannian case isometries and in the symplectic case symplectomorphisms), but this only moves the question to why one would care about symplectomorphisms from a geometric viewpoint. The case of isometries again is rather clear….

REPLY [8 votes]: I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take examples from or around my work.
1. A classic and undeniably geometric result by Cauchy says that the surface area of a convex body $K$ in Euclidean 3-space (to fix dimensions) is $4$ times the average of the areas of its orthogonal projections onto planes.  Equivalently, it is equal to $\frac{1}{\pi}$ the volume of the set of oriented straight lines that cross it.
Is that more geometric than: the open unit codisc bundle for the Riemannian metric on $\partial K$ is symplectomorphic to the set of oriented straight lines that cross the interior of $K$? The latter statement is more basic and not only says more but can be easily generalized to normed spaces and other Finsler manifolds giving you integral geometric formulas in all those spaces.
2. A lovely result by Banchoff states that if a surface $S \subset \mathbb{R}^3$ is diffeomorphic to a sphere and no plane or (round) sphere separates it into more than two connected pieces, then $S$ is itself a round sphere. That has to be as geometric as one can hope for, but I don't think it is much more geometric than
Let $L_S$ be the set of oriented lines perpendicular to $S$ seen as an immersed submanifold in the space of oriented lines in $\mathbb{R^3}$ and let $\hat{x}$ denote the sphere of all oriented lines passing through a point $x \in \mathbb{R}^3$. If every sphere $\hat{x}$ that intersects $L_S$ transversally intersects it in exactly two points, then $L_S$ is itself the set of all oriented lines passing through some point in 3-space.
The second (symplectic) version can be immediately generalized not only to more general Lagrangian submanifolds than normal congruences to smooth surfaces, but also to normed spaces, Hadamard manifolds, metrics arising as solutions of Hilbert's fourth problem, etc.
3. A nice problem by Juan-Jorge Schäffer in the theory of Banach spaces asked whether  the girth of a Banach space (the infimum of the lengths of centrally symmetric curves on its unit sphere) equaled the girth of its dual. It turned out the (positive) solution was easy, IF you had the idea to think the problem "symplectically".
To see the "geometry" in symplectic geometry, you also have to look at its impressive array of applications to other geometries that may look more geometric to you at first, and start building "dictionaries" in your mind. The many books and papers by V. Arnold are a great place to start.