A Hamiltonian system may be understood as a fiber bundle E over time R, with the fiber E t being the position space at time t ∈ R. The Lagrangian is thus a function on the jet bundle J over E ; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T E t, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form.