Higher order conservation laws and a higher order Noether's theorem
type: Item
instance of: scholarly article
title: Higher order conservation laws and a higher order Noether's theorem
zbMATH Open document ID: 0655.58014
DOI: 10.1016/0196-8858(87)90021-2
author: S. H. Smith
published in: Advances in Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: An interesting alternative approach to the global formulation of variational calculus is the object of this work (studies in this direction has been undertaken by authors in these last years, for instance, \textit{H. Goldschmidt} and \textit{S. Sternberg}, Ann. Inst. Fourier 23, No.4, 203-267 (1973; Zbl 0252.49016); \textit{D. Krupka}, Modern developments in analytical mechanics, Vol. I: Geometrical dynamics, Proc. IUTAM-ISIMM Symp., Torino/Italy 1982, 197-238 (1983; Zbl 0572.58003); \textit{P. L. Garcia} and \textit{J. Muñoz}, ibid., 127-147 (1983; Zbl 0569.58008); \textit{M. Horák} and \textit{I. Kolar}, Czech. Math. J. 33(108), 467-475 (1983; Zbl 0545.58004) \textit{J. Munoz Masque}, Rev. Mat. Iberoam. 1, No.4, 85-126 (1985; Zbl 0606.49026), among many others). Here the author gives a sequence to his PhD thesis recovering some results of \textit{P. A. Griffith}'s book [Exterior differential systems and the calculus of variations (1983; Zbl 0512.49003), to the case of several independent variables. The Euler-Lagrange equations associated to general variational problems for functionals whose domain of definition consists of integral manifolds of an exterior differential system are obtained in terms of a procedure that remains to Kuranishi's P.D.E. systems geometric viewpoint [Lectures on Involutive Systems of P.D.E., Publ. Soc. Mat. São Paulo, S.P., Brazil]. To compute the conservation laws the author adopts the ``formal differential geometry'' of Gel'fand and the spectral sequence of Vinogradov for formal jets. The study of this spectral sequence generates higher-order Noether symmetries from which it is possible to obtain a Noether's Theorem of higher-order. The method gives a solution of the inverse problem as well as to certain kinds of P.D.E. systems like the Cauchy-Kowalewski systems. The constraint situation is also studied.
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 70H30
Mathematics Subject Classification ID: 58A20
zbMATH DE Number: 4069872
zbMATH Keywords: Exterior differential systems
zbMATH Keywords: calculus of variations
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: integral manifolds
zbMATH Keywords: jets
zbMATH Keywords: Noether's Theorem of higher-order
zbMATH Keywords: inverse problem
zbMATH Keywords: Cauchy-Kowalewski systems
reviewed by: Paulo R. Rodrigues
MaRDI profile type: Publication
full work available at URL: 0196-8858(87)90021-2
OpenAlex ID: W2170869471
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description: scientific article; zbMATH DE number 4069872