An intrinsic version of the k-harmonic equation
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/39606 |
Resumo: | The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^kM$ to the cotangent bundle $T^*T^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
spelling |
An intrinsic version of the k-harmonic equationK-harmonic curvesRiemannian manifoldsLagrangian and Hamiltonian formalismLegendre transformationThe notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^kM$ to the cotangent bundle $T^*T^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves.MDPI2023-10-24T09:31:03Z2023-01-01T00:00:00Z2023info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/39606eng10.3390/math11173628Abrunheiro, LígiaCamarinha, Margaridainfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T12:17:20Zoai:ria.ua.pt:10773/39606Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T03:09:44.226501Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
An intrinsic version of the k-harmonic equation |
title |
An intrinsic version of the k-harmonic equation |
spellingShingle |
An intrinsic version of the k-harmonic equation Abrunheiro, Lígia K-harmonic curves Riemannian manifolds Lagrangian and Hamiltonian formalism Legendre transformation |
title_short |
An intrinsic version of the k-harmonic equation |
title_full |
An intrinsic version of the k-harmonic equation |
title_fullStr |
An intrinsic version of the k-harmonic equation |
title_full_unstemmed |
An intrinsic version of the k-harmonic equation |
title_sort |
An intrinsic version of the k-harmonic equation |
author |
Abrunheiro, Lígia |
author_facet |
Abrunheiro, Lígia Camarinha, Margarida |
author_role |
author |
author2 |
Camarinha, Margarida |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Abrunheiro, Lígia Camarinha, Margarida |
dc.subject.por.fl_str_mv |
K-harmonic curves Riemannian manifolds Lagrangian and Hamiltonian formalism Legendre transformation |
topic |
K-harmonic curves Riemannian manifolds Lagrangian and Hamiltonian formalism Legendre transformation |
description |
The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^kM$ to the cotangent bundle $T^*T^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves. |
publishDate |
2023 |
dc.date.none.fl_str_mv |
2023-10-24T09:31:03Z 2023-01-01T00:00:00Z 2023 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/39606 |
url |
http://hdl.handle.net/10773/39606 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
10.3390/math11173628 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
MDPI |
publisher.none.fl_str_mv |
MDPI |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
|
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1799137747799113728 |