Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

dc.contributor.author

Bryant, Robert

dc.contributor.author

Griffiths, Phillip

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Grossman, Daniel

dc.date.accessioned

2016-08-25T14:02:12Z

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2016-08-25T14:05:44Z

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.identifier

http://arxiv.org/abs/math/0207039v1

dc.identifier.uri

https://hdl.handle.net/10161/12686

dc.publisher

University of Chicago Press

dc.relation.replaces

http://hdl.handle.net/10161/12684

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10161/12684

dc.subject

math.DG

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math.DG

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math.AP

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58A15 (Primary), 35A30 (Secondary)

dc.title

Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

dc.type

Book

duke.contributor.orcid

Bryant, Robert|0000-0002-4890-2471

pubs.author-url

http://arxiv.org/abs/math/0207039v1

pubs.organisational-group

Duke

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Mathematics

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Trinity College of Arts & Sciences

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