Hamiltonian versus Lagrangian formulations of supermechanic

We take advantage of different generalizations of the tangent manifold to the context of graded manifolds, together with the notion of super section along a morphism of graded manifolds, to obtain intrinsic definitions of the main objects in supermechanics such as, the vertical endomorphism, the canonical and the Cartan's graded forms, the total time derivative operator and the super--Legendre transformation. In this way, we obtain a correspondence between the Lagrangian and the Hamiltonian formulations of supermechanics.