  Using parametrized curves (Section 1) or parametrized sheets (Section 3), and
suitable metrics, we treat the jet bundle of order one as a semi-Riemann
manifold. This point of view allows the description of solutions of DEs as
pregeodesics (Section 1) and the solutions of PDEs as potential maps (Section
3), via Lagrangians of order one or via generalized Lorentz world-force laws.
Implicitly, we solved a problem rised first by Poincar\'e: find a suitable
geometric structure that converts the trajectories of a given vector field into
geodesics (see also [6] - [11]). Section 2 and Section 3 realize the passage
from the Lagrangian dynamics to the covariant Hamilton equations.
