Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

TL;DR

This paper develops a geometric framework using exterior differential systems to analyze scalar first-order Lagrangian functionals and their Euler-Lagrange PDEs, providing new insights into symmetries, invariants, and conservation laws.

Contribution

It introduces a geometric approach to Euler-Lagrange PDEs via exterior differential systems, solving classical problems and identifying invariants and structures without prior EDS or calculus of variations knowledge.

Findings

Solution to the inverse problem for Poincare-Cartan forms

Identification of differential invariants and geometric structures

Conservation laws for nonlinear Poisson and wave equations

Abstract

We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an introduction of the classical Poincare-Cartan form in the context of EDS, followed by proofs of classical results, including a solution to the relevant inverse problem, Noether's theorem on symmetries and conservation laws, and several aspects of minimal hypersurfaces. In the second chapter, the equivalence problem for Poincare-Cartan forms is solved, giving the differential invariants of such a form, identifying associated geometric structures (including a family of affine hypersurfaces), and exhibiting certain "special" Euler-Lagrange equations characterized by their invariants. In the third chapter, we discuss a collection of Poincare-Cartan forms having a naturally associated conformal geometry, and exhibit the conservation laws for non-linear Poisson and wave equations that result from this. The fourth and final chapter briefly discusses additional PDE topics from this viewpoint--Euler-Lagrange PDE systems, higher order Lagrangians and conservation laws, identification of local minima for Lagrangian functionals, and Backlund transformations. No previous knowledge of exterior differential systems or of the calculus of variations is assumed.