  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.
