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.
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