Lepage Equivalents and the Variational Bicomplex
We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A cho...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212110 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Lepage Equivalents and the Variational Bicomplex. David Saunders. SIGMA 20 (2024), 013, 18 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862706400569851904 |
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| author | Saunders, David |
| author_facet | Saunders, David |
| citation_txt | Lepage Equivalents and the Variational Bicomplex. David Saunders. SIGMA 20 (2024), 013, 18 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper, we investigate the rôle of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection, they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections, which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators.
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| first_indexed | 2026-03-19T00:30:45Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212110 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T00:30:45Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Saunders, David 2026-01-28T13:56:30Z 2024 Lepage Equivalents and the Variational Bicomplex. David Saunders. SIGMA 20 (2024), 013, 18 pages 1815-0659 2020 Mathematics Subject Classification: 58A10; 58A20; 83D05 arXiv:2309.01594 https://nasplib.isofts.kiev.ua/handle/123456789/212110 https://doi.org/10.3842/SIGMA.2024.013 We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper, we investigate the rôle of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection, they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections, which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators. I should like to acknowledge correspondence with Nicoleta Voicu, which encouraged me to return to this topic after several years. Some results from this paper were presented at a meeting in Torino in honour of Marco Ferraris in June 2023, and at the International Summer School on Global Analysis and Applications in Prešov in August 2023. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Lepage Equivalents and the Variational Bicomplex Article published earlier |
| spellingShingle | Lepage Equivalents and the Variational Bicomplex Saunders, David |
| title | Lepage Equivalents and the Variational Bicomplex |
| title_full | Lepage Equivalents and the Variational Bicomplex |
| title_fullStr | Lepage Equivalents and the Variational Bicomplex |
| title_full_unstemmed | Lepage Equivalents and the Variational Bicomplex |
| title_short | Lepage Equivalents and the Variational Bicomplex |
| title_sort | lepage equivalents and the variational bicomplex |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212110 |
| work_keys_str_mv | AT saundersdavid lepageequivalentsandthevariationalbicomplex |