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Publications [#318268] of Robert Bryant
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 with Bryant, RL; Griffiths, PA; Grossman, DA, Exterior Differential Systems and EulerLagrange Partial Differential
Equations, Chicago Lectures in Mathematics
(July, 2003),
pp. 213 pages, University of Chicago Press, ISBN 0226077934 (vii+213 pages, ISBN: 0226077942.) [MR1985469], [math.DG/0207039]
(last updated on 2017/12/14)
Author's Comments: This is a book of approximately 160 pages that covers
the work that Phillip Griffiths and I have been doing in
the geometry of the calculus of variations for the last
several years.
Abstract: We use methods from exterior differential systems (EDS) to develop a
geometric theory of scalar, firstorder Lagrangian functionals and their
associated EulerLagrange PDEs, subject to contact transformations. The first
chapter contains an introduction of the classical PoincareCartan 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 PoincareCartan 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" EulerLagrange equations characterized by their invariants. In the
third chapter, we discuss a collection of PoincareCartan forms having a
naturally associated conformal geometry, and exhibit the conservation laws for
nonlinear Poisson and wave equations that result from this. The fourth and
final chapter briefly discusses additional PDE topics from this
viewpointEulerLagrange 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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