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Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

dc.contributor.author Bryant, Robert
dc.contributor.author Griffiths, Phillip A
dc.contributor.author Grossman, DA
dc.date.accessioned 2016-08-25T14:02:12Z
dc.date.accessioned 2016-08-25T14:05:44Z
dc.identifier http://arxiv.org/abs/math/0207039v1
dc.identifier.uri https://hdl.handle.net/10161/12686
dc.description.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.
dc.format.extent 205+xiv pages, latex2e with hyperrefs, xypic
dc.relation.replaces http://hdl.handle.net/10161/12684
dc.relation.replaces 10161/12684
dc.subject math.DG
dc.subject math.DG
dc.subject math.AP
dc.subject 58A15 (Primary), 35A30 (Secondary)
dc.title Exterior Differential Systems and Euler-Lagrange Partial Differential Equations
dc.type Book
pubs.author-url http://arxiv.org/abs/math/0207039v1
pubs.organisational-group Duke
pubs.organisational-group Mathematics
pubs.organisational-group Trinity College of Arts & Sciences


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