Solutions of DEs and PDEs as Potential Maps Using First Order Lagrangians

Details Solutions of DEs and PDEs as Potential Maps Using First Order Lagrangians Solutions of DEs and PDEs as Potential Maps Using First Order Lagrangians

2000-07-10

arxiv.org/abs/math/0007061v1

Mathematics

Dynamical Systems

18 pages

Scientific paper

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.

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