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….