We introduce the notion of an F-manifold, that is a Frolicher space which is locally F-diffeomorphic to subspaces of the Euclidean space R^n, eventually for various dimensions. We restrict the study to the constant dimensional case and, using the technique of lifting to local bases, we show that one can always construct a canonical symplectic structure on the cotangent bundle. Alternatively the structure can be induced by the presence of a pseudo-Riemannian metric, paving the way to the Hamiltonian formalism of classical mechanics.