Equivalence problems for first order Lagrangians on the line

Niky Kamran, Peter J. Olver

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

Complete solutions to and applications of the equivalence problems for first order particle Lagrangians under the pseudo-groups of contact, point, and fiber-preserving transformations, both with and without the addition of divergence terms, are presented.

Original languageEnglish (US)
Pages (from-to)32-78
Number of pages47
JournalJournal of Differential Equations
Volume80
Issue number1
DOIs
StatePublished - Jul 1989

Bibliographical note

Funding Information:
A basic problem in the calculus of variations is to recognize when two variational problems are actually manifestations of the same problem, but expressed in different coordinate systems. The solution of such equivalence problems is potent tool in the analysis and simplification of complicated variational problems, and it is essential if one is to attempt to solve the more difficult problem of determination of canonical forms for Lagrangians. Applications to particle dynamics, elasticity, symmetry groups and conservation laws, and classical invariant theory are but a few of the benefits of such a solution. Elie Cartan (cf. [6]) developed a powerful construction for completely resolving equivalence problems which can be recast into the framework of exterior differential systems. In this paper, we apply the Cartan method to study the case of a first order Lagrangian on the line. Although the simplest of the possible equivalence problems arising in the calculus of variations, nevertheless this problem already embodies many of * Research supported in part by NSF and NSERC grants. Present address: Departmenotf Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl. + Research supported in part by NSF Grant DMS 86-02004.

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