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 conceptional 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?*