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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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2012 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2012
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148385 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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| _version_ | 1862713442928951296 |
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| author | Constantinescu, O. |
| author_facet | Constantinescu, O. |
| citation_txt | 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-12-07T17:44:13Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148385 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:44:13Z |
| publishDate | 2012 |
| publisher | Інститут математики НАН України |
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| spelling | Constantinescu, O. 2019-02-18T11:18:45Z 2019-02-18T11:18:45Z 2012 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. 1815-0659 2010 Mathematics Subject Classification: 49N45; 58E30; 34A26; 37J30 DOI: http://dx.doi.org/10.3842/SIGMA.2012.059 https://nasplib.isofts.kiev.ua/handle/123456789/148385 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. The author express his thanks to Ioan Bucataru for the many interesting discussions about the pape en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations Article published earlier |
| spellingShingle | Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations Constantinescu, O. |
| title | Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations |
| title_full | Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations |
| title_fullStr | Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations |
| title_full_unstemmed | Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations |
| title_short | Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations |
| title_sort | formal integrability for the nonautonomous case of the inverse problem of the calculus of variations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148385 |
| work_keys_str_mv | AT constantinescuo formalintegrabilityforthenonautonomouscaseoftheinverseproblemofthecalculusofvariations |