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 SIGMA, 2012, Volume 8, 059, 17 pages (Mi sigma736)

Formal integrability for the nonautonomous case of the inverse problem of the calculus of variations

Oana Constantinescu

Faculty of Mathematics, Alexandru Ioan Cuza University, Bd. Carol no. 11, 700506, Iasi, Romania

Abstract: We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan–Kähler theorem. We consider a linear partial differential operator $P$ given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that $P$ is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor $R$ of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.

Keywords: formal integrability; partial differential operators; Lagrangian semisprays; Helmholtz conditions.

DOI: https://doi.org/10.3842/SIGMA.2012.059

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Bibliographic databases:

ArXiv: 1203.1716
MSC: 49N45; 58E30; 34A26; 37J30
Received: March 16, 2012; in final form September 3, 2012; Published online September 6, 2012
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Citation: Oana Constantinescu, “Formal integrability for the nonautonomous case of the inverse problem of the calculus of variations”, SIGMA, 8 (2012), 059, 17 pp.

Citation in format AMSBIB
\Bibitem{Con12} \by Oana Constantinescu \paper Formal integrability for the nonautonomous case of the inverse problem of the calculus of variations \jour SIGMA \yr 2012 \vol 8 \papernumber 059 \totalpages 17 \mathnet{http://mi.mathnet.ru/sigma736} \crossref{https://doi.org/10.3842/SIGMA.2012.059} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2970769} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000308736800001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84867025865} 

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This publication is cited in the following articles:
1. Bucataru I. Constantinescu O., “Generalized Helmholtz Conditions For Non-Conservative Lagrangian Systems”, Math. Phys. Anal. Geom., 18:1 (2015), 25
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