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

TL;DR

This paper presents a geometric framework using first order Lagrangians and semi-Riemannian structures on jet bundles to interpret solutions of differential equations as geodesics and potential maps, unifying DEs and PDEs.

Contribution

It introduces a novel geometric approach that treats solutions of DEs and PDEs as geodesics and potential maps within a semi-Riemannian jet bundle framework.

Findings

Solutions of DEs are described as pregeodesics.

Solutions of PDEs are characterized as potential maps.

The framework unifies Lagrangian dynamics with covariant Hamiltonian formalism.

Abstract

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.