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

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 Poincaré: 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.

18 pages

References in corpus (3)

• Covariant Hamiltonian field theory
• Nonclassical Lagrangian Dynamics and Potential Maps
• From flows and metrics to dynamics

Cited by in corpus (9)

• Nonclassical Lagrangian Dynamics and Potential Maps
• A Relativistic Approach on 1-Jet Spaces of the Rheonomic Berwald-Moor Metric
• The Generalized Metrical Multi-Time Lagrange Space of Relativistic Geometrical Optics
• Generalized Metrical Multi-Time Lagrange Model for General Relativity and Electromagnetism
• Canonical Nonlinear Connections on Jet Bundles of First Order
• Poisson-Gradient Dynamical Systems with Convex Potential
• Periodical Solutions of Poisson-Gradient Dynamical Systems with Periodical Potential
• Poisson-Gradient Dynamical Systems with Bounded Non-Linearity
• Harmonicity and submanifold maps