There is now strong evidence that the symplectic geometry of Hamiltonian dynamic al systems is deeply connected to Cartan geometry, the dual of Finsler geometry. For people working in local theory it is clear that there are similarities betwe en the geometry of a Banach space and the geometry of its dual. Using the Legendre transformation we can shift from metric to symplectic, i.e. from Finsler manifol ds to Cartan manifolds. In this presentation I will explain how the differential geom etry of a Lagrange (Finsler) manifold can be related to the differential geometry of a Ham ilton (Cartan) manifold, via Legendre transformation.