Higher order conservation laws and a higher order Noether's theorem
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.
scientific article; zbMATH DE number 4069872
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