Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

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-f...

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Збережено в:
Бібліографічні деталі
Дата:2012
Автор: Constantinescu, O.
Формат: Стаття
Мова:English
Опубліковано: Інститут математики НАН України 2012
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/148385
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations / O. Constantinescu // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 40 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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Резюме: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.