Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form

Autores
Anarella, Mateo; Salvai, Marcos Luis
Año de publicación
2022
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Let G be the manifold of all (unparametrized) oriented lines of R3. We study the controllability of the control system in G given by the condition that a curve in G describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed α. Actually, we pose the analogous more general problem by means of a control system on the manifold Gκ of all the oriented complete geodesics of the three dimensional space form of curvature κ: R3 for κ= 0 , S3 for κ= 1 and hyperbolic 3-space for κ= − 1. We obtain that the system is controllable if and only if α2≠ κ. In the spherical case with α= ± 1 , an admissible curve remains in the set of fibers of a fixed Hopf fibration of S3. We also address and solve a sort of Kendall’s (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.
Fil: Anarella, Mateo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Katholikie Universiteit Leuven; Bélgica
Fil: Salvai, Marcos Luis. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Materia
CONTROL SYSTEM
HELICOID
HOPF FIBRATION
JACOBI FIELD
OXFORD PROBLEM
SPACE OF ORIENTED GEODESICS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/214960
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space FormAnarella, MateoSalvai, Marcos LuisCONTROL SYSTEMHELICOIDHOPF FIBRATIONJACOBI FIELDOXFORD PROBLEMSPACE OF ORIENTED GEODESICShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let G be the manifold of all (unparametrized) oriented lines of R3. We study the controllability of the control system in G given by the condition that a curve in G describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed α. Actually, we pose the analogous more general problem by means of a control system on the manifold Gκ of all the oriented complete geodesics of the three dimensional space form of curvature κ: R3 for κ= 0 , S3 for κ= 1 and hyperbolic 3-space for κ= − 1. We obtain that the system is controllable if and only if α2≠ κ. In the spherical case with α= ± 1 , an admissible curve remains in the set of fibers of a fixed Hopf fibration of S3. We also address and solve a sort of Kendall’s (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.Fil: Anarella, Mateo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Katholikie Universiteit Leuven; BélgicaFil: Salvai, Marcos Luis. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaSpringer2022-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/214960Anarella, Mateo; Salvai, Marcos Luis; Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form; Springer; Acta Applicandae Mathematicae; 179; 1; 5-2022; 1-190167-8019CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s10440-022-00493-yinfo:eu-repo/semantics/altIdentifier/doi/10.1007/s10440-022-00493-yinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-12-23T13:38:19Zoai:ri.conicet.gov.ar:11336/214960instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-12-23 13:38:19.409CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
title Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
spellingShingle Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
Anarella, Mateo
CONTROL SYSTEM
HELICOID
HOPF FIBRATION
JACOBI FIELD
OXFORD PROBLEM
SPACE OF ORIENTED GEODESICS
title_short Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
title_full Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
title_fullStr Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
title_full_unstemmed Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
title_sort Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form
dc.creator.none.fl_str_mv Anarella, Mateo
Salvai, Marcos Luis
author Anarella, Mateo
author_facet Anarella, Mateo
Salvai, Marcos Luis
author_role author
author2 Salvai, Marcos Luis
author2_role author
dc.subject.none.fl_str_mv CONTROL SYSTEM
HELICOID
HOPF FIBRATION
JACOBI FIELD
OXFORD PROBLEM
SPACE OF ORIENTED GEODESICS
topic CONTROL SYSTEM
HELICOID
HOPF FIBRATION
JACOBI FIELD
OXFORD PROBLEM
SPACE OF ORIENTED GEODESICS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Let G be the manifold of all (unparametrized) oriented lines of R3. We study the controllability of the control system in G given by the condition that a curve in G describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed α. Actually, we pose the analogous more general problem by means of a control system on the manifold Gκ of all the oriented complete geodesics of the three dimensional space form of curvature κ: R3 for κ= 0 , S3 for κ= 1 and hyperbolic 3-space for κ= − 1. We obtain that the system is controllable if and only if α2≠ κ. In the spherical case with α= ± 1 , an admissible curve remains in the set of fibers of a fixed Hopf fibration of S3. We also address and solve a sort of Kendall’s (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.
Fil: Anarella, Mateo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Katholikie Universiteit Leuven; Bélgica
Fil: Salvai, Marcos Luis. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
description Let G be the manifold of all (unparametrized) oriented lines of R3. We study the controllability of the control system in G given by the condition that a curve in G describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed α. Actually, we pose the analogous more general problem by means of a control system on the manifold Gκ of all the oriented complete geodesics of the three dimensional space form of curvature κ: R3 for κ= 0 , S3 for κ= 1 and hyperbolic 3-space for κ= − 1. We obtain that the system is controllable if and only if α2≠ κ. In the spherical case with α= ± 1 , an admissible curve remains in the set of fibers of a fixed Hopf fibration of S3. We also address and solve a sort of Kendall’s (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joining two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.
publishDate 2022
dc.date.none.fl_str_mv 2022-05
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/214960
Anarella, Mateo; Salvai, Marcos Luis; Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form; Springer; Acta Applicandae Mathematicae; 179; 1; 5-2022; 1-19
0167-8019
CONICET Digital
CONICET
url http://hdl.handle.net/11336/214960
identifier_str_mv Anarella, Mateo; Salvai, Marcos Luis; Infinitesimally Helicoidal Motions with Fixed Pitch of Oriented Geodesics of a Space Form; Springer; Acta Applicandae Mathematicae; 179; 1; 5-2022; 1-19
0167-8019
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s10440-022-00493-y
info:eu-repo/semantics/altIdentifier/doi/10.1007/s10440-022-00493-y
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.952241

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