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Uniqueness results for the Euler and Helmholtz operators from the variational calculus
Journal
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
100
Author
Mikulski Włodzimierz
Volume
119
Number
3
Article ID
85
ISSN
1578-7303
eISSN
1579-1505
Keywords in English
fibered manifold
Lagrangian
Euler map
Helmholtz map
natural operator
The Euler operator
The Helmholtz operator
Access date
2025-11-03
Language
English
Journal language
English
Abstract in English
Let
Affiliation
Wydział Matematyki i Informatyki : Instytut Matematyki
| dc.abstract.en | Let $\textit{m}, \textit{n}, \textit{s}$ be positive integers. Let $\boldsymbol{FM}_{\textit{m},\textit{n}}$ denotes the category of fibered manifolds with $\textit{m}$-dimensional bases and $\textit{n}$-dimensional fibres and their fibered local diffeomorphisms. We prove that if $\textit{m}\geq 3$ than any $\boldsymbol{FM}_{\textit{m},\textit{n}}$-natural operator C transforming pairs $\left ( \lambda , \textit{X} \right )$ of Lagrangians $\lambda : \textit{J}^{\textit{8}}\textit{Y}\rightarrow \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\boldsymbol{FM}_{\textit{m},\textit{n}}$-object $\textit{Y}\rightarrow \textit{M}$ and vector fields $\textit{X}$ on $\textit{M}$ into Euler maps $\textit{C}\left ( \lambda , \textit{X} \right ):\textit{J}^{2\textit{s}}\textit{Y}\rightarrow \textit{V}^{*}\textit{Y}\otimes \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\textit{Y}$ is of the form $\textit{C}\left ( \lambda ,\textit{X} \right )= \textit{cE}\left ( \lambda \right ), \textit{c}\in \textbf{R}$, where $\mathit{E}$ is the Euler operator. We also prove that if $\textit{m}\geq 2$ and $\textit{n}\geq 2$, than any $\boldsymbol{FM}_{\textit{m},\textit{n}}$-natural operator $\textit{D}$ transforming tuples $\left ( \epsilon , \textit{X} \right )$ of Euler maps $\epsilon :\textit{J}^{\textit{8}}\textit{Y}\to \textit{V}^{*}\textit{Y}\otimes \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\boldsymbol{FM}_{\textit{m},\textit{n}}$-object $\textit{Y}\to \textit{M}$ and vector fields $\textit{X}$ on $\textit{M}$ into Helmholtz maps $\textit{D}\left ( \epsilon , \textit{X} \right ):\textit{J}^{2\textit{s}}\textit{Y}\rightarrow \textit{V}^{*}\textit{J}^{\textit{8}}\textit{Y}\otimes \textit{V}^{*}\textit{Y}\otimes \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\textit{Y}\rightarrow \textit{M}$ is of the form $\textit{D}\left ( \epsilon ,\textit{X} \right )= \textit{cH}\left ( \epsilon \right )$ for a real number $\textit{c}$, where $\textit{H}$ is the Helmholtz operator. | |
| dc.affiliation | Wydział Matematyki i Informatyki : Instytut Matematyki | |
| dc.contributor.author | Mikulski, Włodzimierz - 130617 | |
| dc.date.accession | 2025-11-03 | |
| dc.date.accessioned | 2025-11-03T06:18:36Z | |
| dc.date.available | 2025-11-03T06:18:36Z | |
| dc.date.createdat | 2025-10-20T08:42:31Z | en |
| dc.date.issued | 2025 | |
| dc.date.openaccess | 0 | |
| dc.description.accesstime | w momencie opublikowania | |
| dc.description.number | 3 | |
| dc.description.version | ostateczna wersja wydawcy | |
| dc.description.volume | 119 | |
| dc.identifier.articleid | 85 | |
| dc.identifier.doi | 10.1007/s13398-025-01751-y | |
| dc.identifier.eissn | 1579-1505 | |
| dc.identifier.issn | 1578-7303 | |
| dc.identifier.project | DRC AI | |
| dc.identifier.uri | https://ruj.uj.edu.pl/handle/item/564702 | |
| dc.identifier.weblink | https://link.springer.com/article/10.1007/s13398-025-01751-y#citeas | |
| dc.language | eng | |
| dc.language.container | eng | |
| dc.rights | Udzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa | |
| dc.rights.licence | CC-BY | |
| dc.rights.uri | http://creativecommons.org/licenses/by/4.0/legalcode.pl | |
| dc.share.type | inne | |
| dc.subject.en | fibered manifold | |
| dc.subject.en | Lagrangian | |
| dc.subject.en | Euler map | |
| dc.subject.en | Helmholtz map | |
| dc.subject.en | natural operator | |
| dc.subject.en | The Euler operator | |
| dc.subject.en | The Helmholtz operator | |
| dc.subtype | Article | |
| dc.title | Uniqueness results for the Euler and Helmholtz operators from the variational calculus | |
| dc.title.journal | Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas | |
| dc.type | JournalArticle | |
| dspace.entity.type | Publication | en |
dc.abstract.en
Let $\textit{m}, \textit{n}, \textit{s}$ be positive integers. Let $\boldsymbol{FM}_{\textit{m},\textit{n}}$ denotes the category of fibered manifolds with $\textit{m}$-dimensional bases and $\textit{n}$-dimensional fibres and their fibered local diffeomorphisms. We prove that if $\textit{m}\geq 3$ than any $\boldsymbol{FM}_{\textit{m},\textit{n}}$-natural operator C transforming pairs $\left ( \lambda , \textit{X} \right )$ of Lagrangians $\lambda : \textit{J}^{\textit{8}}\textit{Y}\rightarrow \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\boldsymbol{FM}_{\textit{m},\textit{n}}$-object $\textit{Y}\rightarrow \textit{M}$ and vector fields $\textit{X}$ on $\textit{M}$ into Euler maps $\textit{C}\left ( \lambda , \textit{X} \right ):\textit{J}^{2\textit{s}}\textit{Y}\rightarrow \textit{V}^{*}\textit{Y}\otimes \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\textit{Y}$ is of the form $\textit{C}\left ( \lambda ,\textit{X} \right )= \textit{cE}\left ( \lambda \right ), \textit{c}\in \textbf{R}$, where $\mathit{E}$ is the Euler operator. We also prove that if $\textit{m}\geq 2$ and $\textit{n}\geq 2$, than any $\boldsymbol{FM}_{\textit{m},\textit{n}}$-natural operator $\textit{D}$ transforming tuples $\left ( \epsilon , \textit{X} \right )$ of Euler maps $\epsilon :\textit{J}^{\textit{8}}\textit{Y}\to \textit{V}^{*}\textit{Y}\otimes \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\boldsymbol{FM}_{\textit{m},\textit{n}}$-object $\textit{Y}\to \textit{M}$ and vector fields $\textit{X}$ on $\textit{M}$ into Helmholtz maps $\textit{D}\left ( \epsilon , \textit{X} \right ):\textit{J}^{2\textit{s}}\textit{Y}\rightarrow \textit{V}^{*}\textit{J}^{\textit{8}}\textit{Y}\otimes \textit{V}^{*}\textit{Y}\otimes \wedge ^{\textit{m}}\textit{T}^{*}\textit{M}$ on $\textit{Y}\rightarrow \textit{M}$ is of the form $\textit{D}\left ( \epsilon ,\textit{X} \right )= \textit{cH}\left ( \epsilon \right )$ for a real number $\textit{c}$, where $\textit{H}$ is the Helmholtz operator. dc.affiliation
Wydział Matematyki i Informatyki : Instytut Matematyki dc.contributor.author
Mikulski, Włodzimierz - 130617 dc.date.accession
2025-11-03 dc.date.accessioned
2025-11-03T06:18:36Z dc.date.available
2025-11-03T06:18:36Z dc.date.createdaten
2025-10-20T08:42:31Z dc.date.issued
2025 dc.date.openaccess
0 dc.description.accesstime
w momencie opublikowania dc.description.number
3 dc.description.version
ostateczna wersja wydawcy dc.description.volume
119 dc.identifier.articleid
85 dc.identifier.doi
10.1007/s13398-025-01751-y dc.identifier.eissn
1579-1505 dc.identifier.issn
1578-7303 dc.identifier.project
DRC AI dc.identifier.uri
https://ruj.uj.edu.pl/handle/item/564702 dc.identifier.weblink
https://link.springer.com/article/10.1007/s13398-025-01751-y#citeas dc.language
eng dc.language.container
eng dc.rights
Udzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa dc.rights.licence
CC-BY dc.rights.uri
http://creativecommons.org/licenses/by/4.0/legalcode.pl dc.share.type
inne dc.subject.en
fibered manifold dc.subject.en
Lagrangian dc.subject.en
Euler map dc.subject.en
Helmholtz map dc.subject.en
natural operator dc.subject.en
The Euler operator dc.subject.en
The Helmholtz operator dc.subtype
Article dc.title
Uniqueness results for the Euler and Helmholtz operators from the variational calculus dc.title.journal
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas dc.type
JournalArticle dspace.entity.typeen
Publication Affiliations
Wydział Matematyki i Informatyki
Mikulski, Włodzimierz
mathematics
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