On Lagrangians with Reduced-Order Euler-Lagrange Equations
If a Lagrangian defining a variational problem has order k, then its Euler-Lagrange equations generically have order 2k. This paper considers the case where the Euler-Lagrange equations have order strictly less than 2k, and shows that in such a case the Lagrangian must be a polynomial in the highest...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209761 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Lagrangians with Reduced-Order Euler-Lagrange Equations / D. Saunders // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862724410885013504 |
|---|---|
| author | Saunders, D. |
| author_facet | Saunders, D. |
| citation_txt | On Lagrangians with Reduced-Order Euler-Lagrange Equations / D. Saunders // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | If a Lagrangian defining a variational problem has order k, then its Euler-Lagrange equations generically have order 2k. This paper considers the case where the Euler-Lagrange equations have order strictly less than 2k, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial. The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such k-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way.
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| first_indexed | 2025-12-07T18:46:52Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209761 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:46:52Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Saunders, D. 2025-11-26T11:22:56Z 2018 On Lagrangians with Reduced-Order Euler-Lagrange Equations / D. Saunders // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58E30 arXiv: 1801.06888 https://nasplib.isofts.kiev.ua/handle/123456789/209761 https://doi.org/10.3842/SIGMA.2018.089 If a Lagrangian defining a variational problem has order k, then its Euler-Lagrange equations generically have order 2k. This paper considers the case where the Euler-Lagrange equations have order strictly less than 2k, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial. The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such k-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way. The author would like to thank the referees for their helpful suggestions regarding the presentation of some technical aspects of this work. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Lagrangians with Reduced-Order Euler-Lagrange Equations Article published earlier |
| spellingShingle | On Lagrangians with Reduced-Order Euler-Lagrange Equations Saunders, D. |
| title | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_full | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_fullStr | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_full_unstemmed | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_short | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_sort | on lagrangians with reduced-order euler-lagrange equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209761 |
| work_keys_str_mv | AT saundersd onlagrangianswithreducedordereulerlagrangeequations |